Applied Functional Analysis Lecture Notes Fall, 2010

Dr. H. T. Banks Center for Research in Scientiﬁc Computation Department of Mathematics N. C. State University

August 18, 2010 1Preface

1.1 Features of These Notes In these lectures, we shall present functional analysis for partial diﬀerential equations (PDE’s) or distributed parameter systems (DPS) as the basis of modern PDE techniques.This is in contrast to classical PDE techniques such as separation of variables, Fourier transforms, Sturm-Louiville problems, etc.It is also somewhat diﬀerent from the emphasis in usual functional analysis courses where one learns functional analysis as a subdiscipline in its own right.Here we treat functional analysis as a tool to be used in understanding and treating distributed parameter systems.We shall also motivate our discussions with numerous application examples from biology, electromagnetics and materials/mechanics.

∙ Sometimes we give proofs, sometimes not!!–Our choice–if proof sheds light or is not readily found in literature, we might give some or all of arguments behind a result-

∙ examples are quite diverse and details given varying widely -for example, heat,, transport, beam examples are widely found and well documented in literature-don’t always give much in formulation—on other hand, static two-player min-max example not widely found, cellular HIV leading to delay systems, population models with aggregate data, relaxed or chattering controls vs mixing distributions vs Preisach hysteresis and relation to Prohorov metric, some of applications of FA to IP and control not so widely found–

1.2 Functional Analysis for Inﬁnite Dimensional Systems As we shall see, functional analysis techniques can often provide powerful tools for for treatment of PDEs and FDEs in areas of:

∙ Modeling

∙ Qualitative analysis

∙ Inverse problems

∙ Feedback Control

∙ Engineering analysis

1 ∙ Approximation and Computation (such as ﬁnite element and spectral methods)

2 2 Introduction to Functional Analysis

2.1 Goals of this Course In these lectures, we shall present functional analysis for partial diﬀerential equations (PDE’s) or distributed parameter systems (DPS) as the basis of modern PDE techniques.This is in contrast to classical PDE techniques such as separation of variables, Fourier transforms, Sturm-Louiville problems, etc.It is also somewhat diﬀerent from the emphasis in usual functional analysis courses where one learns functional analysis as a subdiscipline in its own right.Here we treat functional analysis as a tool to be used in understanding and treating distributed parameter systems.We shall also motivate our discussions with numerous application examples from biology, electromagnetics and materials/mechanics.

2.2 Uses of Functional Analysis for PDEs As we shall see, functional analysis techniques can often provide powerful tools for insight into a number of areas including:

∙ Modeling

∙ Qualitative analysis

∙ Inverse problems

∙ Control

∙ Engineering analysis

∙ Computation (such as ﬁnite element and spectral methods)

2.3 Example 1: Heat Equation

∂𝑦 ∂ ∂𝑦 (𝑡, 𝜉)= (𝐷(𝜉) (𝑡, 𝜉)) + 𝑓(𝑡, 𝜉)0<𝜉<𝑙, 𝑡>0(1) ∂𝑡 ∂𝜉 ∂𝜉 where 𝑦(𝑡, 𝜉) denotes the temperature in the rod at time 𝑡 and position 𝜉 and 𝑓(𝑡, 𝜉) is the input from a source, e.g. heat lamp, laser, etc. B.C.

𝜉 =0: 𝑦(𝑡, 0) = 0 Dirichlet B.C. This indicates temperature is held constant.

3 ∂𝑦 𝜉 = 𝑙 : 𝐷(𝑙) ∂𝜉 (𝑡, 𝑙) = 0 Neumann B.C. This indicates an insulated end and results in heat ﬂux being zero. ∂𝑦 Here 𝑗(𝑡, 𝜉)=𝐷(𝜉) ∂𝜉 (𝑡, 𝜉) is called the heat ﬂux.

I.C.

𝑦(0,𝜉)=Φ(𝜉)0<𝜉<𝑙

Here Φ denotes the initial temperature distribution in the rod.

2.4 Some Preliminary Operator Theory Let 𝑋, 𝑌 be normed linear spaces.Then

ℒ(𝑋, 𝑌 )={𝑇 : 𝑋 → 𝑌 ∣bounded, linear} is also a normed linear space, and

∣𝑇 ∣ℒ(𝑋,𝑌 ) =sup∣𝑇𝑥∣𝑌 . ∣𝑥∣𝑋 =1 Recall that linear spaces are closed under addition and scalar multiplication. A normed linear space 𝑋 is complete if for any Cauchy sequence {𝑥𝑛} there exists 𝑥 ∈ 𝑋 such that lim𝑛→∞ 𝑥𝑛 = 𝑥. Note: ℒ(𝑋, 𝑌 )iscompleteif𝑌 is complete. If 𝑋 is a Hilbert or Banach space (complex), then 𝑇 is a bounded linear operator from 𝑋1 → 𝑋2,i.e.,𝑇 ∈ℒ(𝑋1,𝑋2) ⇐⇒ 𝑇 is continuous.

Diﬀerentiation in Normed Linear Spaces [HP] Suppose 𝑓 : 𝑋 → 𝑌 is a (nonlinear) transformation (mapping).

𝑓(𝑥0+𝜖𝑧)−𝑓(𝑥0) Definition 1 If lim𝜖→0+ 𝜖 exists, we say 𝑓 has a directional derivative at 𝑥0 in direction 𝑧.This is denoted 𝛿𝑓(𝑥0; 𝑧), and is called the Gateaux differential at 𝑥0 in direction 𝑧. If the limit exists for any direction 𝑧,wesay𝑓 is Gateaux differentiable at 𝑥0 and 𝑧 → 𝛿𝑓(𝑥0; 𝑧) is Gateaux derivative.Note that 𝑧 → 𝛿𝑓(𝑥0; 𝑧)isnot necessarily linear, however it is homogeneous of degree one (i.e., 𝛿𝑓(𝑥0; 𝜆𝑧)= 𝜆𝛿𝑓(𝑥0; 𝑧) for scalars 𝜆).Moreover, it need not be continuous in 𝑧. The following definition of 𝑜(∣𝑧∣) will be useful in defining the Fréchet derivative:

4 ∣𝑔(𝑧)∣ Deﬁnition 2 𝑔(𝑧)is𝑜(∣𝑧∣)if ∣𝑧∣ → 0as𝑧 → 0.

Deﬁnition 3 If there exists 𝑑𝑓 (𝑥0; ⋅) ∈ℒ(𝑋, 𝑌 ) such that ∣𝑓(𝑥0 + 𝑧) − 𝑓(𝑥0) − 𝑑𝑓 (𝑥0; 𝑧)∣𝑌 = 𝑜(∣𝑧∣𝑋 ), then 𝑑𝑓 (𝑥0; ⋅)istheFr´echet derivative of 𝑓 𝑓(𝑥0+𝜖𝑧)−𝑓(𝑥0) at 𝑥0.Equivalently, lim 𝜖→0 𝜖 = 𝑑𝑓 (𝑥0; 𝑧) for every 𝑧 ∈ 𝑋 with ′ 𝑧 → 𝑑𝑓 (𝑥0; 𝑧) ∈ℒ(𝑋, 𝑌 ).We write 𝑑𝑓 (𝑥0; 𝑧)=𝑓 (𝑥0)𝑧.

Results 1.If 𝑓 has a Fr´echet derivative, it is unique.

2.If 𝑓 has a Fr´echet derivative at 𝑥0, then it also has a Gateaux derivative at 𝑥0 and they are the same.

3.If 𝑓 : 𝐷𝑜𝑝𝑒𝑛 ⊂ 𝑋 → 𝑌 has a Fr´echet derivative at 𝑥0,then𝑓 is continuous at 𝑥0.

Examples 1. 𝑋 = ℝ𝑛, 𝑌 = ℝ𝑚,and𝑓 : 𝑋 → 𝑌 such that each component of 𝑓 has partials at 𝑥 in the form ∂𝑓𝑖 (𝑥 ).Then 0 ∂𝑥𝑗 0 ( ) ∂𝑓𝑖 𝑓 ′(𝑥 )= (𝑥 ) 0 ∂𝑥𝑗 0

and hence ( ) ∂𝑓𝑖 𝑑𝑓 (𝑥 ; 𝑧)= (𝑥 ) 𝑧. 0 ∂𝑥𝑗 0

2.Now we consider the case where 𝑋 = ℝ𝑛 and 𝑌 = ℝ1.Then

𝑑𝑓 (𝑥0; 𝑧)=▽𝑓(𝑥0) ⋅ 𝑧.

3.We also look at the case where 𝑋 = 𝑌 = ℝ1.Here,

′ 𝑑𝑓 (𝑥0; 𝑧)=𝑓 (𝑥0)𝑧.

1 ′ Normally 𝑧 = 1 because that is the only direction in ℝ .Then 𝑓 (𝑥0) ′ is called the derivative, but in actuality 𝑧 → 𝑓 (𝑥0)𝑧 is the derivative. ′ 1 1 1 1 Note that 𝑓 (𝑥0) ∈ℒ(ℝ , ℝ ), but the elements of ℒ(ℝ , ℝ )arejust numbers.

5 Homework Exercises ∙ Ex.1 :Consider푓 : ℝ2 → ℝ1. { 𝑥 𝑥2 1 2 , if 푥 ∕=0 𝑥2+𝑥4 푓(푥1,푥2)= 1 2 0, if 푥 =0

Show 푓 is Gateaux differentiable at 푥 =(푥1,푥2) = 0, but not continuous at 푥 = 0, (and hence cannot be Fréchet differentiable).

2.5 Transforming the Initial Boundary Value Problem If we can transform the initial boundary value problem (IBVP) for equation (1) into something of the form { 푥˙(푡)=퐴푥(푡)+퐹 (푡) (2) 푥(0) = 푥0 then conceptually it might be an easier problem with which to work.That is, formally it looks like an ordinary diﬀerential equation problem for which 푒𝐴𝑡 is a solution operator. To rigorously make this transformation and develop a corresponding con- ceptual framework, we need to undertake several tasks: 1.Find a space 푋 of functions and an operator 퐴 : 𝒟(퐴) ⊂ 푋 → 푋 such that the IBVP can be written in the form of equation (2).We may ﬁnd 푋 = 퐿2(0,푙)or퐶(0,푙). We want a solution 푥(푡)=푦(푡, ⋅) to (2), but in what sense - mild, weak, strong, classical? This will answer questions about regularity (i.e., smoothness) of solutions. 2.We also want solution operators or “semigroups” 푇 (푡) which play the role of 푒𝐴𝑡.But what does “ 푒𝐴𝑡”meaninthiscase?If퐴 ∈ 푅𝑛 × 푅𝑛 is a constant matrix, then

∑∞ (퐴푡)𝑛 퐴2푡2 푒𝐴𝑡 ≡ = 퐼 + 퐴푡 + + ... 푛! 2! 𝑛=0 where the series has nice convergent properties.In this case, by the “variation of constants” or “variation of parameters” representation, we can then write ∫ 𝑡 𝐴𝑡 𝐴(𝑡−𝑠) 푥(푡)=푒 푥0 + 푒 퐹 (푠)푑푠. 0

6 We want a similar variation of constants representation in 푋 with operators 푇 (푡) ∈ℒ(푋)=ℒ(푋, 푋) such that 푇 (푡) ∼ 푒𝐴𝑡,sothat

∫ 𝑡 푥(푡)=푇 (푡)푥0 + 푇 (푡 − 푠)퐹 (푠)푑푠. 0 holds and represents the PDE solutions in some appropriate sense and can be used for qualitative (stability, asymptotic behavior, control) and quantitative (approximation and numerics) analyses.

7 3 Semigroups and Inﬁnitesimal Generators

3.1 Basic Principles of Semigroups [HP, Pa, Sh, T]

Deﬁnition 4 A semigroup is a one parameter set of operators {푇 (푡): 푇 (푡) ∈ℒ(푋)},where푋 is a Banach or Hilbert space, such that 푇 (푡)satisﬁes

1. 푇 (푡 + 푠)=푇 (푡)푇 (푠) (semigroup or Markov or translation property)

2. 푇 (0) = 퐼.(identity property)

Classiﬁcation of semigroups by continuity ∙ 푇 (푡)isuniformly continuous if lim ∣푇 (푡) − 퐼∣ =0. 𝑡→0+ This is not of interest to us, because 푇 (푡) is uniformly continuous if and only if 푇 (푡)=푒𝐴𝑡 where 퐴 is a bounded linear operator.

∙ 푇 (푡)isstrongly continuous, denoted 퐶0,ifforeach푥 ∈ 푋, 푡 → 푇 (푡)푥 is continuous on [0,훿) for some positive 훿.

Note 1: All continuity statements are in terms of continuity from the right at zero.For ﬁxed 푡

푇 (푡 + ℎ) − 푇 (푡)=푇 (푡)[푇 (ℎ) − 푇 (0)] = 푇 (푡)[푇 (ℎ) − 퐼] and 푇 (푡) − 푇 (푡 − 휖)=푇 (푡 − 휖)[푇 (휖) − 퐼] so that continuity from the right at zero is equivalent to continuity at any 푡 for operators that are uniformly bounded on compact intervals. Note 2: 푇 (푡) uniformly continuous implies 푇 (푡) strongly continuous (∣푇 (푡)푥 − 푥∣≤∣푇 (푡) − 퐼∣∣푥∣), but not conversely.

3.2 Return to Example 1 : Heat Equation To begin the process of writing the system of Example 1 in the form of equation (2), take 푋 = 퐿2(0,푙) and deﬁne

𝒟(퐴)={휑 ∈ 퐻2(0,푙)∣휑(0) = 0,휑′(푙)=0}

8 to be the domain of 퐴.(Assume 퐷 is smooth for now, e.g., 퐷 is at least 퐻1.) Then we can deﬁne 퐴 : 𝒟(퐴) ⊂ 푋 → 푋 by 퐴[휑](휉)=(퐷(휉)휑′(휉))′. (3) Notation: In general, 푊 𝑘,𝑝(푎, 푏)={휑 ∈ 퐿𝑝∣휑′,휑′′,...,휑(𝑘) ∈ 퐿𝑝}. 𝑘,2 𝑘 If 푝 =2,wewrite푊 (푎, 푏)=퐻 (푎, 푏).퐴퐶(푎, 푏)={휑 ∈ 퐿2(푎, 푏)∣휑 is absolutely continuous (퐴퐶)on[푎, 푏]}.

Homework Exercises ∙ Ex.2 : Show that 퐴 : 𝒟(퐴) ⊂ 푋 → 푋 of the heat example (Example 1) is not bounded 푋 → 푋. ∙ Ex.3 : Give an example when 푋 is a Hilbert space, but ℒ(푋)isnot a Hilbert space. ∙ Ex.4 :Consider퐻1(푎, 푏)=푊 1,2(푎, 푏),푊1,1(푎, 푏)and퐴퐶(푎, 푏).Find the relationships for each pair of spaces in terms of ⊂.That is, establish 푋 ⊂ 푌 or 푋 ⊊ 푌 ,etc.

3.3 Example 2 : General Transport Equation Problems in which the transport equation is used ∙ Insect dispersion ∙ Growth and decline of population-a special case-see Example 4 below ∙ Flow problems (convective/diﬀusive transport)

Speciﬁc applications as described in [BKa, BK] include the “cat brain problem” (which involves in vivo labeled transport) and population dispersal problems.

Description of various ﬂuxes The quantity 푦(푡, 휉) is the population or amount of a substance at location 휉 in the habitat.

∙ 푗1(푡, 휉) is the ﬂux due to dispersion (random foraging or moments; random molecular collisions). ∂푦 푗 (푡, 휉)=퐷(휉) (푡, 휉) 1 ∂휉

9 ∙ 푗2(푡, 휉) is the advective/convective/directed bulk movement.

푗2(푡, 휉)=휈(휉)푦(푡, 휉)

∙ 푗3(푡, 휉) is the net loss due to “birth/death”; immigration/emigration, label decay. 푗3(푡, 휉)=휇(휉)푦(푡, 휉)

General transport equation for a 1-dimensional habitat

∂푦 ∂ ∂ ∂푦 (푡, 휉)+ (휈(휉)푦(푡, 휉)) = (퐷(휉) (푡, 휉)) − 휇(휉)푦(푡, 휉)(4) ∂푡 ∂휉 ∂휉 ∂휉

0 <휉<푙, 푡>0

B.C.

휉 =0: [푦(푡, 0) = 0 (essential) ] 휉 = 푙 : 퐷(휉) ∂𝑦 (푡, 휉) − 휈(휉)푦(푡, 휉) = 0 (natural) ∂𝜉 𝜉=𝑙

I.C. 푦(0,휉)=Φ(휉)

To write this example in operator form (2), we ﬁrst let 푋 = 퐿2(0,푙) be the usual complex Hilbert space of square integrable functions, and the domain of 퐴 be deﬁned by 𝒟(퐴)={휑 ∈ 퐻2(0,푙)∣휑(0) = 0, [퐷휑′ − 휈휑](푙)= 0} where 퐴휑(휉)=(퐷(휉)휑′(휉))′ − (휈(휉)휑(휉))′ − 휇휑(휉). (5) Note: We will assume 퐷 and 휈 are smooth for now.

3.4 Example 3 : Delay Systems–Insect/Insecticide Models Delay systems have been of interest for the past 70 or more years, arising in applications ranging from aerospace engineering to biology (biochemical pathways, etc.), population models, ecology, HIV and other disease progression models to viscoelastic and smart hysteretic materials, as well as network models [BBH, BBJ, BBPP, BKurW1, BKurW2, BRS, Hutch, KP, Ma, Vis, Warga, Wright].We describe here one arising in insect/insecrticide investigations [BBJ].

10 Mathematical models that are suitable for field data with mixed populations should consider reproductive effects and should also account for multiple generations, containing neonates (juveniles) and adults and their interconnectedness.This suggests the need at the minimum for a coupled system of equations describing two separate age classes.Additionally, due to individual differences within the insect population, it is biologically un- realistic to assume that all neonate aphids born on the same day reach the adult age class at the same time.In fact, the age at which the insects reach adulthood varies from as few as five to as many as seven days.Hence one must include a term in any model to account for this variability, leading one to develop a coupled delay differential equation model for the insect population dynamics.We consider the delay between birth and adulthood for neonate pea aphids and present a first mathematical model that treats this delay as a random variable.For a careful derivation of models with similar structure in HIV progression at the cellular level, see [BBH]. Let 푎(푡)and푛(푡) denote the number of adults and neonates, respectively, in the population at time 푡.We lump the mortality due to insecticide into one parameter 푝𝑎 for the adults, 푝𝑛 for the neonates, and denote by 푑𝑎 and 푑𝑛 the background or natural mortalities for adults and neonates, respectively. We let 푏 be the rate at which neonates are born into the population. We suppose that there is a time delay for maturation of a neonate to adult life stage.We further assume that this time delay varies across the insect population according to a probability distribution 푃 (휏)for휏 ∈ [−푇𝑛, 0] 𝑑𝑃 (𝜏) with corresponding density 푘(휏)= 𝑑𝜏 .Here we tacitly assume an upper bound on 푇𝑛 for the maturation period of neonates into adults.Thus, we have that 푘(휏), 휏<0, is the probability per unit time that a neonate who has been in the population −휏 time units becomes an adult.Then the rate at which such neonates become adults is 푛(푡 + 휏)푘(휏).Summing ∫over all such 휏’s, we obtain that the rate at which neonates become adults is 0 푛(푡+휏)푘(휏)푑휏.Using the biological knowledge that the maturation pro- −𝑇𝑛 cess varies between five and seven days (i.e., 푘 vanishes outside [−7, −5]), we obtain the functional differential equation (FDE) (see [JKH1, JKH2, JKH3] for the widespread interest and use of such systems) system

∫ −5 𝑑𝑎 𝑑𝑡 (푡)= 푛(푡 + 휏)푘(휏)푑휏 − (푑𝑎 + 푝𝑎) 푎(푡) −7 ∫ −5 𝑑𝑛 (6) 𝑑𝑡 (푡)=푏푎(푡) − (푑𝑛 + 푝𝑛) 푛(푡) − 푛(푡 + 휏)푘(휏)푑휏 −7 푎(휃)=Φ(휃),푛(휃)=Ψ(휃),휃∈ [−7, 0) 푎(0) = 푎0,푛(0) = 푛0,

11 where 푘 is now a probability density kernel which we have assumed has the property 푘(휏) ≥ 0for휏 ∈ [−7, −5] and 푘(휏)=0for휏 ∈ (−∞, −7) ∪ (−5, 0]. The system of functional diﬀerential equations described in (6) can be written in terms of semigroups [BBu1, BBu2, BKap1]. Let 푧(푡)=(푎(푡),푛(푡))𝑇 and 푧𝑡(휏)=푧(푡 + 휏), −7 ≤ 휏 ≤ 0. (7) 2 2 We deﬁne the Hilbert space 푋 ≡ ℝ × 퐿2(−7, 0; ℝ ) with inner product

( ∫ 0 )1/2 2 2 ∣(휂, 휑)∣𝑋 = ∣휂∣ + ∣휑(휃)∣ 푑휃 , (휂, 휑) ∈ 푋, −7 and let 푥(푡)=(푧(푡),푧𝑡) ∈ 푋.Then our system (6) can be written as

𝑑𝑧 𝑑𝑡 (푡)=퐿 (푧𝑡)for0≤ 푡 ≤ 푇, 2 (8) (푧(0),푧0) = (Φ(0), Φ) ∈ 푋, Φ ∈𝒞(−7, 0; ℝ ), where 푇<∞ and for 휑 =(휓, 휁)𝑇 ∈𝒞(−7, 0; ℝ2) [ ] [ ] ∫ −푑 − 푝 0 01 −5 퐿(휑)= 𝑎 𝑎 휑(0) + 휑(휏)푘(휏)푑휏. (9) 푏 −푑𝑛 − 푝𝑛 0 −1 −7 We now deﬁne a linear operator 𝒜 : 𝒟(𝒜) ⊂ 푋 → 푋 with domain { } 𝒟(𝒜)= (휑(0),휑) ∈ 푋 ∣ 휑 ∈ 퐻1(−7, 0; ℝ2) (10) by 𝒜(휑(0),휑)=(퐿(휑),퐷휑) . (11) Then the delay system (6) can be formulated as

푥˙(푡)=𝒜푥(푡) (12) 푥(0) = 푥0, ( ) 0 0 𝑇 𝑇 where 푥0 = (푎 ,푛 ) , (Φ, Ψ) .

