Stability of Input/Output Dynamical Systems on Metric Spaces: Theory and Applications

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Hugo Gonzalez Villasanti, B.S., M.S.

Graduate Program in Electrical & Computer Engineering

The Ohio State University

2019

Dissertation Committee:

Kevin M. Passino, Advisor Andrea Serrani Atilla Eryilmaz c Copyright by

Hugo Gonzalez Villasanti

2019 Abstract

The first part of this dissertation extends the applicability of stability-preserving mappings to dynamical systems whose evolutionary processes explicitly consider the effect of external perturbations (inputs) and measurements (outputs), via multi- valued operators. We provide definitions for input-to-state stability and input-to- output stability for a general class of systems whose trajectories lie in arbitrary metric spaces, indexed by “hybrid” time sets. Novel proofs of results such as the

ISS-Lyapunov and the small-gain theorem are developed with the use of stability- preserving mappings.

The second part, where we employ the theory to model and analyze the complex dynamics found in the interplay of the determinants of mood disorders. The model integrates biopsychosocial findings of the bipolar and depressive spectra, modeling attractors corresponding to mood states such as euthymia, mania, depression, the mixed state, anhedonia, hedonia, and flat or blunted affect, as well as the transitions among these attractors caused by external influences, like stress and medication.

Conditions for global stability of euthymia, obtained via a stability analysis, are supported by studies in the neuropsychology literature, while computational analyses provide a novel explanation of the mechanism underlying mood stabilizers.

ii To Oliva and Rafael, with them, for them.

iii Acknowledgments

First, I give thanks to my adviser, Professor Kevin M. Passino, whose commitment as a mentor allowed