12 3.5 Example 4 : Probability Measure Dependent Systems - Maxwell’s Equations We consider Maxwell’s equations in a complex, heterogenerous material (see [BBL] for details): −∂퐵 ▽ × 퐸 = ∂푡 ∂퐷 ▽ × 퐻 = + 퐽 ∂푡 ▽ ⋅ 퐷 = 휌 ▽ ⋅ 퐵 =0 where 퐸 is the electric field (force), 퐻 is the magnetic field (force), 퐷 is the electric flux density (also called the electric displacement), 퐵 is the magnetic flux density, and 휌 is the density of charges in the medium. To complete this system, we need constituitive (material dependent) relations:

퐷 = 휖0퐸 + 푃

퐵 = 휇0퐻 + 푀

퐽 = 퐽𝑐 + 퐽𝑠

where 푃 is electric polarization, 푀 is magnetization, 퐽𝑐 is the conduction current density, 퐽𝑠 is the source current density, 휖0 is the dielectric permittivity, and 휇0 is the magnetic permeability. General Polarization: ∫ 𝑡 푃 (푡, 푥¯)=[푔 ∗ 퐸](푡, 푥¯)= 푔(푡 − 푠, 푥¯)퐸(푠, 푥¯)푑푠 0 Here, 푔 is the polarization susceptibility kernel, or dielectric response function (DRF).

Several examples of polarization are widely used: ∙ Debye Model for Polarization This describes reasonably well a polar material.This is also called dipolar or orientational polarization.The DRF is deﬁned by ( ) −(𝑡−𝑠) 휖0(휖𝑠 − 휖∞) 푔(푡 − 푠, 푥,¯ 휏)=푒 𝜏 휏

13 and corresponds to 1 푃˙ + 푃 = 휖 (휖 − 휖 )퐸. 휏 0 𝑠 ∞ It is important to note that 푃 represents macroscopic polarization, and when we refer to microscopic polarization we will instead use 푝.

∙ Lorentz Polarization This is also called electronic polarization or the electronic cloud model. The DRF is given by

2 휖0휔𝑝 −(𝑡−𝑠) 푔(푡 − 푠, 푥,¯ 휏)= 푒 2𝜏 sin(휈0(푡 − 푠)), 휈0 and corresponds to 1 푃¨ + 푃˙ + 휔2푃 = 휖 휔2퐸 휏 0 0 𝑝 √ √ where 휔 = 휔 휖 − 휖 and 휈 = 휔2 − 1 . 𝑝 0 𝑠 ∞ 0 0 4𝜏 2

Note: When we allow for instantaneous polarization, we ﬁnd that 퐷 = 휖0휖𝑟퐸 + 푃 where 휖𝑟 =1+휒 ≥ 1 is a relative permittivity. For complex composite materials, the standard Debye or Lorentz polarization model is not adequate, e.g., we need multiple relaxation times 휏’s in some kind of distribution [BBo, BG1, BG2].Then the multiple Debye model becomes ∫ 푃 (푡, 푥¯; 퐹 )= 푝(푡, 푥¯; 휏)푑퐹 (휏) 𝒯 where 𝒯 is a set of possible relaxation parameters 휏 and

퐹 ∈ℱ(𝒯 )={퐹 : 𝒯→ℝ1∣퐹 is a probability distribution 𝒯}.

Note: Here the microscopic polarization is given by

∫ 𝑡 푝(푡, 푥¯; 휏)= 푔(푡 − 푠, 푥¯; 휏)퐸(푠, 푥¯)푑푠 0

−(𝑡−𝑠) 𝜖0(𝜖𝑠−𝜖∞) where 푔(푡 − 푠, 푥¯; 휏)= 𝜏 푒 𝜏 which corresponds to 1 푝˙ + 푝 = 휖 (휖 − 휖 )퐸. 휏 0 𝑠 ∞

14 Here, 퐸 is the total electric ﬁeld.Thus,

∫ ∫ 𝑡 푃 (푡, 푥¯; 퐹 )= 푔(푡 − 푠, 푥¯; 휏)퐸(푠, 푥¯)푑푠푑퐹 (휏) 𝒯 0 ∫ 𝑡 [∫ ] = 푔(푡 − 푠, 푥¯; 휏)푑퐹 (휏) 퐸(푠, 푥¯)푑푠 0 𝒯 ∫ 𝑡 = 퐺(푡 − 푠, 푥¯; 퐹 )퐸(푠, 푥¯)푑푠 0 Assuming 푀 = 0 (nonmagnetic materials), we ﬁnd this system becomes

∂ ▽ × 퐸 = − (휇 퐻) ∂푡 0 [ ∫ ] ∂ 𝑡 ▽ × 퐻 = 휖0휖𝑟퐸 + 퐺(푡 − 푠, 푥¯; 퐹 )퐸(푠, 푥¯)푑푠 + 퐽 ∂푡 0 ▽ ⋅ 퐷 = 0 (13) ▽ ⋅ 퐻 =0 where 퐹 ∈ℱ(𝒯 )and퐽 = 퐽𝑐 + 퐽𝑠.Note: 퐽𝑐 is usually also a convolution on 퐸 although Ohm’s law uses 퐽𝑐 = 휎퐸 where 휎 is the conductivity of the material.In general, one should treat 퐽𝑐 as a convolution, e.g.,

∫ 𝑡 퐽𝑐 = 휎𝑐 ∗ 퐸 = 휎𝑐(푡 − 푠, 푥¯)퐸(푠, 푥¯)푑푠. 0

For the measure dependent system (13), existence and uniqueness via a semigroup formulation have not been established.Comparison of solutions via semigroups versus weak solution have not been done.Nothing has yet been done in two or three dimensions.For the one-dimensional case only, existence, uniqueness, and continuous dependence have been established via a weak formulation (see [BG1]).For continuous dependence of solutions on 퐹 , a metric is needed on ℱ(𝒯 ).More generally, we may need to treat other 4 material parameters 푞 =(휏,휖𝑠,휖∞,휎), where 푞 ∈ 푄 ⊂ ℝ and we look for 퐹 ∈ℱ(푄).

∙ Special Case

We can consider a physically meaningful special case of the Maxwell system in a dielectric material which has a general polarization convolution relationship.Detailed derivations given in Section 2.3of [BBL] lead to a

15 one-dimensional version we next present and will use for an example in our subsequent discussions.Among the assumptions are some homogeneity in the medium (in planes parallel to that of an interrogating polarized planar wave) and a polarized sheet antenna source 퐽𝑠.The resulting model leads to only nontrivial 퐸 fields in the 푥 direction, and 퐻 fields in the 푦 direction, each depending only on 푡 and 푧.Assuming Ohm’s law for 퐽𝑐, we find the system reduces to

∂퐸 ∂퐻 = −휇 ∂푧 0 ∂푡 ∂퐻 ∂퐷 = + 휎퐸 + 퐽 (14) ∂푧 ∂푡 𝑠 퐷 = 휖퐸 + 푃 or in second order form for 퐸(푡, 푧):

∂2퐸 ∂2푃 ∂퐸 ∂2퐸 ∂퐽 휇 휖 + 휇 + 휇 휎 − = −휇 𝑠 0 ∂푡2 0 ∂푡2 0 ∂푡 ∂푧2 0 ∂푡 or ∂2퐸 1 ∂2푃 휎 ∂퐸 ∂2퐸 −1 ∂퐽 휖 + + − 푐2 = 𝑠 , 𝑟 2 2 2 ∂푡 휖0 ∂푡 휖0 ∂푡 ∂푧 휖0 ∂푡 2 1 where 푐 = and 휖𝑟 is a relative permittivity that is material and geom- 𝜇0𝜖0 etry dependent.Typical boundary conditions (say on Ω = [0 , 1]) are [ ] 1 ∂퐸 ∂퐸 − = 0 (absorbing B.C. at 푧 =0) 푐 ∂푡 ∂푧 𝑧=0 and 퐸∣𝑧=1 = 0 (supraconductive B.C. at 푧 =1).

If we assume a general polarization relationship

∫ 𝑡 푃 (푡, 푧)= 푔(푡 − 푠,푧)퐸(푠,푧)푑푠 0 and initial conditions ∂퐸(0,푧) 퐸(0,푧)=Φ(푧), =Ψ(푧), ∂푡 then it can be argued that the system becomes ∫ ∂2퐸 ∂퐸 0 ∂2퐸 + 훾 + 훽퐸 + 푔˜(푠)퐸(푡 + 푠)푑푠 − 푐2 = 𝒥 (푡), 2 2 ∂푡 ∂푡 −∞ ∂푧

16 (without loss of generality we may take 휖𝑟 = 1 for theoretical conditions) where we have tacitly assumed 퐸(푡, 푧)=0for푡<0.If we approximate the “memory” term by assuming only a finite past history is significant, we obtain the integro-partial differential equation ∫ ∂2퐸 ∂퐸 0 ∂2퐸 + 훾 + 훽퐸 + 푔˜(푠)퐸(푡 + 푠)푑푠 − 푐2 = 𝒥 (푡). (15) 2 2 ∂푡 ∂푡 −𝑟 ∂푧 As with the first two examples, this example can also be written as 𝑑𝑥 𝑑𝑡 = 퐴푥 + 퐹 in an appropriate function space setting.In this direction we define an operator 퐴 in appropriate state spaces.In this example one might choose the electric field 퐸 and the magnetic field (퐻 in (14)) as “natural” state spaces along with some type of hysteresis state to account for the memory in (15).However, in second order (in time) systems, it is ∂𝐸 also sometimes natural to choose the state 퐸 and velocity ∂𝑡 as states. We do that in this example.We first define an auxiliary variable 푤(푡)in 2 2 푊 ≡ 퐿𝑔˜(−푟, 0; 퐿 (Ω)) by 푤(푡)(휃)=퐸(푡) − 퐸(푡 + 휃), −푟 ≤ 휃 ≤ 0.

For the inner product in 푊 , we choose the weighted inner product

∫ 0 ⟨휂, 푤⟩𝑊 ≡ 푔˜(휃)⟨휂(휃),푤(휃)⟩𝐿2(Ω)푑휃 −𝑟 and then (15) can be written as ∫ ∂2퐸 ∂퐸 0 ∂2퐸 + 훾 +(훽 + 푔 )퐸 − 푔˜(푠)푤(푡)(푠)푑푠 − 푐2 = 𝒥 (푡), 2 11 2 ∂푡 ∂푡 −𝑟 ∂푧

∫ 0 where 푔11 ≡ −𝑟 푔˜(푠)푑푠 and 푤(푡)(푠)=퐸(푡) − 퐸(푡 + 푠), −푟 ≤ 푠 ≤ 0.For a semigroup formulation we consider the state space

1 2 2 푍 = 푉 × 퐻 × 푊 = 퐻𝑅(Ω) × 퐿 (Ω) × 퐿𝑔˜(−푟, 0; 퐻)

1 1 (here 퐻𝑅(Ω) = {휙 ∈ 퐻 ∣휙(1) = 0}) with states ∂퐸(푡) ∂퐸(푡) (휙, 휓, 휂)=(퐸(푡), ,푤(푡)) = (퐸(푡), ,퐸(푡) − 퐸(푡 + ⋅)). ∂푡 ∂푡

To define what we will later see is an infinitesimal generator of a 퐶0 semigroup, we first define several component operators.Let 퐴ˆ ∈ℒ(푉,푉 ★) be defined by ˆ 2 ′′ 2 ′ 퐴휙 ≡ 푐 휙 − (훽 + 푔11)휙 + 푐 휙 (0)훿0

17 ★ where 훿0 is the Dirac operator 훿0휓 = 휓(0).Here 푉 is the dual space of 푉 (e.g. the space of continuous linear functionals on 푉 , to be discussed in detail later).We also deﬁne 퐵 ∈ℒ(푉,푉 ★)and퐾ˆ ∈ℒ(푊, 퐻)by

퐵휙 = −훾휙 − 푐휙(0)훿0 and, for 휂 ∈ 푊 ,

∫ 0 퐾휂ˆ (푧)= 푔˜(푠)휂(푠,푧)푑푠, 푧 ∈ Ω. −𝑟 Moreover, we introduce the operator 퐶 : 𝒟(퐶) ⊂ 푊 → 푊 deﬁned on

𝒟(퐶)={휂 ∈ 퐻1(−푟, 0; 퐿2(Ω)) ∣ 휂(0) = 0} by ∂ 퐶휂(휃)= 휂(휃). ∂휃 We then deﬁne the operator 𝒜 on

𝒟(𝒜)={(휙, 휓, 휂) ∈ 푍∣휓 ∈ 푉,휂 ∈𝒟(퐶), 퐴휙ˆ + 퐵휓 ∈ 퐻} by ⎛ ⎞ 0 퐼 0 𝒜 = ⎝ 퐴퐵ˆ 퐾ˆ ⎠ . 0 퐼퐶 That is, 𝒜Φ=(휓, 퐴휙ˆ + 퐵휓 + 퐾휂,휓ˆ + 퐶휂) forΦ=(휙, 휓, 휂) ∈𝒟(𝒜).

3.6 Example 5: Structured Population Models We consider the special case of “transport” models or Example 2 with 퐷 = 0 but with so-called renewal or recruitment boundary conditions. The “spatial” variable 휉 is actually “size” in place of spatial location (see [BT]) and such models have been effectively used to model marine populations such as mosquitofish [BBKW, BF, BFPZ] and, more recently, shrimp [GRD-FP, BDEHAD, GRD-FP2].Such models have also been the basis of labeled cell proliferation models in which 휉 represents label intensity [BSTBRSM].The early versions of these size structured population models were first proposed by Sinko and Streifer [SS] in 1967.Cell population versions were proposed by Bell and Anderson [BA] almost simultaneously.

18 Other versions of these models called “age structured models”, where age can be “physiological age” as well as chronical age, are discussed in [MetzD]. The Sinko-Streifer model (SS) for size-structured mosquitoﬁsh populations is given by

∂푣 ∂ + (푔푣)=−휇푣, 휉 <휉<휉 ,푡>0 (16) ∂푡 ∂휉 0 1 푣(0,휉)=Φ(휉) ∫ 𝜉1 푔(푡, 휉0)푣(푡, 휉0)= 퐾(푡, 푠)푣(푡, 푠)푑푠 𝜉0 푔(푡, 휉1)=0.

Here 푣(푡, 휉) represents number density (given in numbers per unit length) or population density, where 푡 represents time and 휉 represents the length of the mosquitofish.The growth rate of individual mosquitofish is assumed given by 푔(푡, 휉), where 푑휉 = 푔(푡, 휉) (17) 푑푡 for each individual (all mosquitofish of a given size have the same growth rate). In the SS 휇(푡, 휉)representsthemortality rate of mosquitofish, and the function Φ(휉) represents initial size density of the population, while 퐾 represents the fecundity kernel. The boundary condition at 휉 = 휉0 is recruitment, or birth rate, while the boundary condition at 휉 = 휉1 = 휉𝑚𝑎𝑥 ensures the maximum size of the mosquitofish is 휉1. The SS model cannot be used as formulated above to model the mosquitofish population because it does not predict dispersion or bifurcation of the population in time under biologically reasonable assumptions [BBKW, BF, BFPZ].As we shall see, we will subsequently replace the growth rate 푔 by a family 𝒢 of growth rates and reconsider the model with a probability distribution 푃 on this family.The population density is then given by summing “cohorts” of subpopulations where individuals belong to the same subpopulation if they have the same growth rate [BD, BDTR, GRD-FP, BDEHAD, GRD-FP2].Thus, in the so-called Growth Rate Distribution (GRD) model, the population density 푢(푡, 휉; 푃 ), first discussed in [BBKW] and developed in [BF], is actually given by ∫ 푢(푡, 휉; 푃 )= 푣(푡, 휉; 푔)푑푃 (푔), (18) 𝒢

19 where 𝒢 is a collection of admissible growth rates, 푃 is a probability measure on 𝒢, and 푣(푡, 휉; 푔) is the solution of the (SS) with growth rate 푔.This model assumes the population is made up of collections of subpopulations with individuals in the same subpopulation having the same size dependent growth rate.This example can also be formulated in terms of semigroups [BKa, BKW1, BKW2], but the details are somewhat more diﬃcult than those for the ﬁrst three examples.In some cases it is advantageous to use a weak formulation (to be developed below) instead of a semigroup formulation.

3.7 Inﬁnitesimal Generators

Definition 5 Assume {푇 (푡) ∈ℒ(푋); 0 ≤ 푡<∞} is a 퐶0 semigroup in 푋 or on 푋, then the infinitesimal generator 퐴 of the semigroup 푇 (푡) is defined by 푇 (푡)푥 − 푥 𝒟(퐴)={푥 ∈ 푋 : lim exists in X} 𝑡→0+ 푡 and 푇 (푡)푥 − 푥 퐴푥 = lim 𝑡→0+ 푡 with 퐴 : 𝒟(퐴) ⊂ 푋 → 푋.

Theorem 1 Let 푇 (푡) bea퐶0 semigroup in X. Then there exist constants 휔 ≥ 0 and 푀 ≥ 1 such that ∣푇 (푡)∣≤푀푒𝜔𝑡 푡 ≥ 0. See [Pa]: Theorem 2.2. Outline of proof: We want to show there exists 휂>0and푀>0such + that ∣푇 (푡)∣≤푀 on [0,휂].If not, then there exists 푡𝑛 → 0 such that ∣푇 (푡𝑛)∣≥푛, which implies ∣푇 (푡𝑛)푥∣ is unbounded for some 푥 ∈ 푋 (by the uniform boundedness principle).This contradicts the fact that 푡 → 푇 (푡)푥 1 continuously on [0,훿]foreach푥 ∈ 푋.Then deﬁne 휔 ≡ 𝜂 푙푛(푀).Then 푒𝜔𝑡 = 푀 𝑡/𝜂 for any 푡>0. Let 푡 = 훿 + 푘휂 for some integer 푘 and 훿 ∈ [0,휂].Then ∣푇 (푡)∣ = ∣푇 (훿)푇 (휂)𝑘∣≤푀푀𝑘 ≤ 푀푀𝑡/𝜂 = 푀푒𝜔𝑡. Notation:Write퐴 ∈ 퐺(푀,휔).For 푀 =1and휔 =0,퐴 ∈ 퐺(1, 0) is the inﬁnitesimal generator of the contraction semigroup: ∣푇 (푡)푥 − 푇 (푡)푦∣≤ ∣푥 − 푦∣.

20 Theorem 2 Let 푇 (푡) bea퐶0 semigroup and let A be its inﬁnitesimal generator. Then 1. For 푥 ∈ 푋, ∫ 1 𝑡+ℎ lim 푇 (푠)푥푑푠 = 푇 (푡)푥. ℎ→0 ℎ 𝑡 2. For 푥 ∈ 푋, ∫ 𝑡 (∫ 𝑡 ) 푇 (푠)푥푑푠 ∈𝒟(퐴) and 퐴 푇 (푠)푥푑푠 = 푇 (푡)푥 − 푥. 0 0 3. If 푥 ∈𝒟(퐴),then푇 (푡)푥 ∈𝒟(퐴) and 푑 푇 (푡)푥 = 퐴푇 (푡)푥 = 푇 (푡)퐴푥. 푑푡

In other words, 𝒟(퐴) is{ invariant under 푇 (푡), and on 𝒟(퐴)atleast, 푥˙(푡)=퐴푥(푡) 푇 (푡)푥0 is a solution of 푥(0) = 푥0.

4. For 푥 ∈𝒟(퐴), ∫ 𝑡 ∫ 𝑡 푇 (푡)푥 − 푇 (푠)푥 = 푇 (휏)퐴푥푑휏 = 퐴푇 (휏)푥푑휏. 𝑠 𝑠 See [Pa]: Theorem 2.4.

Corollary 1 퐴 ∈ 퐺(푀,휔) ⇒𝒟(퐴) is dense in 푋 and 퐴 is a closed linear operator. See [Pa]: Corollary 2.5. Recall: By deﬁnition, a linear operator 퐴 is closed is equivalent to the property that 퐴 has a closed graph in 푋 × 푋. That is, 퐺푟(퐴)={(푥, 푦) ∈ 푋 × 푋∣푥 ∈𝒟(퐴),푦 = 퐴푥} is closed or for any (푥𝑛,푦𝑛) ∈ 퐺푟(퐴), if 푥𝑛 → 푥 and 푦𝑛 → 푦,then푥 ∈𝒟(퐴)and푦 = 퐴푥. Question: Does there exist a one-to-one relationship between a semigrouup and its inﬁnitesimal generator?

Theorem 3 Let 푇 (푡) and 푆(푡) be 퐶0 semigroups on 푋 with inﬁnitesimal generators 퐴 and 퐵 respectively. If 퐴 = 퐵,then푇 (푡)=푆(푡), i.e., the inﬁnitesimal generator uniquely determines the semigroup on all of X. See [Pa]: Theorem 2.6.

21 4 Generators

4.1 Introduction to Generation Theorems

How do we tell when 퐴, derived from a PDE (recall Example 1: the heat equation), is actually a generator of a 퐶0 semigroup? This is important, because it leads to the idea of well-posedness and continuous dependence of solutions for an IBVPDE. Well-posedness of a PDE is equivalent to saying that a unique solution exists in some sense and is continuous with respect to data.In other words, { 푥˙(푡)=퐴푥(푡)+퐹 (푡) 푥(0) = 푥0, is satisfied in some sense and the corresponding semigroup generated solution ∫ 𝑡 푥(푡)=푇 (푡)푥0 + 0 푇 (푡−푠)퐹 (푠)푑푠, yields the map (푥0,퐹) → 푥(⋅; 푥0,퐹), that is then continuous in some sense (depending on the spaces used). Probably the best known generation theorem is the Hille-Yosida theorem.First, however, we review resolvents.In the study of 퐶0-semigroups, one frequently encounters the operators 휆퐼 − 퐴, for 휆 ∈ ℂ (also denoted −1 by 휆 − 퐴), and their inverse 푅𝜆(퐴)=(휆 − 퐴) . Here ℂ is the field of complex scalars.For a linear operator 퐴 in 푋,wedenotetheresolvent set by 휌(퐴)={휆 ∈ ℂ∣휆 − 퐴 has range ℛ(휆 − 퐴)densein푋 and 휆 − 퐴 has a continuous inverse on ℛ(휆 − 퐴)}.We recall that any continuous densely defined linear operator in 푋 can be extended continuously to all of 푋.For −1 휆 ∈ 휌(퐴), we denote the resolvent operator in ℒ(푋)by푅𝜆(퐴)=(휆 − 퐴) . The spectrum 휎(퐴) of a linear operator 퐴 is the complement in ℂ of the resolvent set 휌(퐴).

4.2 Hille-Yosida Theorems [HP, Pa, Sh, T] Theorem 4 (Hille - Yosida) For 푀 ≥ 1,휔 ∈ ℝ, we have 퐴 ∈ 퐺(푀,휔) if and only if

1. A is closed and densely deﬁned (i.e., 퐴 closed and 퐷(퐴)=푋).

2. For real 휆>휔, we have 휆 ∈ 휌(퐴) and 푅𝜆(퐴) satisﬁes 푀 ∣푅 (퐴)𝑛∣≤ ,푛=1, 2,.... 𝜆 (휆 − 휔)𝑛

22 Note: The equivalent version of Hille-Yosida in [Pa] is stated diﬀerently:

Theorem 5 (Hille - Yosida) 퐴 ∈ 퐺(1, 0) ⇐⇒

1. A is closed and densely deﬁned.

+ 1 2. 푅 ⊂ 휌(퐴) and for every 휆>0, ∣푅𝜆(퐴)∣≤ 𝜆 . See [Pa]: Theorem 3.1.

Homework Exercises ∙ Ex.5 :

a) Study the proof of Theorem 3.1 in [Pa] and read Section 1.3 carefully. b) Show that the version of Hille-Yosida in [Pa] is completely equivalent to the version stated previously.(Hint: This is not an exercise in proving Hille-Yosida.It is an exercise in comparing 퐴 ∈ 퐺(푀,휔)and퐴 ∈ 퐺(1, 0) with regard to exponential rates and norms.)

4.3 Results from the Hille-Yosida proof Results of the necessity portion of the proof Out of the necessity portion of the proof of Hille-Yosida we ﬁnd the representation: ∫ ∞ −𝜆𝑡 푅𝜆(퐴)푥 = 푅(휆, 퐴)푥 ≡ 푒 푇 (푡)푥푑푡 for 푥 ∈ 푋, 휆 > 휔 0 This says that the Laplace transform of the semigroup is the resolvent operator. We now ask the question: can we recover the semigroup 푇 (푡)fromthe resolvent 푅𝜆(퐴) through some kind of inverse Laplace transform? Yes - later!!

23 Homework Exercises ∙ Ex.6 : Show that the heat equation operator of Example 1 generates a 퐶0 semigroup in 푋 = 퐿2(0,푙). (That is, show that 퐴휑 =(퐷(휉)휑′(휉))′ on 𝒟(퐴)={휑 ∈ 퐻2(0,푙)∣휑(0) = ′ 0,휑(푙)=0} is the inﬁnitesimal generator of a 퐶0 semigroup in 푋.) Note: If you want to, you can take 퐷(휉)=퐷 constant for this exercise.

Results of the suﬃciency portion of the proof In the suﬃciency proof of Hille-Yosida, we encounter the Yosida approximation: For 휆>휔, 2 퐴𝜆 ≡ 휆퐴푅𝜆(퐴)=휆 푅𝜆(퐴) − 휆퐼.

From this deﬁnition, we can see that 퐴𝜆 is bounded and deﬁned on all of 푋.To see the above relationship, note that

휆퐴푅𝜆(퐴)=휆[(퐴 − 휆퐼)푅𝜆(퐴)+휆푅𝜆(퐴)]. We also found that for 퐴 satisfying the hypothesis of Hille-Yosida,wehave

lim 퐴𝜆푥 = 퐴푥 for all 푥 ∈𝒟(퐴). 𝜆→∞

This says that for large 휆, 퐴𝜆 acts like 퐴.Moreover, under this hypothesis, 퐴𝜆 is the inﬁnitesimal generator of the uniformly continuous semigroup of contraction operators : 푒𝑡𝐴𝜆 .Then we can argue that 푇 (푡)푥 ≡ lim 푒𝑡𝐴𝜆 푥, 𝜆→∞ for all 푥 ∈ 푋, is the desired semigroup that 퐴 generates.(The proof is constructive.) See Corollary 3.5 in [Pa].

4.4 Corollaries to Hille-Yosida

Corollary 2 Let 퐴 be the inﬁnitesimal generator of a 퐶0 semigroup in 푋, and 퐴𝜆 be the Yosida approximation, then

푇 (푡)푥 = lim 푒𝑡𝐴𝜆 푥 for all 푥 ∈ 푋. 𝜆→∞

Corollary 3 If 퐴 is the inﬁnitesimal generator of a 퐶0 semigroup in 푋, then we actually have {휆∣푅푒(휆) >휔}⊂휌(퐴) and for such 휆 푀 ∣푅 (퐴)𝑛∣≤ . 𝜆 (푅푒휆 − 휔)𝑛

24 Corollary 4 퐴 ∈ 퐺(푀,휔) implies 𝒟(퐴) is dense in 푋, and moreover, we ∩∞ ﬁnd 𝒟(퐴𝑛) is also dense in 푋. 𝑛=1

25 5 Dissipative Operators

Deﬁnition 6 Let 푋 be a Hilbert space, then a linear operator 퐴 : 𝒟(퐴) ⊂ 푋 → 푋 is dissipative if Re ⟨퐴푥, 푥⟩≤0 for all 푥 ∈𝒟(퐴).

Theorem 6 Alinearoperator퐴 is dissipative if and only if ∣(휆퐼 − 퐴)푥∣≥ 휆∣푥∣ for all 푥 ∈𝒟(퐴) and 휆>0.

This result yields that for dissipative operators we have 휆퐼 − 퐴 is continuously invertible on ℛ(휆 − 퐴)when휆>0.Thus, 휆>0 implies 휆 ∈ 휌(퐴) whenever ℛ(휆 − 퐴)isdensein푋.Note: Proof will come later.

Theorem 7 (Lumer-Phillips) Suppose 퐴 is a linear operator in a Hilbert space 푋. 1. If 퐴 is densely deﬁned, 퐴 − 휔퐼 is dissipative for some real 휔,and ℛ(휆0 − 퐴)=푋 for some 휆0 with Re (휆0) >휔,then퐴 ∈ 퐺(1,휔). 2. If 퐴 ∈ 퐺(1,휔),then퐴 is densely deﬁned, 퐴 − 휔퐼 is dissipative and ℛ(휆0 − 퐴)=푋 for all 휆0 with Re (휆0) >휔.

Proof of Theorem 6 퐴 is dissipative means Re ⟨퐴푥, 푥⟩≤0 for all 푥 ∈𝒟(퐴).This implies Re⟨−퐴푥, 푥⟩≥0.So we have:

∣(휆 − 퐴)푥∣∣푥∣≥∣⟨(휆 − 퐴)푥, 푥⟩∣ ≥ Re⟨(휆 − 퐴)푥, 푥⟩ ≥ 휆⟨푥, 푥⟩ = 휆∣푥∣2. Conversely, suppose ∣(휆퐼 − 퐴)푥∣≥휆∣푥∣ for all 푥 ∈𝒟(퐴)and휆>0. Let 푥 ∈𝒟(퐴). 𝑦𝜆 Deﬁne 푦𝜆 =(휆 − 퐴)푥 and 푧𝜆 = .Then we have ∣푧𝜆∣ =1and ∣𝑦𝜆∣ 휆∣푥∣≤∣(휆 − 퐴)푥∣

= ⟨(휆 − 퐴)푥, 푧𝜆⟩

= 휆Re⟨푥, 푧𝜆⟩−Re⟨퐴푥, 푧𝜆⟩ (19)

≤ 휆∣푥∣∣푧𝜆∣−Re⟨퐴푥, 푧𝜆⟩

= 휆∣푥∣−Re⟨퐴푥, 푧𝜆⟩.

26 This implies

Re⟨퐴푥, 푧𝜆⟩≤0. (20) We always have the relationship

−Re⟨퐴푥, 푧𝜆⟩≤∣퐴푥∣. (21) Combining equations (19) and (21), we obtain

휆∣푥∣≤휆Re⟨푥, 푧𝜆⟩−Re⟨퐴푥, 푧𝜆⟩

≤ 휆Re⟨푥, 푧𝜆⟩ + ∣퐴푥∣. (22) Rearranging equation (22), we have

휆Re⟨푥, 푧𝜆⟩≥휆∣푥∣−∣퐴푥∣ or 1 Re⟨푥, 푧 ⟩≥∣푥∣− ∣퐴푥∣. (23) 𝜆 휆

Observethatwehave∣푧𝜆∣ =1for휆>0.This implies that as 휆𝑘 → ∞,푧 ⇀ 푧˜ with ∣푧˜∣≤1.(Note: The norm is weakly lower semi-continuous; 𝜆𝑘 therefore, 푧𝜆 ⇀ 푧˜ implies ∣푧˜∣ = ∣w lim 푧𝜆 ∣≤lim∣푧𝜆 ∣ = 1.This is discussed 𝑘 𝑘 𝑘 𝑘 in detail in Section 5.2.) Taking the limit of equation (50), we obtain

Re⟨퐴푥, 푧˜⟩≤0. (24)

Similarly, taking the limit of equation (23), we have

Re⟨푥, 푧˜⟩≥∣푥∣. (25) Thus because ∣푧˜∣≤1 we can make the arguments

∣푥∣≤Re⟨푥, 푧˜⟩ ≤∣⟨푥, 푧˜⟩∣ ≤∣푥∣∣푧˜∣ ≤∣푥∣. Therefore, we actually have the equality

Re⟨푥, 푧˜⟩ = ∣⟨푥, 푧˜⟩∣ = ⟨푥, 푧˜⟩ = ∣푥∣. (26)

27 Deﬁne푥 ˜ =˜푧∣푥∣;then⟨푥, 푥˜⟩ = ∣푥∣2 by (26) as ⟨푥, 푧˜⟩ = ∣푥∣ implies ⟨푥, 푧˜∣푥∣⟩ = ∣푥∣⟨푥, 푧˜⟩ = ∣푥∣2.From equation (24),

Re⟨퐴푥, 푧˜⟩≤0 implies Re⟨퐴푥, 푧˜∣푥∣⟩ ≤ 0 or Re⟨퐴푥, 푥˜⟩≤0. We also have ∣푥˜∣≤∣푥∣ because ∣푥˜∣ = ∣푧˜∣∣푥∣≤∣푥∣.

We claim that푥 ˜ = 푥.If this is true, then (27) would imply Re ⟨퐴푥, 푥⟩≤ 0 for all 푥 ∈𝒟(퐴) and we are ﬁnished.We need the following lemma to prove this.

Lemma 1 (Duality Set Lemma) Let 푥 ∈ 푋 be arbitrary in a Hilbert space 푋 and 푥˜ ∈ 푋 such that ⟨푥,˜ 푥⟩ = ∣푥∣2 and ∣푥˜∣≤∣푥∣;then푥˜ = 푥.

Proof 0 ≤⟨푥 − 푥,˜ 푥 − 푥˜⟩ = ∣푥∣2 −⟨푥,˜ 푥⟩−⟨푥,˜ 푥⟩ + ∣푥˜∣2 ≤∣푥∣2 −∣푥∣2 −∣푥∣2 + ∣푥∣2 =0 Therefore, ∣푥 − 푥˜∣ = 0 implies 푥 =˜푥; i.e., in a Hilbert space, 퐹 (푥) ≡ duality set = {푥}.For 푥 ∈ 푋, 퐹(푥)={푥∗ ∈ 푋∣⟨푥, 푥∗⟩ = ∣푥∣2 = ∣푥∗∣2}. As we shall see below, for a general Banach space 푋, the duality set is a subset of the dual space 푋∗.Because Hilbert spaces are self-dual, in a Hilbert space 푋 the concept reduces to one involving subsets of the Hilbert space 푋 itself.

Deﬁnition 7 Let 푋 be a Hilbert space with 퐴 : 𝒟(퐴) ⊂ 푋 → 푋 with 𝒟(퐴)densein푋..Thentheadjoint of 퐴, denoted 퐴∗, is deﬁned by

𝒟(퐴∗)={푦 ∈ 푋∣ there exists 푤 ∈ 푋 satisfying ⟨퐴푥, 푦⟩ = ⟨푥, 푤⟩ for all 푥 ∈𝒟(퐴)} and 퐴∗푦 = 푤.

Corollary 5 (to Lumer-Phillips) Let 푋 be a Hilbert space and 퐴 : 𝒟(퐴) ⊂ 푋 → 푋 with 𝒟(퐴) dense and 퐴 closed in 푋.Ifboth퐴 and 퐴∗ are dissipative, then 퐴 is an inﬁnitesimal generator of a 퐶0 semigroup of contractions on 푋.

28 Proof By Lumer-Phillips, it suﬃces to argue that ℛ(휆 − 퐴)=푋 for some 휆>0. Take 휆 =1: Recall that in a Hilbert space we have

⟨퐴푥, 푦⟩ = ⟨푥, 퐴∗푦⟩ for 푥 ∈𝒟(퐴),푦 ∈𝒟(퐴∗).

But by assumption 퐴 is dissipative and closed, which means

{(푥, 퐴푥)∣푥 ∈𝒟(퐴)} is closed in 푋 × 푋.

Therefore, ℛ(퐼 − 퐴) is a closed subspace of 푋. Claim: ℛ(퐼 − 퐴)=푋. Suppose ℛ(퐼 −퐴) ∕= 푋.Then by the Hahn-Banach theorem in a Hilbert space, there exists푥 ˜ ∈ 푋, 푥˜ ∕= 0, such that

⟨푥,˜ (퐼 − 퐴)푥⟩ =0.

In other words

⟨푥,˜ 푥⟩ = ⟨푥,˜ 퐴푥⟩ for all 푥 ∈𝒟(퐴). Therefore

푥˜ ∈𝒟(퐴∗)and⟨퐴∗푥,˜ 푥⟩ = ⟨푥,˜ 푥⟩, or

⟨푥˜ − 퐴∗푥,˜ 푥⟩ =0 forall푥 ∈𝒟(퐴). However, 𝒟(퐴) is dense, which implies

푥˜ − 퐴∗푥˜ =0or˜푥 = 퐴∗푥.˜ (27) By assumption, 퐴∗ is also dissipative.Therefore

∣(퐼 − 퐴∗)˜푥∣≥∣푥˜∣ for all푥 ˜ ∈𝒟(퐴∗). (28) Combining (27) and (28) we have

∣푥˜∣≤∣푥˜ − 퐴∗푥˜∣ ∣푥˜∣≤∣푥˜ − 푥˜∣ ∣푥˜∣≤0. This is a contradiction since푥 ˜ ∕= 0.Therefore, ℛ(퐼 − 퐴)=푋.

29 Theorem 8 In a Hilbert space 푋,if퐴 is dissipative and ℛ(휆 − 퐴)=푋 for some 휆>0,then퐴 is densely deﬁned. See [Pa]: Theorem 4.6. Note: This also holds in a reﬂective Banach space (more on this later).

Important practical results: 1.In a Hilbert space, we only need 퐴 to be dissipative and ℛ(휆퐼−퐴)=푋 for some 휆>0 in order to use Lumer-Phillips theorem.

2.In a real Hilbert space (such as real 퐿2(0,푙)), we can get by with ⟨퐴푥, 푥⟩≤0.

Lemma 2 (A Practical Lemma) Let 푋 be a complex Hilbert space and suppose that for 푢 ∈𝒟(퐴),퐴푢is real whenever 푢 is real (for example, an operator 퐴 coming from a PDE deﬁned by real coeﬃcients), then 퐴 is dissipative, i.e., Re⟨퐴푥, 푥⟩≤0 for all 푥 ∈𝒟(퐴) if and only if

⟨퐴푢, 푢⟩≤0 for all real valued 푢 ∈𝒟(퐴).

Proof: We have for 푥 = 푢 + 푖푣,with푢, 푣,real:

⟨퐴푥, 푥⟩ = ⟨퐴(푢 + 푖푣),푢+ 푖푣⟩ = ⟨퐴푢, 푢⟩ + ⟨퐴푣, 푣⟩ + 푖[⟨퐴푣, 푢⟩−⟨퐴푢, 푣⟩].

We want to argue that ⟨퐴푣, 푢⟩−⟨퐴푢, 푣⟩ is real; however, for any complex Hilbert space we have the polarization identity [GP, p.168] 1 ⟨푢, 푣⟩ = {∣푢 + 푣∣2 + 푖∣푢 + 푖푣∣2 − 푖∣푢 − 푖푣∣2 −∣푢 − 푣∣2}. 4 But if 푢 and 푣 are real-valued we have ∣푢 + 푖푣∣ = ∣푢 − 푖푣∣ and hence 1 ⟨푢, 푣⟩ = {∣푢 + 푣∣2 −∣푢 − 푣∣2} 4 (note that this is the polarization identity in a real Hilbert space) which is real valued.Since 푢 and 푣 are real, then by assumption 퐴푢 and 퐴푣 are real. Therefore, ⟨퐴푣, 푢⟩−⟨퐴푢, 푣⟩ is real-valued and hence

30 Re ⟨퐴푥, 푥⟩ = ⟨퐴푢, 푢⟩ + ⟨퐴푣, 푣⟩. Thus to have Re ⟨퐴푥, 푥⟩≤0, it suﬃces to argue

⟨퐴푢, 푢⟩≤0 for all real valued 푢 in 𝒟(퐴).

Remark: Any Banach space is a Hilbert space if and only if the parallelogram law ∣푥 + 푦∣2 + ∣푥 − 푦∣2 =2∣푥∣2 +2∣푦∣2 holds.If this holds in a Banach space, the polarization identity can be used to deﬁne a norm compatible inner product. To illustrate the use of the Lumer-Phillips theorem, we return to some of the previous examples.

Examples using Lumer-Phillips Theorem Return to Example 1: The Heat Equation We return to the heat diﬀusion example where we found the PDE ( ) ∂푦 ∂ ∂푦 (푡, 휉)= 퐷(휉) + 푓(푡, 휉) ∂푡 ∂휉 ∂휉 ∂푦 푦(푡, 0) = 0,퐷(푙) (푡, 푙)=0 ∂휉 gave rise to an operator 퐴 in 푋 = 퐿2(0,푙)givenby

2 ′ ′ ′ 𝒟(퐴)={휑 ∈ 퐻 (0,푙)∣(퐷휑 ) ∈ 퐿2(0,푙),휑(0) = 0,휑(푙)=0}, and 퐴휑 =(퐷휑′)′. Remark:Noticethatif퐷 is smooth, we have

𝒟(퐴)={휑 ∈ 퐻2(0,푙)∣휑(0) = 0,휑′(푙)=0}.

In view of the remarks following the Lumer-Phillips theorem, to show that 퐴 generates a 퐶0 semigroup in 푋, it suﬃces to show that 퐴 is dissipative and that ℛ(휆 − 퐴)=푋;i.e.휆 (− 퐴)𝒟(퐴)=푋 for some 휆>0.

31 We have for 휑 ∈𝒟(퐴),휑 real-valued:

⟨퐴휑, 휑⟩ = ⟨(퐷휑′)′,휑⟩ ∫ 𝑙 ′ ′ = 0(퐷휑 ) 휑 = −⟨퐷휑′,휑′⟩ + 퐷휑′휑∣𝑙 ∫ 0 1 ′ 2 = − 0 퐷(휑 ) ≤ 0 for 퐷 ≥ 0 (a natural assumption since 퐷 is the coeﬃcient of thermal diﬀu- sion). To argue that ℛ(휆 − 퐴)=퐿2(0,푙)forsome휆>0, we note that this is equivalent to arguing that for some 휆>0, the boundary value problem

휆휑 − (퐷휑′)′ = 휓, 0 <푥<푙, 휑(0) = 0,휑′(푙)=0,

2 has a solution 휑 ∈ 퐻 (0,푙) for any 휓 ∈ 퐿2(0,푙).But one can answer this latter question in the affirmative (assuming 퐷>0 is sufficiently regular, 1 e.g., 퐷 ∈ 푊∞) using classical Sturm-Liouville theory.(See [BR, CH, D, S].) Thus, we readily see that 퐴 is a densely-defined generator of a 퐶0 solution semigroup for the system above.

Homework Exercises ∙ Ex.7 : Consider the general transport example, Example 2, and show that it generates a 퐶0 semigroup.

Return to Example 2 : The General Transport Equation We next reconsider the population model given by ( ) ∂푦 ∂ ∂ ∂푦 (푡, 휉)+ (휈(휉)푦(푡, 휉)) = 퐷(휉) (푡, 휉) − 휇(휉)푦(푡, 휉) ∂푡 ∂휉 ∂휉 ∂휉 [ ] ∂푦 푦(푡, 0) = 0, 퐷(휉) (푡, 휉) − 휈(휉)푦(푡, 휉) =0. ∂휉 𝜉=𝑙

The corresponding operator equation in 푋 = 퐿2(0,푙)isintermsofthe operator 퐴 given by

𝒟(퐴)= [ ] 2 ′ ′ ′ ′ {휑 ∈ 퐻 (0,푙) ∣(퐷휑 ) − (휈휑) ∈ 퐿2(0,푙),휑(0) = 0, 퐷휑 − 휈휑 (푙)=0}

32 퐴휑 =(퐷휑′)′ − (휈휑)′ − 휇휑.

Again we show that 퐴 is dissipative in 퐿2(0,푙) (actually we shall argue that a translation of 퐴 is dissipative).For real valued 휑 ∈𝒟(퐴), we have

⟨퐴휑, 휑⟩ = ⟨(퐷휑′ − 휈휑)′,휑⟩−⟨휇휑, 휑⟩,

′ ′ ′ 𝑙 = −⟨(퐷휑 − 휈휑),휑⟩ +(퐷휑 − 휈휑)휑∣0 −⟨휇휑, 휑⟩,

= −⟨퐷휑′,휑′⟩−⟨휇휑, 휑⟩ + ⟨휈휑,휑′⟩. √ Hence, using 푎푏 = √1 푎 2휖푏 ≤ 1 푎2 + 휖푏2, we ﬁnd 2𝜖 4𝜖

′ ′ 1 2 ′ 2 ⟨퐴휑, 휑⟩≤−⟨퐷휑 ,휑⟩−⟨휇휑, 휑⟩ + 4𝜖 ∣휈휑∣ + 휖∣휑 ∣

′ ′ 𝜈2 = ⟨(−퐷 + 휖)휑 ,휑⟩ + ⟨(−휇 + 4𝜖 )휑, 휑⟩

𝑘 2 ≤ 4𝜖 ∣휑∣ for 휖 suﬃciently small so that 퐷(휉) ≥ 휖 and 푘 chosen so that −4휖휇+휈2(휉) ≤ 𝑘 푘.Thus,wehavethat퐴 − ( 4𝜖 )퐼 is dissipative in 퐿2(0,푙). 𝑘 The range statement ℛ(휆 + 4𝜖 − 퐴)=푋 again reduces to a question for Sturm-Liouville problems.For any 휓 ∈ 퐿2(0,푙), we seek an 퐿2(0,푙)solution of 푘 (휆 + + 휇)휑 +(휈휑)′ − (퐷휑′)′ = 휓, 0 <휉<푙, 4휖

휑(0) = 0, [퐷휑′ − 휈휑](푙)=0. For 휆 suﬃciently large, the Sturm-Liouville theory guarantees existence of 𝑘 solutions.Thus 퐴 − 4𝜖 is the generator of a 퐶0-semigroup 푆(푡)sothat 𝑘 𝑡 푇 (푡)=푒 4𝜖 푆(푡) is the solution semigroup generated by 퐴.

Return to Example 3 : Delay Systems We return to the delay system equation of Example 3 which is a special case of more general vector delay systems.We consider the Cauchy problem

푧˙(푡)=퐿(푧𝑡), (푧(0),푧0)=(휂, 휙), (29) where 푧𝑡(휉)=푧(푡 + 휉), 휉 ∈ [−푟, 0] and

33 ∑𝑚 ∫ 0 퐿(푧𝑡)= 퐴𝑖푧(푡 − 휏𝑖)+ 퐾(휉)푧(푡 + 휉)푑휉, 𝑖=0 −𝑟 where 0 = 휏0 <휏1 <... <휏𝑚 = 푟,and퐴𝑖 and 퐾(휉)are푛 × 푛 matrices.For general functions 휙 ∈ 퐶(−푟, 0; ℝ𝑛), the mapping 퐿(⋅):퐶(−푟, 0; ℝ𝑛) → ℝ𝑛 has the form

∑𝑚 ∫ 0 퐿(휙)= 퐴𝑖휙(−휏𝑖)+ 퐾(휉)휙(휉)푑휉. (30) 𝑖=0 −𝑟 𝑛 𝑛 Let 푥(푡)=(푧(푡),푧𝑡) ∈ 푋, 푋 ≡ ℝ × 퐿2(−푟, 0; ℝ ), where the Hilbert space 푋 has the inner product

∫ 0 < (휂, 휙), (휁,휓) >𝑋 =<휂,휁>ℝ𝑛 + 휙(휉)휓(휉)푑휉. (31) −𝑟 Define 𝒜 : 𝒟(𝒜) ∈ 푋 → 푋,where𝒟(𝒜)={휙ˆ =(휙(0),휙) ∈ 푋∣휙 ∈ 퐻1(−푟, 0; ℝ𝑛)}, which is dense in 푋.Then 𝒜휙ˆ = 𝒜(휙(0),휙)=(퐿(휙),퐷휙) for 휙ˆ =(휙(0),휙) ∈𝒟(𝒜).We use Lumer-Phillips to show that 𝒜 is an infinitesimal generator of the 퐶0 semigroup 푆(푡)where푆(푡)(휂, 휙)=(푧(푡),푧𝑡) for solutions of (29).We do this in a space 푋𝑔 that is topologically equivalent to 푋,sothatwehavethesemigroupon푋 as well as on 푋𝑔. First, we show that 𝒜 is dissipative in a space topologically equivalent to 푋.Renorm 푋 by the weighting function 푔 defined on [−푟, 0], where 푔(휉)=푗 for 휉 ∈ [−휏𝑚−𝑗+1, −휏𝑚−𝑗), 푗 =1, 2,...,푚.Define the Hilbert 𝑛 𝑛 space 푋𝑔 ≡ ℝ × 퐿2(−푟, 0; 푔; ℝ ) to be the elements of 푋 with this new inner product

∫ 0 𝑛 < (휂, 휙), (휁,휓) >𝑋𝑔 =<휂,휁>ℝ + 휙(휉)휓(휉)푔(휉)푑휉. (32) −𝑟 This gives rise to an equivalent topology to that of 푋 as long as 푔(휉) > 0 for all 휉 ∈ [−푟, 0] (see [BBu2, p.186], [BKap1, Webb] for more details). Then 𝒜 is dissipative if

< 𝒜푥, 푥 > ≤ 휔∣푥∣2 (33) 𝑋𝑔 𝑋𝑔 for all 푥 ∈𝒟(𝒜)where푥 =(휙(0),휙).To show this, we argue

34 < 𝒜푥, 푥 > = < (퐿(휙),퐷휙), (휙(0),휙) >

= <퐿(휙),휙(0) > 𝑛 + <퐷휙,휙> 𝑛 ℝ 𝐿2(−𝑟,0;𝑔;ℝ ) ∑𝑚 ∫ 0 = < 퐴𝑖휙(−휏𝑖)+ 퐾(휉)휙(휉)푑휉, 휙(0) >ℝ𝑛 (34) 𝑖=0 −𝑟 ∫ 0 + 퐷휙(휉)휙(휉)푔(휉)푑휉 −𝑟 ∑𝑚 ∫ 0 = < 퐴𝑖휙(−휏𝑖)+ 퐾(휉)휙(휉)푑휉, 휙(0) >ℝ𝑛 (35) 𝑖=0 −𝑟 𝑚 ∫ ∑ −𝜏𝑚−𝑗 + 퐷휙(휉)휙(휉)푔(휉)푑휉. (36) 𝑗=1 −𝜏𝑚−𝑗+1

Consider the last term and denote 휙𝑚−𝑗 = 휙(−휏𝑚−𝑗)

35 𝑚 ∫ ∑ −𝜏𝑚−𝑗 퐷휙(휉)휙(휉)푔(휉)푑휉 𝑗=1 −𝜏𝑚−𝑗+1 ∑𝑚 1 = 푗휙(휉)2∣𝜉=−𝜏𝑚−𝑗 2 𝜉=−𝜏𝑚−𝑗+1 𝑗=1 ( ) ∑𝑚 푗 푗 = ∣휙 ∣2 − ∣휙 ∣2 2 𝑚−𝑗 2 𝑚−𝑗+1 𝑗=1 1 ∑𝑚 1 𝑚∑−1 = 푗∣휙 ∣2 − (푘 +1)∣휙 ∣2 (for푘 = 푗 − 1) 2 𝑚−𝑗 2 𝑚−𝑘 𝑗=1 𝑘=0 1 1 𝑚∑−1 = 푚∣휙 ∣2 + 푗∣휙 ∣2 2 0 2 𝑚−𝑗 𝑗=1 1 𝑚∑−1 1 − (푘 +1)∣휙 ∣2 − ∣휙 ∣2 2 𝑚−𝑘 2 𝑚 𝑘=1 1 1 𝑚∑−1 1 = 푚∣휙 ∣2 − ∣휙 ∣2 − ∣휙 ∣2 2 0 2 𝑚−𝑗 2 𝑚 𝑗=1 1 1 𝑚∑−1 = 푚∣휙 ∣2 − ∣휙 ∣2 2 0 2 𝑚−𝑗 𝑗=0 푚 +1 1 ∑𝑚 = ∣휙 ∣2 − ∣휙 ∣2 2 0 2 𝑚−𝑗 𝑗=0

Returning to (36), we have for 푥 =(휙(0),휙),

36 ∑𝑚 ∫ 0 𝑛 < 𝒜푥, 푥 >𝑋𝑔 ≤ ∣퐴𝑖∣∣휙𝑖∣∣휙0∣+ < 퐾(휉)휙(휉)푑휉, 휙(0) >ℝ 𝑖=0 −𝑟 푚 +1 1 ∑𝑚 + ∣휙 ∣2 − ∣휙 ∣2 2 0 2 𝑚−𝑗 𝑗=0 ( ) ∫ ∑𝑚 1 1 0 ≤ ∣퐴 ∣2∣휙 ∣2 + ∣휙 ∣2 + ∣ 퐾(휉)휙(휉)푑휉∣∣휙(0)∣ 2 𝑖 0 2 𝑖 𝑖=0 −𝑟 푚 +1 1 ∑𝑚 + ∣휙 ∣2 − ∣휙 ∣2 2 0 2 𝑚−𝑗 𝑗=0 ∫ 1 ∑𝑚 1 ∑𝑚 0 = ∣퐴 ∣2∣휙 ∣2 + ∣휙 ∣2 + ∣ 퐾(휉)휙(휉)푑휉∣∣휙(0)∣ 2 𝑖 0 2 𝑖 𝑖=0 𝑖=0 −𝑟 푚 +1 1 ∑𝑚 + ∣휙 ∣2 − ∣휙 ∣2 2 0 2 𝑚−𝑗 𝑗=0 ( ) ∑𝑚 1 2 푚 +1 2 1 2 2 1 2 ≤ ∣퐴𝑖∣ + ∣휙0∣ + ∣퐾∣ ∣휙∣ + ∣휙(0)∣ 2 2 2 𝐿2 𝐿2 2 ( 𝑖=0 ) ∑𝑚 1 2 푚 +1 1 1 2 2 ≤ ∣퐴𝑖∣ + + + ∣퐾∣ ∣푥∣ 2 2 2 2 𝐿2 𝑋 𝑖=0 2 ≤ 휔∣푥∣𝑋 ≤ 휔∣푥∣2 , 𝑋𝑔 using the deﬁnition of the inner product and the parallelogram identity; therefore choosing

∑𝑚 1 2 푚 1 2 휔 = ∣퐴𝑖∣ + +1+ ∣퐾∣ , 2 2 2 𝐿2 𝑖=0 we have that 𝒜 is dissipative in 푋𝑔. To use Lumer-Phillips, it only remains to be shown that for some 휆>0, given any 휓ˆ =(휓(0),휓) ∈ 푋,(휆퐼 − 퐴)휙ˆ = 휓ˆ, can be solved for some 휙ˆ = (휙(0),휙) ∈𝒟(𝒜).This is equivalent to solving 휆휙(0)−퐿(휙)=휓(0) and 휆휙− 퐷휙 = 휓 on [−푟, 0).But this follows immediately from [BBu2, Lemma 3.3, p.178].

37 6 Adjoint Operators and Dual Spaces

6.1 Adjoint Operators Recall that 퐴∗ : 𝒟(퐴∗) ⊂ 푋 → 푋 is defined in Definition 7 and we required that 퐴 be densely defined in 푋.We argue that this defines 퐴∗ uniquely if 𝒟(퐴)isdensein푋.To show this, suppose there exist 푤1 and 푤2 such that

⟨퐴푥, 푦⟩ = ⟨푥, 푤1⟩ for all 푥 ∈𝒟(퐴) and ⟨퐴푥, 푦⟩ = ⟨푥, 푤2⟩ for all 푥 ∈𝒟(퐴). Then, ⟨푥, 푤1 − 푤2⟩ =0 forall푥 ∈𝒟(퐴).

If 𝒟(퐴)isdensein푋,then푤1 − 푤2 must equal zero.Therefore, 푤1 = 푤2, and the adjoint of A is uniquely deﬁned.

6.1.1 Computation of 퐴∗: an Example from the Heat Equation

∗ The computation of 퐴 can be easy or impossible.Let 푋 = 퐿2(0,푙), then the operator 퐴 from the heat equation is deﬁned by 퐴휑 =(퐷휑′)′ on 2 ′ ′ ′ 𝒟(퐴)={휑 ∈ 퐻 (0,푙) ∣ (퐷휑 ) ∈ 퐿2(0,푙),휑(0) = 휑 (푙)=0}.

For 휑 ∈𝒟(퐴), 퐷 real, and 휓 ∈ 퐿2(0,푙)=푋,wehave ∫ ⟨퐴휑, 휓⟩ = 𝑙(퐷휑′)′휓¯ 0∫ = − 𝑙 퐷휑′휓¯′ + 퐷휑′휓¯∣𝑙 ∫ 0 0 𝑙 ¯′ ′ ¯′ 𝑙 ′ ¯ 𝑙 = 0 휑(퐷휓 ) − 휑퐷휓 ∣0 + 퐷휑 휓∣0 ′ ′ ¯′ 𝑙 ′ ¯ 𝑙 = ⟨휑, (퐷휓 ) ⟩−휑퐷휓 ∣0 + 퐷휑 휓∣0 = ⟨휑, 푤⟩ for all 휑 ∈𝒟(퐴) if and only if 휓′(푙)=0and휓(0) = 0.(We must, of course, have 휓 ∈ 퐻2(0,푙) ′ ′ and (퐷휓 ) ∈ 퐿2(0,푙) to carry out the operations.) Hence ∗ 2 ′ ′ ′ 𝒟(퐴 )={휓 ∈ 퐻 (0,푙) ∣ (퐷휓 ) ∈ 퐿2(0,푙),휓(0) = 휓 (푙)=0}. and 퐴∗휓 =(퐷휓′)′. In this case, ⟨퐴휑, 휓⟩ = ⟨휑, 퐴∗휓⟩; therefore 퐴 = 퐴∗, 𝒟(퐴)=𝒟(퐴∗), and hence 퐴 is a self adjoint operator.

38 6.1.2 Self Adjoint Operators and Dissipativeness For self adjoint linear operators, if 휑 ∈𝒟(퐴)=𝒟(퐴∗), we have

Re ⟨퐴휑, 휑⟩ =Re⟨휑, 퐴∗휑⟩ =Re⟨퐴∗휑, 휑⟩.

Therefore, whenever 퐴 is self adjoint, 퐴 is dissipative if and only if 퐴∗ is dissipative.Thus, for any closed, densely-deﬁned, dissipative self adjoint operator in 푋,both퐴 and 퐴∗ are inﬁnitesimal generators (see Corollary 𝐴𝑡 5 to the Lumer-Phillips theorem) of 퐶0 semigroups 푇 (푡) ∼ 푒 and 푆(푡) ∼ 푒𝐴∗𝑡.From a fundamental Hilbert space result, given below, we know that 푆(푡)=푇 (푡)∗.

Theorem 9 If 퐴 is an inﬁnitesimal generator of a 퐶0 semigroup 푇 (푡) on a ∗ Hilbert space 푋,then푇 (푡) = 푆(푡) is a 퐶0 semigroup on 푋 with inﬁnitesimal generator 퐴∗.

Homework Exercises ∙ Ex.8 (Project 1) :Consider𝒟(퐴∗) in a Hilbert space 푋 where 퐴 is densely deﬁned in 푋.

– What can you say about 𝒟(퐴∗) with respect to closed, dense, nonempty, all of 푋, relationship with 퐴−1 and (퐴∗)−1 when they exist, and symmetric versus self adjoint? – Now consider all the same questions in a Banach space 푋.(Note: In a Banach space 𝒟(퐴∗) ⊂ 푋∗.)

Give complete examples and counter examples.Also, give complete references. For relevant material, see [DS, HP, K, RN, Y].

6.2 Dual Spaces and Strong, Weak, and Weak* Topologies Before discussing dissipative operators in a Banach space 푋,wemustdis- cuss topological dual spaces 푋∗.We talk about strong, weak, and weak* topologies in a Banach space 푋.For relevant material, see [DS, K, Y]. Let 푋 be a Banach space.Then 푋∗, called the dual space or “topological” dual space, is a Banach space whenever 푋 is a normed linear space and the scalar field is complete.It is defined as the Banach space of continuous (conjugate or anti-) linear functionals on 푋, where a continuous linear functional satisfies ∣푓(푥)∣≤푘∣푥∣ for all 푥 ∈ 푋 and a conjugate linear functional

39 satisﬁes 푓(훼푥 + 훽푦)=¯훼푓(푥)+훽푓¯ (푦) for all 푥, 푦 ∈ 푋.The elements of 푋∗ ∗ ∗ ∗ are denoted by 푥 where ∣푥 ∣∗ =sup∣푥 (푥)∣,where∣⋅∣∗ denotes the norm ∣𝑥∣≤1 in 푋∗.This of course gives rise to a strong or normed topology of 푋∗. If 푋 is a complex Hilbert space, then 푋 ≈ 푋∗, and we have the Riesz theorem.

Theorem 10 (Riesz theorem) If 푋 is a Hilbert space, then for every 푓 ∈ 푋∗, there is a corresponding 푓˜ ∈ 푋 such that

푓(푥)=⟨푓,푥˜ ⟩ and

푓(훼푥 + 훽푦)=⟨푓,훼푥˜ + 훽푦⟩ =¯훼⟨푓,푥˜ ⟩ + 훽¯⟨푓,푦˜ ⟩ for all 푥 ∈ 푋.

Deﬁnition 8 Weak convergence of a sequence {푥𝑛}⊂푋, 푥𝑛 ⇀푥,isde- ﬁned by

∗ ∗ ∗ ∗ 푥𝑛 ⇀푥if and only if 푥 (푥𝑛) → 푥 (푥) for all 푥 ∈ 푋 .

We write 푥 = wlim 푥𝑛.

Deﬁnition 9 푋 is said to be weakly sequentially complete if and only if every weakly Cauchy sequence in 푋 converges weakly to an element 푥 ∈ 푋.

Result Section 1 There are a number of useful results related to weak convergence. 1.Weak limits are unique. ∗ ∗ To see this, suppose 푥𝑛 ⇀푥and 푥𝑛 ⇀푦,then푥 (푥)=푥 (푦), i.e, 푥∗(푥 − 푦) = 0 for all 푥∗ ∈ 푋∗.However, the Hahn-Banach theorem says if 푧 ∕= 0, then there exists an 푥∗ ∈ 푋∗ such that 푥∗(푧) ∕=0. Therefore, 푥 = 푦.

2.(a) 푥𝑛 → 푥 implies 푥𝑛 ⇀푥, but the converse is not true.

(b) 푥𝑛 ⇀푥implies {푥𝑛} is bounded in 푋 and ∣wlim 푥𝑛∣ = ∣푥∣≤lim inf ∣푥𝑛∣. Note: This says that the norm is weakly lower semi-continuous. Recall if 푓 is continuous, then ∣푓(푥) − 푓(푥0)∣ <휖or 푓(푥0) − 휖<

40 푓(푥) <푓(푥0)+휖 for 푥 near 푥0 .Lower semi-continuous means we just have 푓(푥0) − 휖<푓(푥)for푥 suﬃciently near 푥0.This has important applications in optimization (see the remark below). ∗ ∗ (c) 푥𝑛 → 푥 if and only if 푥 (푥𝑛) converges uniformly for ∣푥 ∣∗ ≤ 1. Remark: Application of Semi-continuity in Optimization. Weak lower semi-continuity plays a fundamental role in many optimization settings.Suppose we’re in a Hilbert space 푋 and we want to minimize a function 퐽(푥)over퐾 ⊂ 푋.Further suppose we have a minimizing sequence {푥𝑛} such that 퐽(푥𝑛) ↘ inf 퐽(푥)and{푥𝑛} is bounded, i.e., ∣푥𝑛∣≤푀. 𝑥∈𝐾 As we shall see below, bounded sequences possess weakly convergent sub- sequences.Therefore we can choose a subsequence converging weakly, i.e, 푥 ⇀ 푥˜.If 퐽 is weakly lower semi-continuous, then we have 퐽(wlim 푥 ) ≤ 𝑛𝑘 𝑛𝑘 lim inf 퐽(푥𝑛 )or퐽(˜푥) ≤ inf 퐽(푥).Whenever 퐾 is weakly closed, this 𝑘 𝑥∈𝐾 yields existence of a minimum (not just an inﬁmum) when minimizing a weakly lower semi-continuous 퐽.These facts are often used in constructive arguments to develop computational algorithms.

Deﬁnition 10 Weak convergence generates a topology in 푋 (use generalized sequences or nets) called the weak topology of 푋 or 푋∗ topology of 푋. Note: Weakly closed implies strongly closed, but not conversely in general. However, we do have an important case for which weak and strong closure are equivalent.

Theorem 11 (Mazur - 1933) A convex set 퐸 ⊂ 푋 is 푋∗ closed if and only if 푋 is closed. Thus, for convex sets, a set is weakly closed if and only if it is strongly closed.See [DS] for relevant information.

Definition 11 Since 푋∗ is a Banach space, we can define its dual, 푋∗∗. There is a natural embedding 푋M→ 푋∗∗ defined by 퐽 : 푋 → 푋∗∗ where

퐽(푥)(푥∗)=푥∗(푥) for all 푥∗ ∈ 푋∗. In general, 퐽(푋) ⊂ 푋∗∗, but 퐽(푋) ∕= 푋∗∗.

Deﬁnition 12 If 퐽(푋)=푋∗∗,then푋 is called a reﬂexive Banach space.

41 Result Section 2 The following elementary results can be argued. 1.Hilbert spaces are reflexive. 2. 푋 is a reflexive Banach space implies 푋 is weakly sequentially complete. 3. 푋 is a reflexive Banach space if and only if {푥 ∈ 푋∣∣푥∣≤1} is compact in the weak topology.Therefore, in a Banach space, bounded sets are weakly, sequentially pre-compact. 4.Bounded sets in a Hilbert space are weakly, sequentially pre-compact. (We have this from Results 1.and 3.) 5.Let 푋 be a Hilbert space.Then

푥𝑛 ⇀푥and ∣푥𝑛∣→∣푥∣ implies 푥𝑛 → 푥. Conversely,

푥𝑛 ⇀푥and 푥𝑛 → 푥 implies ∣푥𝑛∣→∣푥∣. Therefore,

푥𝑛 ⇀푥implies 푥𝑛 → 푥 if and only if ∣푥𝑛∣→∣푥∣.

This is easy to see as 2 2 2 ∣푥𝑛 − 푥∣ = ⟨푥𝑛 − 푥, 푥𝑛 − 푥⟩ = ∣푥𝑛∣ −⟨푥𝑛,푥⟩−⟨푥, 푥𝑛⟩ + ∣푥∣ .

Deﬁnition 13 For a Banach space 푋, 푋∗ is itself a Banach space; therefore, the weak topology of 푋∗ can be deﬁned and is called the 푋∗∗ topology. We can now consider weak convergence in 푋∗ which can be characterized by

∗ ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗∗ ∗∗ 푥𝑛 ⇀푥 ∈ 푋 if and only if 푥 (푥𝑛) → 푥 (푥 ) for all 푥 ∈ 푋 . Deﬁnition 14 We can put an even weaker topology on 푋∗.We can deﬁne ∗ ∗ ∗ ∗ 𝑤 ∗ a weak* convergence of {푥𝑛}⊂푋 , denoted as 푥𝑛 ⇀푥 ,by

∗ 𝑤∗ ∗ ∗ ∗ 푥𝑛 ⇀푥 ≡ 푥𝑛(푥) → 푥 (푥) for all x ∈ 푋. (i.e., the pointwise convergence of (anti) linear functionals.) Note that this is in general a weaker convergence than weak convergence since 퐽(푋) ⊂ 푋∗∗ with 퐽(푋) ∕= 푋∗∗ in general.

42 Result Section 3 We have the following results. ∗ 𝑤∗ ∗ ∗ 1. 푥𝑛 ⇀푥 implies ∣푥𝑛∣∗ ≤ 푀. ∗ ∗ ∗ ∗ 𝑤∗ ∗ ∗ 2. 푥𝑛 → 푥 in 푋 implies 푥𝑛 ⇀푥 in 푋 (i.e., strong convergence implies weak* convergence). The converse does not hold. 3.If 푋 is a Banach space, then 푋∗ is weak* sequentially complete and the norm ∣⋅∣∗ is weak* lower semi-continuous.In other words, ∗ ∗ ∣w*lim inf 푥𝑛∣∗ ≤ lim inf ∣푥𝑛∣∗.

∗ ∗ ∗ ∗ ∗ 𝑤 ∗ ∗ 4. 푥𝑛 ⇀푥 in 푋 implies 푥𝑛 ⇀푥 in 푋 (i.e., weak convergence implies weak* convergence).

The weak* convergence in 푋∗ can be used to deﬁne a topology for 푋∗, called the 푋 topology of 푋∗.In general, the 푋 topology of 푋∗ is weaker than the 푋∗∗ topology of 푋∗.In other words, in 푋∗ strong convergence implies weak convergence implies weak* convergence; however, the converses do not hold.

6.2.1 Summary of topologies on 푋 and 푋∗ 푋 푋∗ strong strong weak weak ( 푋∗ topology of 푋) ( 푋∗∗ topology of 푋∗) – weak * ( 푋 topology of 푋∗) Note: For any Banach space 푌 , we can talk about weak convergence, but we can only talk about weak* convergence in the case that 푌 = 푋∗ = the dual of some space 푋 !!

Theorem 12 (Alaoglu’s Theorem) Let 푋 be a Banach space. Then the unit ball (closed unit sphere) in 푋∗ is compact in the weak* topology (i.e., the 푋 topology of 푋∗).

Corollary 6 Bounded sets in 푋∗ are weak* sequentially compact.

43 Result Section 4 We can summarize additional ﬁndings that are consequences of the above results. 1.Let 퐽 : 푋 → 푋∗∗ be the natural embedding so that for a reﬂex- ive Banach space, 퐽(푋)=푋∗∗.Hence we have that the weak and ∗ ∗∗ ∗ weak* topologies of 푋 arethesame.Toseethisweargue푥 (푥𝑛)= ∗ ∗ 퐽(푥)(푥𝑛)=푥𝑛(푥) and hence ∗∗ ∗ ∗∗ ∗ 푥 (푥𝑛) → 푥 (푥 ) is weak sense convergence with 푥∗∗(푥∗)=퐽(푥)(푥∗)=푥∗(푥).

∗ ∗ Since 푥𝑛(푥) → 푥 (푥) is weak* convergence, these are thus equivalent. Conversely, if 푋 is not reflexive, then 퐽(푋) ∕= 푋∗∗ and the 푋 topology of 푋∗ is weaker than the 푋∗∗ topology of 푋∗.Thus in a reflexive Banach space (which includes Hilbert spaces), bounded sets are weakly sequentially compact.(This is a special case of Alaoglu’s theorem.) 2.A Banach space is reflexive if and only if the unit ball is compact in weak topology.

6.3 Examples of Spaces and Their Duals

∙ We ﬁrst consider 퐿𝑝(0, 1) and recall 1 1 퐿∗(0, 1) = 퐿 (0, 1), 1 ≤ 푝<∞, + =1. 𝑝 𝑞 푝 푞

∗ In other words, 퐿∞ is the dual of a space, 퐿1 = 퐿∞; however, we do ∗ not have a satisfactory representation for the dual 퐿∞ of 퐿∞.In other words, we cannot write 퐿1 as the dual of 퐿∞ or any other known space. As a result, we can consider the weak* topology in 퐿𝑝 for 1 <푝≤∞, but we cannot talk about the weak* topology in 퐿1.

∙ 퐿𝑝(0,푇; 푋) are important spaces because they are very useful in study- ing the system푥 ˙(푡)=퐴푥(푡)+퐹 (푡), 0 <푡<푇, in 푋.We have (to be discussed later) ∗ ∼ ∗ 퐿𝑝(0,푇; 푋) = 퐿𝑞(0,푇; 푋 ), 1 ≤ 푝<∞.

See [E] for relevant details on Banach space valued functions of time on intervals (0,푇).

44 ∙ The Banach space of (bounded) continuous functions on [푎, 푏]isde- noted by 퐶[푎, 푏].It has a norm deﬁned by ∣휑∣ =sup∣휑(휉)∣ for 𝜉∈[𝑎,𝑏] 휑 ∈ 퐶[푎, 푏].Then we have the following theorem from [DS, Chap- ter IV].

Theorem 13 The following equivalences hold:

퐶∗[푎, 푏] =∼ rba[푎, 푏] =∼ NBV[푎, 푏]

where NBV[푎, 푏] stands for normalized bounded variation functions and rba[푎, 푏] stands for regular bounded additive set functions.

Therefore we can consider the weak* topology for NBV[푎, 푏]; however, 퐶[푎, 푏] is not the dual of another space, and speciﬁcally NBV[푎, 푏]∗ ∕= 퐶[푎, 푏] (i.e., we do not have satisfactory characterizations or representations– there is no known space 푋 such 퐶[푎, 푏]=푋∗).Thus we cannot talk about the weak* topology in 퐶[푎, 푏].

Discussion of rba[푎, 푏]: Suppose 휇 is a regular set function on Borel subsets of [푎, 푏].Then by deﬁnition of regular, given a set 퐹 ⊂ [푎, 푏] and 휖>0, there exists a closed set 퐸 andanopenset푂 such that 퐸 ⊂ 퐹 ⊂ 푂¯ and 휇(푂¯ − 퐸) <휖.Additive means that for any two Borel subsets 퐴 and 퐵 contained in [푎, 푏] such that 퐴∩퐵 = ∅,wehave 휇(퐴 ∪ 퐵)=휇(퐴)+휇(퐵).

Discussion of NBV[푎, 푏]: BV[푎, 푏] has a weak norm ∣푓∣ = ∣푓(푎+)∣+ var(푓) where var(푓) is the total variation

∑𝑁 var(푓)= sup ∣푓(푥𝑖) − 푓(푥𝑖−1)∣, 𝜋[𝑎,𝑏] 𝑖=1 and 휋[푎, 푏] denotes the partitions of [푎, 푏].Note that var( 푓)providesa semi-norm on BV and BV can be written as BV𝑜 ⊕ 푁 where BV𝑜 = {푓 ∈ BV∣푓(푎+) = 0 and N is the 1-D space of constant functions}. NBV[푎, 푏] is normalized by making 푓 right-continuous at the interior points and 푓(푎+)=0with∣푓∣ =var(푓). Suppose 푓 ∈ NBV[푎, 푏]; then we can write [Ru1, p.101], [Ru2, p.163]

푓(푥)=푓(푎)+푝𝑓 (푥) − 푛𝑓 (푥)

푣(푥)=푝𝑓 (푥)+푛𝑓 (푥),

45 where 푝𝑓 , 푛𝑓 and 푣 are nondecreasing monotone functions deﬁned by ∑𝑁 + 푝𝑓 (푥)= sup (푓(푥𝑖) − 푓(푥𝑖−1)) , 𝜋[𝑎,𝑥] 𝑖=1 ∑𝑁 − 푛𝑓 (푥)= sup ∣(푓(푥𝑖) − 푓(푥𝑖−1)) ∣, and 𝜋[𝑎,𝑥] 𝑖=1 ∑𝑁 푣(푥)= sup ∣푓(푥𝑖) − 푓(푥𝑖−1∣, 𝜋[𝑎,𝑥] 𝑖=1 where (⋅)+ and (⋅)− are the positive and negative parts of (⋅), respectively.Using the monotone functions 푝𝑓 and 푛𝑓 , one can generate (positive) Lebesgue-Stieltjes measures 휇 and 휇 [DS, Hal, HeSt, 𝑝𝑓 𝑛𝑓 McS, Ru1, Ru2, Smir] and subsequently a signed measure 휇 = 휇 − 휇 , 𝑓 𝑝𝑓 𝑛𝑓 ∗ ∼ where 휇𝑓 is regular.From the above theorem we have 퐶 [푎, 푏] = rba(푎, 푏), which means there is a one-to-one correspondence 푥∗ ↔ 휇,

given by ∫ 푥∗(휑)= 휑푑휇.

We also have 퐶∗[푎, 푏] =∼ NBV[푎, 푏], which means we have ∗ 푥 ↔ 휇𝑓 ,

given by ∫ ( ∫ ) ∗ 푥 (휑)= 휑푑휇𝑓 = 휑푑푓 where the integrals are Lebesque-Stieltjes integrals. ∙ We also need to consider Sobolev spaces, 푊 𝑘,𝑝(Ω).Recall 푊 𝑘,𝑝(Ω) is reflexive for 1 <푝<∞ and, moreover, 푊 𝑘,2(Ω) is a Hilbert space.We 𝑘,𝑝 can define 푊0 (Ω) as Sobolev spaces with compact support (“func- 𝑘,𝑝 tions vanish at the boundary”), or we can think of 푊0 (Ω) as the ∞ 𝑘,𝑝 closure of 퐶0 (Ω) in 푊 .We can consider their dual spaces and find 1 1 푊 𝑘,𝑝(Ω)∗ =∼ 푊 −𝑘,𝑝(Ω) 1 <푝<∞, + =1. 0 푝 푞 For relevant material, see [A, DS].

46 6.4 Return to Dissipativeness for General Banach Spaces Deﬁnition 15 Let 푋 be a Banach space and 푋∗ be its conjugate dual. Then 푥∗(푥)=⟨푥∗,푥⟩ is called the duality product.

Deﬁnition 16 For each 푥 in a Banach space 푋, we deﬁne the duality set 퐹 (푥) ⊂ 푋∗ by

∗ ∗ ∗ 2 ∗ 2 퐹 (푥)={푥 ∈ 푋 ∣⟨푥 ,푥⟩ = ∣푥∣𝑋 = ∣푥 ∣𝑋∗ }. We know 퐹 (푥) ∕= ∅ for each 푥 ∈ 푋 by the Hahn-Banach theorem.

Result If 푋 is a Hilbert space, one can show 퐹 (푥)={푥} under the usual isomor- phism 푋 ≃ 푋∗.(This is the Duality Set Lemma, Section 4)

Deﬁnition 17 Let 퐴 : 𝒟(퐴) ⊂ 푋 → 푋 for a Banach space 푋.Then 퐴 is called dissipative if for every 푥 ∈𝒟(퐴)thereexistssome푥∗ ∈ 퐹 (푥)such that Re⟨푥∗,퐴푥⟩≤0.

Result The theorems stated previously for Hilbert spaces are still true in a Banach space with the deﬁnition of dissipativeness given above.(Lumer-Phillips, characterizations, etc.) See [Pa, p.13-16].

6.5 More on Adjoint Operators Let 푋 and 푌 be normed linear spaces with 푋∗ and 푌 ∗ as their conjugate duals, respectively.For 𝒟(퐴)densein푋, suppose we have the operator 퐴 : 𝒟(퐴) ⊂ 푋 → 푌 .Then we can deﬁne the adjoint operator 퐴∗ : 𝒟(퐴∗) ⊂ 푌 ∗ → 푋∗ with 𝒟(퐴∗)={푦∗ ∈ 푌 ∗∣ 푥 → 푦∗(퐴푥) is continuous on 𝒟(퐴)}. Given 푦∗ ∈ 푌 ∗,weseethat푦∗ ∈𝒟(퐴∗) if and only if there exists 푥∗ ∈ 푋∗ such that 푦∗(퐴푥)=푥∗(푥) for all 푥 ∈𝒟(퐴).Therefore, we deﬁne 퐴∗ by 퐴∗푦∗ = 푥∗.

47 In other words, (퐴∗푦∗)(푥)=푥∗(푥).

6.5.1 Adjoint Operator in a Hilbert Space In a Hilbert space, 푋∗ ∼ 푋 and 푌 ∗ ∼ 푌 ; therefore,

∗ ⟨퐴푥, 푦⟩𝑌 = ⟨푥, 퐴 푦⟩𝑋 .

6.5.2 Special Case In general, we do not need to have 푇 ∈ℒ(푋, 푌 )for푇 ∗ to be deﬁned; however, if we do have 푇 ∈ℒ(푋, 푌 ), then 푇 ∗ ∈ℒ(푌 ∗,푋∗).

Theorem 14 Let 푋 and 푌 be reﬂexive Banach spaces. If 푇 is a closed linear operator 푇 : 푋 → 푌 with 𝒟(푇 ) dense, then 𝒟(푇 ∗) is dense in 푌 ∗(∼ 푌 ) and 푇 ∗∗ = 푇 .

Note: This does not say anything about the relationship between 𝒟(푇 )and 𝒟(푇 ∗).

6.6 Examples of Computing Adjoints In this series of examples, we illustrate how the domain, and in particular, boundary conditions, can affect the definition of an operator.In this case we study operators which are all essentially the operations of second order differentiation, i.e., 퐴휑 = 휑′′, arising in the heat equation operator of Ex- ample 1.As we shall see, the boundary conditions affect dramatically the nature of the adjoint 퐴∗. 2 2 ′ ′ We define 퐻0 (0, 1) = {휑 ∈ 퐻 (0, 1)∣휑(0) = 휑(1) = 휑 (0) = 휑 (1) = 0}.

2 1.Let 푉 = 퐻0 (0, 1) and 퐻 = 퐿2(0, 1).We deﬁne the operator 퐴 by 퐴 : 𝒟(퐴)=푉 ⊂ 퐻 → 퐻

by 퐴휑 = 휑′′. In order to deﬁne the adjoint operator 퐴∗, we need to ﬁnd, for each 휓,anelement푤 such that

2 ⟨퐴휑, 휓⟩ = ⟨휑, 푤⟩ for all 휑 ∈ 퐻0 (0, 1).

48 Therefore, we may carry out the calculations ∫ ⟨퐴휑, 휓⟩ = 1 휑′′휓 0 ∫ = − 1 휑′휓′ + 휑′휓∣1 ∫ 0 0 = 1 휑휓′′ − 휑휓′∣1 + 휑′휓∣1 ∫0 0 0 1 ′′ = 0 휑휓 +0+0 = ⟨휑, 휓′′⟩

which hold if and only if 휓 ∈ 퐻2. Therefore 𝒟(퐴∗)=퐻2(0, 1) with 퐴∗휓 = 휓′′. Observe that 𝒟(퐴∗) ∕= 𝒟(퐴); therefore, 퐴 is not self-adjoint.However, ∗ 2 퐴 = 퐴 on their common domain 퐻0 .Thus we see that there are boundary conditions which immediately render the heat operator non- self-adjoint.

2 ∗ −2 2.Let 퐻 = 퐿2(0, 1), 푉 = 퐻0 (0, 1) and recall that 푉 = 퐻 (0, 1) ≡ ∫ 𝑥 ∫ 𝑠 {휓∣푥 → 0 0 휓(푡)푑푡푑푠 is in 퐿2(0, 1)} (see [A]).We deﬁne the operator 퐴˜ by 퐴˜ : 퐻 → 푉 ∗ by ˜ ′′ (퐴휑)휓 = ⟨휑, 휓 ⟩𝐻 for 휑 ∈ 퐻 and 휓 ∈ 푉 .We claim that 퐴˜ ∈ℒ(퐻, 푉 ∗) and this shows that diﬀerentiation can be bounded (continuous) if the spaces are chosen ˜ in a certain way.To verify our claim show ∣퐴휑∣𝑉 ∗ ≤ 푘∣휑∣𝐻 .We have

∣(퐴휑˜ )(휓)∣ = ∣⟨휑, 휓′′⟩∣ ′′ ≤∣휑∣𝐿2 ∣휓 ∣𝐿2 ≤∣휑∣𝐻 ∣휓∣𝑉 .

Therefore ∣(퐴휑˜ )(휓)∣ ≤∣휑∣𝐻 . ∣휓∣𝑉 However ˜ ˜ ∣(퐴휑)(휓)∣ ∣퐴휑∣𝑉 ∗ =sup . 𝜓∕=0 ∣휓∣𝑉

49 Thus ˜ ∣퐴휑∣𝑉 ∗ ≤∣휑∣𝐻 which implies that 퐴˜ ∈ℒ(퐻, 푉 ∗).From this we know 퐴˜∗ ∈ℒ(푉,퐻). Next we seek to ﬁnd 퐴˜∗ by using

(퐴˜∗휓)(휑)=(퐴휑˜ )(휓)=⟨휑, 휓′′⟩

for 휑 ∈ 퐻.Thus 퐴˜∗ is deﬁned on all of 푉 by

퐴˜∗휓 = 휓′′.

As in the previous example, 퐴˜ is not self adjoint.

3.Let 퐻 = 퐿2(0, 1) and deﬁne the operator 퐴1 by

퐴1 : 𝒟(퐴1) ⊂ 퐻 → 퐻

where 2 1 𝒟(퐴1)=퐻 (0, 1) ∩ 퐻0 (0, 1) with ′′ 퐴1휑 = 휑 .

We know 퐴1 ∈ℒ/ (퐻) from a previous assignment.In order to ﬁnd the ∗ adjoint operator 퐴1, we want for each 휓 to ﬁnd a 푤 such that

⟨퐴1휑, 휓⟩ = ⟨휑, 푤⟩ for all 휑 ∈𝒟(퐴1).

However ∫ ⟨퐴 휑, 휓⟩ = 1 휑′′휓 1 ∫0 = 1 휑휓′′ − 휑휓′∣1 + 휑′휓∣1 ∫0 0 0 = 1 휑휓′′ +0+휑′휓∣1 ∫0 0 1 ′′ = 0 휑휓 2 1 if and only if 휓 ∈ 퐻 ∩ 퐻0 .Hence

∗ ′′ 퐴1휓 = 휓

with ∗ 2 1 𝒟(퐴1)=퐻 (0, 1) ∩ 퐻0 (0, 1) = 𝒟(퐴1).

Thus 퐴1 is self-adjoint; however, it’s not a bounded linear operator on 퐻.

50 1 ∗ −1 4.Let 푉 = 퐻0 (0, 1).Then 푉 = 퐻 (0, 1).We can deﬁne the operator ˜ 퐴1 by ˜ ∗ 퐴1 : 푉 → 푉 with ˜ ′ ′ (퐴1휑)(휓)=−⟨휑 ,휓 ⟩𝐻 .

˜ ∗ ˜ We argue 퐴1 ∈ℒ(푉,푉 ).Toseethiswemustestablish ∣퐴1휑∣𝑉 ∗ ≤ 푘∣휑∣𝑉 for some 푘.We ﬁnd

˜ ′ ′ ∣(퐴1휑)(휓)∣ = ∣⟨휑 ,휓 ⟩∣ ′ ′ ≤∣휑 ∣𝐿2 ∣휓 ∣𝐿2 ≤∣휑∣𝑉 ∣휓∣𝑉 ,

and hence ˜ ∣(퐴1휑)(휓)∣ ≤∣휑∣𝑉 . ∣휓∣𝑉 However ˜ ˜ ∣(퐴1휑)(휓)∣ ∣퐴1휑∣𝑉 ∗ =sup . 𝜓∕=0 ∣휓∣𝑉 Therefore ˜ ∣퐴1휑∣𝑉 ∗ ≤∣휑∣𝑉 ˜ ∗ ˜ ∗ ∗ which yields that 퐴1 ∈ℒ(푉,푉 ).We know that 퐴1 ∈ℒ(푉,푉 ).Next ˜∗ we would like to ﬁnd 퐴1 which must satisfy ˜ ∗ ˜ ′ ′ (퐴1 휓)(휑)=(퐴1휑)(휓)=−⟨휑 ,휓 ⟩ ˜ ˜ ∗ for 휑 ∈𝒟(퐴1).Hence 퐴1 is deﬁned on all of V by

˜ ∗ ′ ′ (퐴1 휓)휑 = −⟨휑 ,휓 ⟩. ˜ Thus we ﬁnd that 퐴1 is a bounded self adjoint linear operator.

51 7 Weak* Convergence, the Prohorov Metric, In- verse Problems and Optimization

7.1 Weak* Convergence and the Prohorov Metric We turn next to discuss functional analysis concepts that are integral to several types of inverse problems.These problems entail parameter estimation using output observations where the parameters to be estimated are probability distributions.Such problems can be readily found in certain areas of applications [BBi, BBH, BBKW, BBPP, BD, BDTR, GRD-FP, BDEHAD, GRD-FP2, BF, BFPZ, BG1, BG2, BGIT].Here we consider two types of measure dependent problems: individual dynamics and aggregate dynamics.

7.1.1 Type I: Individual Dynamics/Aggregate Data Recall, from Example 5 (the size-structured population model for mosquitofish where there is a family 𝒢 of individual growth rates 푔) we have the expected size density 푢(푡, 휉; 푃 )=ℰ[푣(푡, 휉; ⋅)∣푃 ] described for a general probability distribution 푃 ,wehave ∫ 푢(푡, 휉; 푃 )= 푣(푡, 휉; 푔)푑푃 (푔), 𝒢 as the expected value of the density 푣(푡, 휉; 푔) which satisfies for a given 푔 the Sinko-Streiffer. Since in typical inverse problems, we must use aggregate population data 푑𝑖𝑗 to estimate 푃 itself, we are therefore required to understand the qualitative properties (continuity, sensitivity, etc.) of 푢(푡, 휉; 푃 ) as a function of 푃 . Thus, we see that the data for parametric estimation or inverse problems is 푑𝑖𝑗 ∼ 푢(푡, 휉; 푃 ).

7.1.2 Type II: Aggregate Dynamics We return to the electromagnetics example of Example 4 which entails polarization of inhomogeneous dielectric materials.In this case, individual (particle) dynamics are not available.Instead, the dynamics (13) them- selves depend on a probability measure 퐹 ,e.g., ∂푢 ∂2 + 푢 = 푓(푢, 퐹 ). ∂푡 ∂푥2 To describe the behavior of the electric polarization 푃 ,webeginwith the general formulation of Chapter 2 of [BBL] by employing a polarization

52 kernel 푔 in the convolution expression

∫ 𝑡 푃 (푡, 푧)= 푔(푡 − 푠,푧; 휏)퐸(푠,푧)푑푠. (37) 0 As explained in Section 2.5 and [BBL], this general formulation includes as special cases the well known orientational or Debye polarization model, the electronic or Lorentz polarization model, and linear combinations thereof, as well as other higher order models.In the Debye case the kernel is given by −𝑡/𝜏 푔(푡; 휏)=휖0(휖𝑠 − 휖∞)/휏 푒 , while in the Lorentz model (again see [BBL]), it takes the form

2 −𝑡/2𝜏 푔(푡; 휏)=휖0휔𝑝/휈0 푒 sin(휈0푡). However, use of these kernels presupposes that the material may be sufficiently defined by a single relaxation parameter 휏, which is generally not the case.In order to account for multiple relaxation parameters in the polarization mechanisms, we allow for a distribution of relaxation parameters which is conveniently described in terms of a probability measure 퐹 .Thus, we define our polarization model in terms of a convolution operator

∫ 𝑡 푃 (푡, 푧)= 퐺(푡 − 푠,푧; 퐹 )퐸(푠,푧)푑푠 0 where 퐺 is determined by various polarization mechanisms each described by a diﬀerent parameter 휏, and therefore is given by ∫ 퐺(푡, 푧; 퐹 )= 푔(푡, 푧; 휏)푑퐹 (휏) 𝒯 where 𝒯⊂[휏1,휏2]. To obtain the macroscopic polarization, we sum over all the parameters. We cannot separate dynamics to obtain individual dynamics, and therefore we have (13) in Section 2.5 as an example where the dynamics for the 퐸 and 퐻 ﬁelds depend explicitly on the probability measure 퐹 . For inverse problems, data 푑𝑖𝑗 ≈ 푢(푡𝑖,휉𝑗)=푢(푡𝑖,휉𝑗; 푃 )inTypeI,whereas 푑𝑖𝑗 ≈ 퐸(푡𝑖, 푥¯𝑗; 퐹 ) in Type II.Observe that 푃 or 퐹 ∈𝒫(푄), which is the space of probability measures, where 푄 consists of “parameters”, and 𝒫(푄) is a set of probability distributions over 푄. For example, in ordinary least square problems, we have ∑ ∑ 2 2 퐽(푃 )= ∣푑𝑖𝑗 − 푢(푡𝑖,휉𝑗; 푃 )∣ or 퐽(퐹 )= ∣푑𝑖𝑗 − 퐸(푡𝑖, 푥¯𝑗; 퐹 )∣ , 𝑖𝑗 𝑖𝑗

53 to be minimized over 푃 or 퐹 ∈𝒫(푄) or over some subset of 𝒫(푄).We need (for computational and theoretical considerations) a sense of closeness of two probability distributions 푃1 and 푃2.The ideas of local minima, continuity, “gradient”, etc., depend on a metric (or topology) for 푃1 and 푃2. More fundamentally for Type II systems, the “well-posedness of systems”, including existence and continuous dependence of solutions on “parameters”, depends on the concept of 퐹1 and 퐹2 being close, i.e., 퐹 → 퐸(푡, 푥¯; 퐹 ) is continuous.

54 7.2 Metrics on Probability Spaces We consider 𝒫(푄) as the space of probability measures or distributions on 푄.Let 푃1 and 푃2 be probability distributions.If working on the real line (−∞, ∞)or(0, ∞) we will sometimes not distinguish between the probability measure 푃 and its cumulative distribution function 퐹 . We need to understand the concept of a metric or distance 휌(푃1,푃2) between 푃1 and 푃2. There are several metrics that metrize (i.e.,generate an equivalent topology for) the weak* topology on 𝒫 including

i) Levy −−denoted here by 휌𝐿(푃1,푃2),

ii) Prohorov −−denoted by 휌𝑃𝑅(푃1,푃2),

iii) Bounded Lipschitz −−denoted by 휌𝐵𝐿(푃1,푃2), and also some that do not metrize the weak* topology on 𝒫 including

iv) Total variation −−denoted by 휌𝑇𝑉 (푃1,푃2),

v) Kolmogorov −−denoted by 휌𝐾 (푃1,푃2). For relevant material, see [B, H, P]. Previous developments in probability theory provide helpful results in the pursuit of a possible complete computational methodology.One of the most important tools in probability theory is the Prohorov metric, which we will now formally deﬁne.Let 푄 be a complete metric space with metric 푑. Given a closed subset 퐴 of 푄, deﬁne the 휖-neighborhood of 퐴 as

퐴𝜖 = {푞 ∈ 푄 : 푑(ˆ푞,푞) ≤ 휖 for some푞 ˆ ∈ 퐴} = {푞 ∈ 푄 :inf푑(ˆ푞,푞) ≤ 휖}. 𝑞ˆ∈𝐴 We deﬁne the Prohorov metric 휌 : 𝒫(푄) ×𝒫(푄) → 푅+ by

𝜖 휌(푃1,푃2) ≡ inf{휖>0:푃1(퐴) ≤ 푃2(퐴 )+휖, 퐴 closed ⊂ 푄}. This can be shown to be a metric on 𝒫(푄) and has many properties including (푎.)If푄 is complete, then (𝒫(푄),휌) is a complete metric space;

(푏.)If푄 is compact, then (𝒫(푄),휌) is a compact metric space. Note that the deﬁnition of 휌 is not intuitive.For example, we do not necessarily know what 푃𝑘 → 푃 in 휌 means.We have the following important characterizations [B].Given 푃𝑘,푃 ∈𝒫(푄), the following convergence statements are equivalent:

55 1. 휌(푃𝑘,푃) → 0; ∫ ∫ 1 2. 𝑄 푓푑푃𝑘(푞) → 𝑄 푓푑푃(푞) for all bounded, continuous 푓 : 푄 → 푅 ;

3. 푃𝑘[퐴] → 푃 [퐴] for all Borel sets 퐴 ⊂ 푄 with 푃 [∂퐴]=0. Thus, we immediately obtain the following results:

∙ Convergence in the 휌 metric is equivalent to convergence in distribution or so-called “weak” convergence of measures encountered in probability theory.

∗ ∙ Let 퐶𝐵(푄) denote the topological dual of 퐶𝐵(푄), where 퐶𝐵(푄)isthe usual space of bounded continuous functions on 푄 with the supremum ∗ norm.If we view 𝒫(푄) ⊂ 퐶𝐵(푄), convergence in the 휌 topology is equivalent to weak* convergence in 𝒫(푄).Note the misnomer “weak convergence of measures” used by probabilists which is actually weak* convergence in the functional analysis sense deﬁned here and in almost all standard texts.

More importantly, ∫ ∫

휌(푃𝑘,푃) → 0isequivalentto 푥(푡𝑖; 푞)푑푃𝑘(푞) → 푥(푡𝑖; 푞)푑푃 (푞), 𝑄 𝑄 for any continuous functions 푞 → 푥(푡𝑖; 푞).Thus 푃𝑘 → 푃 in 휌 metric is equivalent to ℰ[푥(푡𝑖; 푞)∣푃𝑘] →ℰ[푥(푡𝑖; 푞)∣푃 ] or “convergence in expectation”.This yields that

∑𝑛 ˆ 2 푃 → 퐽(푃 )= ∣ℰ[푥(푡𝑖; 푞)∣푃 ] − 푑𝑖∣ 𝑖=1 is continuous in the 휌 topology.Continuity of 푃 → 퐽(푃 ) allows us to assert the existence of a solution to min 퐽(푃 )over푃 ∈𝒫(푄). These results give us the following theorem.

Theorem 15 The Prohorov metric metrizes the weak* topology of 𝒫(푄). Now we may consider whether there are other (equivalent) metrics that metrize the weak* topology on 𝒫.But ﬁrst we point out an application in statistics where the need for metrics on distributions arises.

56 Robust Statistics Prohorov (also sometimes found as Prokhorov in translations) was interested in finding a useful metric for the frequently used “convergence in distribution” or “convergence in measure” in probability theory.Later theoretical statisticians became interested in (and concerned with) the underlying foundations for statistical testing and in particular inference procedures [H]. Specifically, attention was given to the “stability” of assumptions under which statistical asymptotic theories may be valid.This led to “robustness” (insensitivity to small deviations in underlying assumptions) or Robust Statistics.These ideas focus on “distributional robustness” as embodied in insensitivity to “small” derivations in distributions (often from the normal or Gaussian distribution assumptions found in many formulations) and also how lack of this robustness might affect the validity of statistical tests.This led to a renewed interest in the use of distributional metrics such as that of Prohorov.As formulated by Prohorov this led to attempts to find a metric for the “weak” topology on 𝒫 defined as the weakest topology on 𝒫 such that for every bounded continuous 휓 (휓 ∈𝒞𝐵(푄)) the map ∫ 푃 → 휓푑푃 𝑄

∗ is continuous, i.e. consider 𝒫⊂퐶𝐵(푄). To describe the various metrics that have found use by statisticians and probabilitists, it is desirable to recall the concept of a Polish space which is deﬁned as a complete, separable, metrizable topological space.We turn to a brief summary of other metrics of interest.

The Levy Metric The Levy metric is deﬁned in the special case for 푄 = 푅1, where one usually does not distinguish between the probability measure or distribution 푃1 and its cumulative distribution function (cdf) 퐹1.The metric is given by

휌𝐿(푃1,푃2)=inf{휖∣ for all 푥, 푃1(푥 − 휖) − 휖 ≤ 푃2(푥) ≤ 푃1(푥 + 휖)+휖}.

One can argue that this is a symmetric function and indeed deﬁnes a metric. Remarks √ ∙ 2휌𝐿(푃1,푃2) is maximum distance between graphs of 푃1,푃2 measured along 45𝑜 direction.It is a theorem that the Levy metric metrizes the weak* topology of 𝒫.

57 ∙ The Prohorov metric is more difficult to visualize but is applicable when 푄 is any complete separable metric space (Polish space), not just the real line (Levy case).For example, when 푄 is a function space such as growth rates or mortality rates in Sinko-Streiffer (mosquito fish problem), the Prohorov metric is applicable whereas the Levy metric is not.

We have already noted that both the Levy and Prohorov metrics metrize the weak* topology on 𝒫, the former only when 푄 = 푅1.Moreover, as noted above, we can argue that for 푄 a complete separable metric space, then (𝒫(푄),휌𝑃𝑅) is a complete separable metric space. Recall separability implies the existence of a subset 푄0 that is a countable dense subset of 푄.One can then obtain a countable dense subset of 𝒫 by deﬁning (see also Theorem 17 below)

𝒫 = {measures with ﬁnite support in 푄 with rational masses} 0 ∑ 0

= {푃 = 푝𝑖훿𝑞𝑖 ∣푝𝑖 rational, {푞𝑖}⊂푄0} ﬁnite

The Bounded Lipschitz Metric: Assume without loss of generality that distance function 푑 on 푄 is bounded by 1.(If necessary, one can replace the metric by an equivalent metric ˜ 𝑑(𝑞1,𝑞2) 푑(푞1,푞2)= .) Then deﬁne 1+𝑑(𝑞1,𝑞2) ∫ ∫ 휌𝐵𝐿(푃1,푃2)=sup 휓푑푃1 − 휓푑푃2 𝜓∈Ψ 𝑄 𝑄

where Ψ = {휓 ∈𝒞(푄):∣휓(푞1) − 휓(푞2)∣≤푑(푞1,푞2)}

A fundamental result of practical importance is

Theorem 16 For al l 푃1,푃2 ∈𝒫(푄),

2 휌𝑃𝑅(푃1,푃2) ≤ 휌𝐵𝐿(푃1,푃2) ≤ 2휌𝑃𝑅(푃1,푃2).

Thus, 휌𝑃𝑅 and 휌𝐵𝐿 deﬁne the same topology and hence 휌𝐵𝐿 also metrizes the weak topology on 𝒫(푄).

Other metrics Other metrics of interest can be found in the literature.These include

58 ∙ Total variation:

휌𝑇𝑉 (푃1,푃2)=sup∣푃1(퐴) − 푃2(퐴)∣, 𝐴∈ℬ and

∙ Kolmogorov: (푄 = 푅1)

휌𝐾(푃1,푃2)= sup∣푃1(푥) − 푃2(푥)∣. 𝑥∈𝑅′

These metrics do not metrize the weak topology, but do satisfy

휌𝐿 ≤ 휌𝑃𝑅 ≤ 휌𝑇𝑉

휌𝐿 ≤ 휌𝐾 ≤ 휌𝑇𝑉 .

Why is the knowledge about 휌𝐵𝐿 versus 휌𝑃𝑅 useful? We have that ∫ ∫ Δ𝑘 ≡ 퐺(푔)푑푃𝑘 (푔) − 퐺(푔)푑푃 (푔) ∫ ∫ = 퐺(푔)푓𝑘(푔)푑푔 − 퐺(푔)푓(푔)푑푔 ∫ = 퐺(푔)[푓𝑘(푔) − 푓(푔)]푑푔

≤ 휌𝐵𝐿(푃𝑘,푃)

≤ 2휌𝑃𝑅(푃𝑘,푃).

Therefore, if we know 푃𝑘 → 푃 in 휌𝑃𝑅, it may be useful in direct estimates.But Δ 𝑘 → 0 is Prohorov metric convergence itself if 퐺 ∈𝒞𝐵(푄).

59 7.3 Populations with Aggregate Data and Uncertainty We consider approximation methods in estimation or inverse problems– quantity of interest is a probability distribution assume we have parameter (푞 ∈ 푄) dependent system with model responses 푥(푡, 푞) describing population of interest For data or observations, we are given a set of values {푦𝑙} for the expected values ∫

ℰ[푥𝑙(푞)∣푃 ]= 푥𝑙(푞)푑푃 (푞) 𝑄 for model 푥𝑙(푞)=푥(푡𝑙,푞) wrt unknown probability distribution 푃 describing distribution of parameters 푞 over population Use data to choose from a given family 𝒫(푄) the distribution 푃 ∗ that gives best fit of underlying model to data Formulate ordinary least squares (OLS) problem–not essential–could equally well use a WLS, MLE, etc., approach Seek to minimize ∑ 2 퐽(푃 )= ∣ℰ[푥𝑙(푞)∣푃 ] − 푦𝑙∣ 𝑙 over 푃 ∈𝒫(푄) Even for simple dynamics for 푥𝑙 is an infinite dimensional optimization problem–need approximations that lead to computationally tractable schemes That is, it is useful to formulate methods to yield finite dimensional sets 𝒫𝑀 (푄) over which to minimize 퐽(푃 ) Of course, we wish to choose these methods so that “𝒫𝑀 (푄) →𝒫(푄)” in some sense In this case we shall use Prohorov metric [BBPP, BBi] of weak* convergence of measures to assure the desired approximation results A general theoretical framework is given in [BBPP] with specific results on the approximations we use here given in [BBi, BPi].Briefly, ideas for the underlying theory are as follows:

1.One argues continuity of 푃 → 퐽(푃 )on𝒫(푄) with the Prohorov metric

2.If 푄 is compact then 𝒫(푄)isacomplete metric space–also compact– when taken with Prohorov metric

3.Approximation families 𝒫𝑀 (푄) are chosen so that elements 푃 𝑀 ∈ 𝒫𝑀 (푄) can be found to approximate elements 푃 ∈𝒫(푄) in Prohorov metric

60 4. Well-posedness (existence, continuous dependence of estimates on data, etc.) obtained along with feasible computational methods

The data {푦𝑙} available (which, in general, will involve longitudinal or time evolution data) determines the nature of the problem.

∙ Type I:InwhatwehavereferredtoaboveasType I problems one has only aggregate or population level longitudinal data available.This is common in marine, insect, etc., catch and release experiments [BK] where one samples at diﬀerent times from the same population but cannot be guaranteed of observing the same set of individuals at each sample time.This type of data is also typical in experiments where the organism or population member being studied is sacriﬁced in the process of making a single observation (e.g., certain physiologically based pharmacokinetic (PBPK) modeling [BPo, Po] and whole organism transport models [BK]).In this case one may still have dynamic (i.e., time course) models for individuals, but no individual data is available.

∙ Type II: The second class of problems which above we called Type II problems involves dynamics which depend explicitly on the probability distribution 푃 itself.In this case one only has dynamics ( aggregate dynamics) for the expected value ∫ 푥¯(푡)= 푥(푡, 푞)푑푃 (푞) 𝑄 of the state variable.No dynamics are available for individual trajectories 푥(푡, 푞)foragiven푞 ∈ 푄.The electromagnetic example of Example 4 is precisely this situation.Such problems also arise in viscoelasticity as well as biology (the HIV cellular models of Banks, Bortz and Holte [BBH]) – see also [BBPP, BG1, BG2, BPi].

While the approximations we discuss below are applicable to both types of problems, we shall illustrate the computational results in the context of size-structured marine populations (mosquitoﬁsh, shrimp) and PBPK problems (TCE) where the inverse problems are of Type I.Finally, we note that in the problems considered here, one can not sample directly from the probability distribution being estimated and this again is somewhat diﬀerent from the usual case treated in some of the statistical literature, e.g., see [Wahba1, Wahba2] and the references cited therein.

61 Example 5: The Growth Rate Distribution Model and Inverse Problem in Marine Populations Our motivating application entails estimation of growth rate distributions for size-structured mosquitofish and shrimp populations.Mosquitofish are used in place of pesticides to control mosquito populations in rice fields.Ma- rine biologists desire to correctly predict growth and decline of mosquitofish population in order to determine the optimal densities of mosquitofish to use to control mosquito populations.A mathematical model that accurately describes the mosquitofish population would be beneficial in this application, as well as in other problems involving population dynamics and age/size- structured data. Based on data collected from rice fields, a reasonable mathematical model would have to predict two key features that are exhibited in the data: dispersion and bifurcation (i.e., a unimodal density becomes a bimodal density) of the population density over time [BBKW, BF, BFPZ]. As mentioned above in Example 5, the Sinko-Streifer model does exhibit either of these under reasonable biological assumptions.However, the Growth rate distribution (GRD) model, developed in [BBKW] and [BF], captures both of these features in its solutions. The model is a modification of the Sinko-Streifer model (used for modeling age/size-structured populations) which allows individuals to have different characteristics or behaviors.

Figure 1: Mosquitoﬁsh data.

Recall that the Sinko-Streifer model (SS) for size-structured mosquitoﬁsh populations is given by

∂푣 ∂ + (푔푣)=−휇푣, 휉 <휉<휉 ,푡>0 (38) ∂푡 ∂휉 0 1 푣(0,휉)=Φ(휉) ∫ 𝜉1 푔(푡, 휉0)푣(푡, 휉0)= 퐾(푡, 휁)푣(푡, 휁)∂휁 𝜉0 푔(푡, 휉1)=0.

Here 푣(푡, 휉) represents size (given in numbers per unit length) or population density, where 푡 represents time and 푥 represents length of mosquitoﬁsh–

62 growth rate of individual mosquitoﬁsh given by 푔(푡, 휉), where

푑휉 = 푔(푡, 휉) (39) 푑푡 for each individual (all mosquitofish of given size have same growth rate) As we have noted, the SS model cannot be used as is to model the mosquitofish population because it does not predict dispersion or bifurcation of the population in time under biologically reasonable assumptions [BBKW, BF].But by modifying the SS model so that the individual growth rates of the mosquitofish vary across the population (instead of being the same for all individuals in the population), one obtains a model, known as the growth rate distribution (GRD) model which does in fact exhibit both dispersal in time and development of a bimodal density from a unimodal density (see [BF, BFPZ]). As noted in Example 5, in the (GRD) model, the population density 푢(푡, 휉; 푃 ) is actually given by ∫ 푢(푡, 휉; 푃 )= 푣(푡, 휉; 푔)푑푃 (푔). (40) 𝒢 where 𝒢 is collection of admissible growth rates, 푃 is probability measure on 𝒢, and 푣(푡, 휉; 푔) is solution of (SS) with 푔.This model assumes that the population is made up of collections of subpopulations –individuals in same subpopulation have same growth rate As demonstrated in [BF], solutions to GRD model exhibit both dispersion and bifurcation of the population density in time.Here we assume that the admissible growth rates 푔 have the form

푔(휉; 푏, 훾)=푏(훾 − 휉) for 휉0 ≤ 휉 ≤ 훾 and zero otherwise, where 푏 is the intrinsic growth rate of the mosquitoﬁsh and 훾 = 휉1 is the maximum size.This choice based on work in [BBKW], where idea of other properties related to the growth rates varying among the mosquitoﬁsh is discussed. Under assumption of varying intrinsic growth rates and maximum sizes, assume that 푏 and 훾 are random variables taking values in the compact sets 퐵 and Γ, respectively.A reasonable assumption is that both are bounded closed intervals.Thus we take

퐺 = {푔(⋅; 푏, 훾)∣푏 ∈ 퐵,훾 ∈ Γ}

63 so that 퐺 is also compact in, for example, 퐶[휉0,휉1]where휉1 = max(Γ).Then 𝒫(퐺) is compact in the Prohorov metric and we are in framework outlined above.In illustrative examples, one may choose a growth rate parameterized by the intrinsic growth rate 푏 with 훾 = 1 , leading to a one parameter family of varying growth rates 푔 among the individuals in the population.We also assume here that 휇 =0and퐾 = 0 in order to focus on only the distribution of growth rates; however, distributions could just as well be placed on 휇 and 퐾. Next, introduce two different approaches that can be used in inverse problem for estimation of distribution of growth rates of the mosquitofish The first approach, which has been discussed and used in [BF] and [BFPZ], involves the use of delta distributions.We assume that probability distributions 𝒫𝑀 placed on growth rates are discrete corresponding to 𝑀 𝑀 𝑀 a collection 퐺 with the form 퐺 = {푔𝑘}𝑘=1 where 푔𝑘(푥)=푏𝑘(1 − 휉), for 푘 =1,...,푀.Here the {푏𝑘} are a discretization of 퐵.For each subpopulation with growth rate 푔𝑘, there is a corresponding probability 푝𝑘 that an individual is in subpopulation 푘. The population density 푢(푡, 푥; 푃 ) in (40) is then approximated by ∑ 푢(푡, 휉; {푝𝑘})= 푣(푡, 휉; 푔𝑘)푝𝑘, 𝑘 where 푣(푡, 휉; 푔𝑘) is the subpopulation density from (SS) with growth rate 푔𝑘. We denote this delta function approximation method as DEL(M), where M is number of elements used in this approximation. While it has been shown that DEL(M) provides a reasonable approximation to (40), a better approach might involve techniques that will provide a smoother approximation of (40) in the case of continuous probability distributions on the growth rates.Thus, as a second approach, we chose to use an approximation scheme based on piecewise linear splines.Here we have assumed that 푃 is a continuous probability distribution on the intrinsic growth ′ 𝑑𝑃 rates.We approximate the density 푃 = 𝑑𝑏 = 푝(푏) using piecewise linear splines, which leads to the following approximation for 푢(푡, 휉; 푃 ) in (40): ∑ ∫ 푢(푡, 휉; {푎𝑘}) ≈ 푎𝑘 푣(푡, 휉; 푔)푙𝑘(푏)푑푏, 𝑘 𝐵 where 푔(휉; 푏)=푏(1−휉), 푝𝑘(푏)=푎𝑘푙𝑘(푏) is the probability density for an individual in subpopulation 푘 and 푙𝑘 represents the piecewise linear spline functions.This spline based approximation method is denoted by SPL(M,N), where M is the number of basis elements used to approximate the growth

64 rate probability distribution and N is the number of quadrature nodes used to approximate the integral in the formula above. One can use the approximation methods DEL(M) and SPL(M,N) in the inverse problem for the estimation of the growth rate distributions.The least squares inverse problem to be solved is ∑ 2 min 퐽(푃 )= ∣푢(푡𝑖,휉𝑗; 푃 ) − 푢ˆ𝑖𝑗∣ (41) 𝑃 ∈𝒫𝑀 (𝐺) 𝑖,𝑗 ∑ 2 2 = (푢(푡𝑖,휉𝑗; 푃 )) − 2푢(푡𝑖,휉𝑗; 푃 )ˆ푢𝑖𝑗 +(ˆ푢𝑖𝑗) , 𝑖,𝑗

𝑀 where {푢ˆ𝑖𝑗 } is the data and 𝒫 (퐺) is the finite dimensional approximation to 𝒫(퐺). When using DEL(M), the finite dimensional approximation 𝒫𝑀 (퐺) to the probability measure space 𝒫(퐺)isgivenby { } ∑ ∑ 𝒫𝑀 (퐺)= 푃 ∈𝒫(퐺)∣ 푃 ′ = 푝 훿 , 푝 =1 , 𝑘 𝑔𝑘 𝑘 𝑘 𝑘 where 훿 is the delta function with an atom at 푔 . However, when using 𝑔𝑘 𝑘 SPL(M,N), the finite dimensional approximation 𝒫𝑀 (퐺)isgivenby { } ∑ ∑ ∫ 𝑀 ′ 𝒫 (퐺)= 푃 ∈𝒫(퐺)∣ 푃 = 푎𝑘푙𝑘(푏), 푎𝑘 푙𝑘(푏)푑푏 =1 . 𝑘 𝑘 𝐵 Furthermore, we note that this least squares inverse problem (41) becomes a quadratic programming problem [BF, BFPZ].Letting p be the vector that contains 푝𝑘, 1 ≤ 푘 ≤ 푀, when using DEL(M) or 푎𝑘, 1 ≤ 푘 ≤ 푀, when using SPL(M,N), we let A be the matrix with entries given by ∑ 퐴𝑘𝑚 = 푣(푡𝑖,휉𝑗; 푔𝑘)푣(푡𝑖,휉𝑗; 푔𝑚), 𝑖,𝑗 let b the vector with entries given by ∑ 푏𝑘 = − 푢ˆ𝑖𝑗푣(푡𝑖,휉𝑗; 푔𝑘), 𝑖,𝑗 and ∑ 2 푐 = (ˆ푢𝑖𝑗) , 𝑖,𝑗 where 1 ≤ 푘, 푚 ≤ 푀. In the place of (41), we now minimize

퐹 (p) ≡ p𝑇 Ap +2p𝑇 b + 푐 (42)

65 over 𝒫𝑀 (퐺). We note when using DEL(M) we also had to include the constraint ∑ 푝𝑘 =1, 𝑘 while when using SPL(M,N) we had to include the constraint ∑ ∫ 푎𝑘 푙𝑘(푏)푑푏 =1. 𝑘 𝐵 However, in both cases, we were able to include these constraints along with non-negativity constraints on the {푝𝑘} and {푎𝑘} in the programming of these two inverse problems.

Other Size-Structured Population Models The Sinko-Streifer (SS) model [SS] and its variations have been widely used to describe numerous age and size-structured populations (see [BBDS1, BBDS2, BBKW, BF, BFPZ, BA, Kot, MetzD] for example). One recent example involves the shrimp growth models of [Shrimp, BDEHAD]. Dispersion in size observed in experimental data for early growth of shrimp has been observed in data from different raceways at Shrimp Mariculture Research Facility, Texas Agricultural Experiment Station in Corpus Christi, TX.Initial sizes were very similar but variability is observed in aggregate type longitudinal data.A reasonable model for these populations must account for variability in size distribution data which perhaps is a result of variability in individual growth rates across population. In summary, the GRD model (40) represents one approach to account- ing for variability in growth rates by imposing a probability distribution on the growth rates in the SS model (38).Individuals in the population grow according to a deterministic growth model (39), but different individuals in the population may have different parameter dependent growth rates in the GRD model.The population is assumed to consist of subpopulations with individuals in the same subpopulation having the same growth rate.The growth uncertainty of individuals in the population is the result of variability in growth rates among the subpopulations.This modeling approach, which entails a stationary probabilistic structure on a family of deterministic dynamic systems, may be most applicable when the growth of individuals is assumed to be the result of genetic variability.

66 7.4 Two Player Min-Max Games with Uncertainty We consider electromagnetic evasion-interrogation games wherein the evader can use ferroelectric material coatings to attempt to avoid detection while the interrogator can manipulate the interrogating frequencies (wave numbers) and angles of incidence of the interrogating inputs to enhance detection and identification.The resulting problems are formulated as two player games in which one player wishes to minimize the reflected signal while the other wishes to maximize it.Simple dete rministic strategies are easily de- feated and hence the players must introduce uncertainty to disguise their intentions and confuse their opponent.Mathematically, the resulting game is carried out over spaces of probability measures which in many cases are appropriately metrized using the Prohorov metric.Details of the results discussed in this section can be found in [BGIT]. In [BIKT] the authors demonstrated that it is possible to design ferroelectric materials with appropriate dielectric permittivity and magnetic permeability to significantly attenuate reflections of electromagnetic interrogation signals from highly conductive targets such as airfoils and missiles. This was done under assumptions that the interrogating input signal is uniformly likely to come from a sector of interrogating angles 훼 ∈ [훼0,훼1] 𝜋 (훼 = 2 − 휙 where 휙 is the angle of incidence of the input signal) but that the evader has knowledge of the interrogator’s input frequency or frequencies (denoted in the presentations below as the interrogator design frequencies 퐼𝐷).These results were further sharpened and illustrated in [BIT] where a series of different material designs were considered to minimize over a given set of input design frequencies 퐼𝐷 the maximum reflected field from input signals.In addition, a second critical finding was obtained in that it was shown that if the evader employed a simple counter interrogation design based on a fixed set (assumed known) of interrogating frequencies 퐼𝐷, then by a rather simple counter-counter interrogation strategy (use an interrogating frequency little more than 10% different from the assumed design frequencies), the interrogator can easily defeat the evader’s material coatings counter interrogation strategy to obtain strong reflected signals. From the combined results of [BIKT, BIT] it is thus rather easily con- cluded that the evader and the interrogator must each try to confuse the other by introducing significant uncertainty in their design and interrogating strategies, respectively.This concept, which we refer to as mixed strategies in recognition of previous contributions to the literature on games (von Neumann’s finite mixed strategies to be explained below), leads to two player non-cooperative games with probabilistic strategy formulations.

67 These can be mathematically formulated as two sided optimization problems over spaces of probability measures, i.e., minmax games over sets of probability measures.

7.4.1 Problem Formulation We consider electromagnetic interrogation of objects in the context of minmax evader-interrogator games where each player has uncertain information about the adversary’s capabilities.The minmax cost functional is based on reﬂected ﬁelds from an object such as an airfoil or missile and can be computed in one of several ways [BIKT, BIT].

Ambient φ

z=0 Dielectric Coating z=d

Perfectly conducting half plane

Figure 2: Interrogating high frequency wave impinging (angle of incidence 휙) on coated (thickness 푑) perfectly conducting surface

The simplest is the reflection coefficient based on a simple planar geometry (e.g., see Figure 7) using Fresnel’s formula for a perfectly conducting half plane which has a coating layer of thickness 푑 with dielectric permittivity 휖 and magnetic permeability 휇.A normally incident ( 휙 = 0) electromagnetic wave with the frequency 푓 is assumed to impinge the half plane.Then the corresponding wavelength 휆 in air is 휆 = 푐/푓, where the speed of light is 푐 =0.3 × 109.The reflection coefficient 푅 for the wave is given by 푎 + 푏 푅 = , (43) 1+푎푏 where √ 휖 − 휖휇 √ 푎 = √ and 푏 = 푒4푖휋 휖휇푓푑/푐. (44) 휖 + 휖휇

68 This expression can be derived directly from Maxwell’s equation by con- sidering the ratio of reflected to incident wave for example in the case of parallel polarized (푇퐸𝑥) incident wave (see [BIKT, Jackson]). An alternative and much more computationally intensive approach (which may be necessitated by some target geometries) employs the far field pattern for reflected waves computed directly using Maxwell’s equations (see Example 4).In two dimensions, for a reflecting body Ω with coating layer (𝑖) Ω1 and computational domain Π with an interrogating plane wave 퐸 ,the scattered field 퐸(𝑠) satisfies the Helmholtz equation [CK1, CK2] ( ) ( ) 1 1 ∇⋅ ∇퐸(𝑠) + 휖휔2퐸(𝑠) = −∇ ⋅ ∇퐸(𝑖) − 휖휔2퐸(𝑖) in Π ∖ Ω¯ 휇 휇

퐸(𝑠) = −퐸(𝑖) on ∂Ω [ ] 1 ∂퐸 =[퐸]=0 on∂Ω ∖ ∂Ω 휇 ∂푛 1 ∂퐸(𝑠) 푖 ∂2퐸(𝑠) − 푖푘퐸(𝑠) − =0 on∂Π ∂푛 2푘 ∂푠2 ∂퐸(𝑠) 3 − 푖푘 퐸(𝑠) =0 at퐶, ∂푠 2 (45) where the Silver-Müller radiation condition has been approximated by a second-order absorbing boundary condition on ∂Π as described in [BJR, HKNT, HRT].The vectors 푛 and 푠 denote the normal and tangential di- rections on the boundary ∂Π, respectively, and 퐶 is the set of the corner points of Π.Here [ ⋅ ] denotes the jumps at interfaces.The incident field in air is given by 퐸(𝑖) = 푒𝑖𝑘(𝑥1 cos 𝛼+𝑥2 sin 𝛼), 𝜋 where 훼 is the interrogation angle (훼 = 2 − 휙)and푘 =2휋푓/푐 = 휔/푐 is the interrogation wave number corresponding to the interrogating frequency 푓. The corresponding far field pattern is given by [BIKT, CK1, CK2]

퐹𝛼(훼 + 휋; 휖, 휇, 훼, 푓)= (√ ) (46) lim 8휋푘푟 푒−𝑖(𝑘𝑟+𝜋/4) 퐸(𝑠)(푟 cos(훼 + 휋),푟sin(훼 + 휋); 휖, 휇, 훼, 푓) , 𝑟→∞

(𝑠) where 퐸 (푥1,푥2; 휖, 휇, 훼, 푓) is the scattered electromagnetic field.This can be used as a measure of the reflected field intensity instead of the reflection coefficient 푅 of (43)–(44).

69 The evader and the interrogator are each subject to uncertainties as to the actions of the other.The evader wants to choose a best coating design (i.e., best 휖’s and 휇’s ) while the interrogator wants to choose best angles of interrogation 훼 and interrogating frequencies 푓.Each player must act in the presence of incomplete information about the other’s action.Partial information regarding capabilities and tendencies of the adversary can be embodied in probability distributions for the choices to be made.That is, we may formalize this by assuming the evader may choose (with an as yet to be determined set of probabilities) dielectric permittivity and magnetic permeability parameters (휖, 휇) from admissible sets ℰ×ℳwhile the interrogator chooses angles of interrogation and interrogating frequencies (훼, 푓) from sets 𝒜×ℱ.The formulation here is based on the mixed strategies pro- posals of von Neumann [Aubin, VN, VNM] and the ideas can be summarized as follows.The evader does not choose a single coating, but rather has a set of possibilities available for choice.He only chooses the probabilities with which he will employ the materials on a target.This, in effect, disguises his intentions from his adversary.By choosing his coatings randomly (according to a best strategy to be determined in, for example, a minmax game), he prevents adversaries from discovering which coating he will use–indeed, even he does not know which coating will be chosen for a given target.The interrogator, in a similar approach, determines best probabilities for choices of frequency and angle in the interrogating signals.Note that such a formulation tacitly assumes that the adversarial relationship persists with multiple attempts at evasion and detection. The associated minmax problem consists of the evader choosing distributions 푃𝑒(휖, 휇)overℰ×ℳ to minimize the reflected field while the interrogator chooses distributions 푃𝑖(훼, 푓)over𝒜×ℱ to maximize the reflected field.If 퐵𝑅(휖, 휇, 훼, 푓) is the chosen measure of reflected field and 𝒫𝑒 = 𝒫(ℰ×ℳ) and 𝒫𝑖 = 𝒫(𝒜×ℱ) are the corresponding sets of probability distributions or measures over ℰ×ℳand 𝒜×ℱ, respectively, then the cost functional for the minmax problem can be defined by ∫ ∫ 2 퐽(푃𝑒,푃𝑖)= ∣퐵𝑅(휖, 휇, 훼, 푓)∣ 푑푃𝑒(휖, 휇)푑푃𝑖(훼, 푓). (47) ℰ×ℳ 𝒜×ℱ The problems thus formulated are special cases of classical static zero– sum two player non-cooperative games [Aubin, Basar] where the evader min- imizes over 푃𝑒 ∈𝒫𝑒 and the interrogator maximizes over 푃𝑖 ∈𝒫𝑖.In such games one defines upper and lower values for the game by

퐽 =inf sup 퐽(푃𝑒,푃𝑖) 𝑃 ∈𝒫 𝑒 𝑒 𝑃𝑖∈𝒫𝑖

70 and 퐽 =sup inf 퐽(푃𝑒,푃𝑖). 𝑃 ∈𝒫 𝑃𝑖∈𝒫𝑖 𝑒 𝑒 The first represents a security level (worst case scenario) for the evader while the latter is a security level for the interrogator.It is readily argued that 퐽 ≤ 퐽 and if the equality 퐽∗ = 퐽 = 퐽 holds, then 퐽∗ is called the value of ∗ ∗ the game.Moreover, if there exist 푃𝑒 ∈𝒫𝑒 and 푃𝑖 ∈𝒫𝑖 such that ∗ ∗ ∗ ∗ ∗ 퐽 = 퐽(푃𝑒 ,푃𝑖 )= min 퐽(푃𝑒,푃𝑖 )= max퐽(푃𝑒 ,푃𝑖), 𝑃𝑒∈𝒫𝑒 𝑃𝑖∈𝒫𝑖 ∗ ∗ then (푃𝑒 ,푃𝑖 )isasaddle point solution or non-cooperative equilibrium of the game. To investigate theoretical, computational and approximation issues for these problems, it is necessary to put a topology on the space of probability measures: a natural choice for 𝒫(푄) is the Prohorov metric topology Using its properties and arguments similar to those in [BBi, BG1, BG2, BK, BPi], one can develop well-posedness and approximation results for the minmax problems defined above.Efficient computational methods that correspond to von Neumann’s finite mixed strategies [Aubin] can readily be presented in this context.These can be based on several approximation theories that have been recently developed and used.The first, developed in [BBi] and based on Dirac delta measures, is summarized in the following theorem.

Theorem 17 Let 푄 be a complete, separable metric space, ℬ the class of all Borel subsets of 푄 and 𝒫(푄) the space of probability measures on (푄, ℬ).Let ∞ 푄0 = {푞𝑗}𝑗=1 be a countable, dense subset of 푄. Then the set of 푃 ∈𝒫(푄) such that 푃 has finite support in 푄0 and rational masses is dense in 𝒫(푄) in the 휌 metric. That is, ∑𝑘 ∑𝑘 + 𝒫0(푄) ≡{푃 ∈𝒫(푄):푃 = 푝𝑗Δ𝑞𝑗 ,푘 ∈𝒩 ,푞𝑗 ∈ 푄0,푝𝑗 rational, 푝𝑗 =1} 𝑗=1 𝑗=1 is dense in 𝒫(푄) taken with the 휌 metric, where Δ𝑞𝑗 is the Dirac measure with atom at 푞𝑗. It is rather easy to use the ideas and results associated with this theorem ∪∞ to develop computationally efficient schemes.Given 푄𝑑 = 𝑀=1 푄𝑀 with 𝑀 𝑀 푄𝑀 = {푞𝑗 }𝑗=1 (a “partition” of 푄) chosen so that 푄𝑑 is dense in 푄, define ∑𝑀 ∑𝑀 𝑀 𝑀 𝒫 (푄)={푃 ∈𝒫(푄):푃 = 푝𝑗Δ 𝑀 ,푞𝑗 ∈ 푄𝑀 ,푝𝑗 rational, 푝𝑗 =1}. 𝑞𝑗 𝑗=1 𝑗=1

71 Then we ﬁnd

(i) 𝒫𝑀 (푄)isacompactsubsetof𝒫(푄)inthe휌 metric,

𝑀 𝑀+1 (ii)𝒫 (푄) ⊂𝒫 (푄) whenever 푄𝑀+1 is a reﬁnement of 푄𝑀 , (iii)“𝒫𝑀 (푄) →𝒫(푄)” in the 휌 topology; that is, for 푀 suﬃciently large, elements in 𝒫(푄) can be approximated in the 휌 metric by elements of 𝒫𝑀 .

A second class of approximations, based on linear splines, was developed and used in [BPi] for problems where one assumes that the probability distributions to be approximated possess densities in 퐿2.

Theorem 18 Let ℱ be a weakly compact subset of 퐿2(푄), 푄 compact and ′ let 𝒫ℱ (푄) ≡{푃 ∈𝒫(푄):푃 = 푝, 푝 ∈ℱ}. Then 𝒫ℱ (푄) is compact in 𝒫(푄) in the 휌 metric. Moreover, if we deﬁne {ℓ𝑀 } to be the linear splines on 푄 ∪𝑗 corresponding to the partition 푄𝑀 ,where 𝑀 푄𝑀 is dense 푄, deﬁne ∑ 𝑀 𝑀 𝑀 𝑀 𝑀 𝑀 𝒫 ≡{푝 : 푝 = 푏𝑗 ℓ𝑗 ,푏𝑗 rational } 𝑗 and if ∫ 𝑀 𝑀 𝑀 𝒫ℱ 𝑀 ≡{푃𝑀 ∈𝒫(푄):푃𝑀 = 푝 ,푝 ∈𝒫 }, ∪ we have 𝑀 𝒫ℱ 𝑀 is dense in 𝒫ℱ (푄) taken with the 휌 metric. A study comparing the relative strengths and weaknesses of these two classes of approximation schemes in the context of inverse problems is given in [BD].

7.4.2 Theoretical Results To establish existence of a saddle point solution for the evasion-interrogation problems formulated above, one can employ a fundamental result of von Neumann [VN, VNM] as stated by Aubin (see [Aubin, p.126]).

Theorem 19 (von Neumann) Suppose 푋0,푌0 are compact, convex subsets of metric linear spaces 푋, 푌 respectively. Further suppose that

(i) for all 푦 ∈ 푌0,푥→ 푓(푥, 푦) is convex and lower semi-continuous;

(ii) for all 푥 ∈ 푋0,푦 → 푓(푥, 푦) is concave and upper semi-continuous.

72 Then there exists a saddle point (푥∗,푦∗) such that

푓(푥∗,푦∗)=minmax 푓(푥, 푦)=maxmin 푓(푥, 푦). 𝑋0 𝑌0 𝑌0 𝑋0 A straight forward application of these results to the problems under discussion lead immediately to desired well-posedness results.

Theorem 20 Suppose ℰ, ℳ, Φ, ℱ are compact and the spaces 푋0 = 𝒫(ℰ× ℳ), 푌0 = 𝒫(Φ ×ℱ) are taken with the Prohorov metric. Then 푋0,푌0 ∗ ∗ are compact, convex subsets of 푋 = 퐶𝐵(ℰ×ℳ) and 푌 = 퐶𝐵(Φ ×ℱ), ∗ ∗ respectively. Moreover, there exists (푃𝑒 ,푃𝑖 ) ∈𝒫(ℰ×ℳ) ×𝒫(Φ ×ℱ) such that

∗ ∗ 퐽(푃𝑒 ,푃𝑖 )= min max 퐽(푃𝑒,푃𝑖)= max min 퐽(푃𝑒,푃𝑖). 𝒫(ℰ×ℳ) 𝒫(Φ×ℱ) 𝒫(Φ×ℱ) 𝒫(ℰ×ℳ)

For computations, we may use the “delta” approximations of Theorem 17 and obtain ∑ 𝑗 푑푃𝑒(휖, 휇) ≈ 푝𝑒훿(𝜖 ,𝜇 )(휖, 휇)푑휖푑휇, ∑ 𝑗 𝑗 푑푃 (휙, 푓) ≈ 푝𝑗훿 (휙, 푓)푑휙푑푓. 𝑖 𝑖 (𝜙𝑗 ,𝑓𝑗 )

As noted above, a convergence theory can be found in [BBi].This corresponds precisely to von Neumann’s ﬁnite mixed strategies framework for protection by disguising intentions from opponents, i.e., by introducing uncertainty in the players’ choices. To illustrate the computational framework based on the delta measure approximations, we take

∑𝑀 ∑𝑁 𝑀 𝑀 𝑁 𝑁 푑푃𝑒 (휖, 휇)= 푝𝑗 훿 𝑀 𝑀 푑휖푑휇 and 푑푃𝑖 (휙, 푓)= 푞𝑘 훿 𝑁 𝑁 푑휙푑푓, (𝜖𝑗 ,𝜇𝑗 ) (𝜙𝑘 ,𝑓𝑘 ) 𝑗=1 𝑘=1 which can be represented respectively by

{ } ∑𝑀 푝¯𝑀 = 푝𝑀 𝑀 ∈ 푃 𝑀 ≡{푝¯ ∈ ℝ𝑀 ∣ 푝 ≥ 0, 푝 =1} (48) 𝑗 𝑗=1 𝑗 𝑗 𝑗=1 and { } ∑𝑁 푞¯𝑁 = 푞𝑁 𝑁 ∈ 푄𝑁 ≡{푞¯ ∈ ℝ𝑁 ∣ 푞 ≥ 0, 푞 =1}. (49) 𝑘 𝑘=1 𝑘 𝑘 𝑘=1

73 𝑀 𝑁 We note that in this case the 푝𝑗 ,푞𝑘 are the probabilities associated with 𝑀 𝑀 the use of the material parameters (휖𝑗 ,휇𝑗 ) and interrogating parameters 𝑁 𝑁 (휙𝑘 ,푓𝑘 ) in the mixed strategies of the evader and interrogator, respectively. 𝑀 𝑁 Then 퐽(푃𝑒 ,푃𝑖 ) reduces to

∑𝑁 ∑𝑀 𝑀 𝑁 𝑀 𝑀 𝑀 𝑁 𝑁 2 𝑁 𝒥 (¯푝 , 푞¯ )= 푝𝑗 퐵𝑅(휖𝑗 ,휇𝑗 ,휙𝑘 ,푓𝑘 ) 푞𝑘 𝑘=1 𝑗=1 where 퐵𝑅 is a measure of the reflected field (either the reflection coefficient or the far field scattering intensity).Since 푃 𝑀 ,푄𝑁 are compact, convex subsets of ℝ𝑀 , ℝ𝑁 , respectively, we have the following theorem.

𝑀 𝑁 𝑀 𝑁 Theorem 21 For ﬁxed 푀,푁 there exists (¯푝∗ , 푞¯∗ ) in 푃 × 푄 such that

∗ 𝑀 𝑁 𝑀 𝑁 𝑀 𝑁 𝒥 = 𝒥 (¯푝∗ , 푞¯∗ )= min max 𝒥 (¯푝 , 푞¯ )= max min 𝒥 (¯푝 , 푞¯ ). 𝑝¯𝑀 ∈𝑃 𝑀 𝑞¯𝑁 ∈𝑄𝑁 𝑞¯𝑁 ∈𝑄𝑁 𝑝¯𝑀 ∈𝑃 𝑀

Assume further that (휖, 휇, 휙, 푓) → 퐵𝑅(휖, 휇, 휙, 푓) is continuous on ℰ×ℳ× 𝑀 𝑁 Φ ×ℱ which is assumed compact. Then there exists a sequence (¯푝∗ , 푞¯∗ ) 𝑀 𝑁 𝑀 𝑁 of elements from 푃 × 푄 respectively with corresponding (푃𝑒 ,푃𝑖 ) in ∗ ∗ 𝒫(ℰ×ℳ) ×𝒫(Φ ×ℱ) converging in the Prohorov metric to (푃𝑒 ,푃𝑖 ) which is a saddle point for the original minmax problem. We give the arguments for these results since they are in some sense rep- resentative of the use of functional analysis in such problems.The hypothesis that the map (휖, 휇, 휙, 푓) → 퐵𝑅(휖, 휇, 휙, 푓) be continuous on ℰ×ℳ×Φ ×ℱ 1 implies continuity of the map (푃𝑒,푃𝑖) → 퐽(푃𝑒,푃𝑖)on푋0 × 푌0 → ℝ+,where ∗ 푋0 = 𝒫(ℰ×ℳ)and푌0 = 𝒫(Φ×ℱ) are compact convex in 푋 = 퐶𝐵(ℰ×ℳ) ∗ and 푌 = 퐶𝐵(Φ ×ℱ), respectively.This is due to the compactness of ℰ×ℳ and Φ ×ℱ, and properties of the Prohorov metric. Let 푃 𝑀 ,푄𝑁 be deﬁned as in (48),(49), and observe that these are com- 𝑀 𝑁 pact convex subsets of ℝ , ℝ , respectively.Now let 푃𝑒 ∈ 푋0 and 푃𝑖 ∈ 푌0 be arbitrary.Then by Theorem 17 and (iii) above (see also [BBi]), there 𝑀 𝑁 𝑀 𝑁 𝑀 𝑁 exists a sequence푝 ¯ , 푞¯ ∈ 푃 ,푄 with associated measures 푃𝑒 ,푃𝑖 ,respectively, such that

𝑀 𝑁 푃𝑒 → 푃𝑒 in 푋0 as 푀 →∞ and 푃𝑖 → 푃𝑖 in 푌0 as 푁 →∞.

𝑀∗ 𝑁∗ Let 푃𝑒 , 푃𝑖 (guaranteed to exist by continuity, compactness, and the 𝑀 𝑁 von Neumann theorem) with coordinates푝 ¯∗ , 푞¯∗ , respectively, satisfy

𝑀∗ 𝑁 𝑀∗ 𝑁∗ 𝑀 𝑁∗ 퐽(푃𝑒 ,푃𝑖 ) ≤ 퐽(푃𝑒 ,푃𝑖 ) ≤ 퐽(푃𝑒 ,푃𝑖 )

74 𝑀 𝑁 𝑀 𝑁 for all (푃𝑒 ,푃𝑖 )in푋0 × 푌0 with coordinate representations in 푃 ,푄 , respectively.Now 푋0 × 푌0 compact implies that there exists a subsequence 𝑀𝑘∗ 𝑁𝑘∗ ˜ ˜ 𝑀𝑘∗ 𝑁𝑘∗ ˜ ˜ (푃𝑒 ,푃𝑖 )and(푃𝑒, 푃𝑖)in푋0 × 푌0 such that (푃𝑒 ,푃𝑖 ) → (푃𝑒, 푃𝑖)in 푋0 × 푌0. Consider (50) for indices 푀𝑘,푁𝑘,sothat

𝑀𝑘∗ 𝑁𝑘 𝑀𝑘∗ 𝑁𝑘∗ 𝑀𝑘 𝑁𝑘∗ 퐽(푃𝑒 ,푃𝑖 ) ≤ 퐽(푃𝑒 ,푃𝑖 ) ≤ 퐽(푃𝑒 ,푃𝑖 ). Then given the continuity of 퐽 and taking the limit we ﬁnd ˜ ˜ ˜ ˜ 퐽(푃𝑒,푃𝑖) ≤ 퐽(푃𝑒, 푃𝑖) ≤ 퐽(푃𝑒, 푃𝑖). ˜ ˜ Since 푃𝑒,푃𝑖 are arbitrary in 푋0,푌0 respectively, we have (푃𝑒, 푃𝑖) is a saddle point for the original minmax problem. Further note that since these arguments hold for any subsequence of 𝑀∗ 𝑁∗ (푃𝑒 ,푃𝑖 ),then

𝑀∗ 𝑁 𝑀∗ 𝑁∗ 𝑀 𝑁∗ 퐽(푃𝑒 ,푃𝑖 ) ≤ 퐽(푃𝑒 ,푃𝑖 ) ≤ 퐽(푃𝑒 ,푃𝑖 )

𝑀∗ ˜ 𝑁∗ ˜ 𝑀∗ 𝑁∗ and 푃𝑒 → 푃𝑒 as well as 푃𝑖 → 푃𝑖.That is, any subsequence of 푃𝑒 ,푃𝑖 ˜ ˜ has a convergent subsequence to 푃𝑒, 푃𝑖 so that the sequence itself must ˜ ˜ converge to 푃𝑒, 푃𝑖.

75 8 Weak* Convergence, the Prohorov Metric, Re- laxed Controls, Preisach Hysteresis, and Mixing Distributions in Statistics

UNDER CONSTRUCTION!!!

8.1 Mixing Distributions 8.1.1 Type I: Individual Dynamics/Aggregate Data Recall, from Example 5 (the size-structured population model for mosquitofish where there is a family 𝒢 of individual growth rates 푔) we have the expected size density 푢(푡, 휉; 푃 )=ℰ[푣(푡, 휉; ⋅)∣푃 ] described for a general probability distribution 푃 ,wehave ∫ 푢(푡, 휉; 푃 )= 푣(푡, 휉; 푔)푑푃 (푔), 𝒢 as the expected value of the density 푣(푡, 휉; 푔) which satisfies for a given 푔 the Sinko-Streiffer. Since in typical inverse problems, we must use aggregate population data 푑𝑖𝑗 to estimate 푃 itself, we are therefore required to understand the qualitative properties (continuity, sensitivity, etc.) of 푢(푡, 휉; 푃 ) as a function of 푃 . Thus, we see that the data for parametric estimation or inverse problems is 푑𝑖𝑗 ∼ 푢(푡, 휉; 푃 ).

8.1.2 Type II: Aggregate Dynamics We return to the electromagnetics example of Example 4 which entails polarization of inhomogeneous dielectric materials.In this case, individual (particle) dynamics are not available.Instead, the dynamics (13) them- selves depend on a probability measure 퐹 ,e.g.,

∂푢 ∂2 + 푢 = 푓(푢, 퐹 ). ∂푡 ∂푥2 To describe the behavior of the electric polarization 푃 ,webeginwith the general formulation of Chapter 2 of [BBL] by employing a polarization kernel 푔 in the convolution expression

∫ 𝑡 푃 (푡, 푧)= 푔(푡 − 푠,푧; 휏)퐸(푠,푧)푑푠. (50) 0

76 As explained in Section 2.5 and [BBL], this general formulation includes as special cases the well known orientational or Debye polarization model, the electronic or Lorentz polarization model, and linear combinations thereof, as well as other higher order models.In the Debye case the kernel is given by −𝑡/𝜏 푔(푡; 휏)=휖0(휖𝑠 − 휖∞)/휏 푒 , while in the Lorentz model (again see [BBL]), it takes the form

77 probability distributions 푃1 and 푃2.The ideas of local minima, continuity, “gradient”, etc., depend on a metric (or topology) for 푃1 and 푃2. More fundamentally for Type II systems, the “well-posedness of systems”, including existence and continuous dependence of solutions on “parameters”, depends on the concept of 퐹1 and 퐹2 being close, i.e., 퐹 → 퐸(푡, 푥¯; 퐹 ) is continuous.

8.2 Relaxed Controls Relaxed Controls (sliding regimes, chattering controls) L.C.Young (1937,1938)–generalized curves in calculus of variations, E.J McShane (1940,1967)–relaxed controls in variational problems, control, A.F.Filippov (1962)–sliding regimes in optimal control J.Warga, (1962,1967,..)–relaxed curves and controls- approximation to original controls Idea: Problems lack closure in ordinary function spaces (Re: Sobolev spaces and weak solutions, distributions of Schwartz for PDE) Chattering controls:

Figure 3: Relaxed or generalized curves

In the limit, derivatives approach “ limits of combinations of delta functions” Convexiﬁes and generalizes problem—obtain closure, and hence existence! Corresponding trajectories are called relaxed trajectories or generalized trajectories (in which family of ordinary trajectories are embedded) Ex- istence (inf = min) of optimal controls in a wider class of functions McShane: Under not too restrictive assumptions, inf actually attained by generalized control that happens to be ordinary! Warga: Knowledge of minimizing generalized curve permits approximation by nearby ordinary curve!

8.3 Preisach Hysteresis Smart materials 1) SMA, piezoelectric, magnetostrictive 2) Viscoelastic materials Rubber, polymers, living tissue 3) Electromagnetics Polarization,

78 conductivity Example: Stress-strain in shape memory alloys, rubber, tissue

Figure 4: Simple ideal relay

Preisach Ideal Relay Preisach-1935-(E&M Hysteresis) Krasnoselskii&Pokrovskii-1983 Mayergoyz- 1991 Visintin-1994 Brokate&Sprekels-1996

Figure 5: Super imposed relays

These ideas are the basis of SMA control efforts in: H.T.Banks and A.J.Kurdila, Hysteretic control influence operators representing smart material actuators: Identification and approximation, Proc. Conf. Decision and Control, Kobe, Japan, Dec, 1996, p.3711-3716. H.T.Banks, A.J.Kurdila and G.Webb, Math. Prob. in Engr.,3(1997),p.287-328. H.T.Banks, A.J.Kurdila and G.Webb, J. Intell. Mat. Sys.&Structures,8(1997),p.536- 550. Efforts on inverse problems involving estimation of measures in control input operator: well- posedness and approximation, computational methods– Involves generalization of Krasnoselskii&Pokrovskii operators and kernels–

79 Figure 6: Smoothed relay

Figure 7: Preisach domain

8.4 Mixing distributions This class of problems involves a mathematical model (the system dynamics for the parameter dependent state x(t;q)), a statistical model (in this case,assumptions about the data– i.i.d. normal with constant variance leading to the ordinary least squares criterion MLE) and mixing distributions or random eﬀects to account for variability in individuals L.B.Sheiner,et.al. -Comp. Biomed. Res.5(1972),p.441-459. B.G.Lindsay- Ann. Statist.11(1983),p.86-94. A.Mallet-Biometrika73(1986),p.657-669. M.Davidian and A.R.Gallant-J.Pharm.&Biopharm.20(1992), p.529-556. Texts summarizing some of the statistical literature on this include B.G.Lindsay, Mixture Models: Theory, Geometry, and Applications, NSF-CBMS Regional Conf.Series in Prob.and Statistics, Inst.Math. Stat, Haywood, CA, Vol.5, 1995. M.Davidian and D.Giltinan, Nonlinear Models for Repeated Measure- ment Data, Chapman&Hall, London, 1998.

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83 [BGIT] H.T. Banks, S.L. Grove, K. Ito and J.A. Toivanen, Static two-player evasion-interrogation games with uncertainty, CRSC-TR06-16, June, 2006; Comp. and Applied Math, 25 (2006), 289–306.

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[BI] H.T.Banks and K.Ito, A uniﬁed framework for approximation and inverse problems for distributed parameter systems.Control: Theory and Adv.Tech. 4 (1988), 73-90.

[BI2] H.T.Banks and K.Ito, Approximation in LQR problems for inﬁnite dimensional systems with unbounded input operators, CRSC-TR94-22, N.C. State University, November, 1994; in J. Math. Systems, Estima- tion and Control, 7 (1997), 119–122.

[BIKT] H.T.Banks, K.Ito, G.M.Kepler and J.A.Toivanen, Material surface design to counter electromagnetic interrogation of targets, Tech. Report CRSC-TR04-37, Center for Research in Scientiﬁc Computation, N.C. State University, Raleigh, NC, November, 2004; SIAM J. Appl. Math., 66 (2006), 1027–1049.

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85 icokinetic model, CRSC-TR02-27, N.C. State University, September, 2002; Math. Biosci., 192 (2004), 193–225.

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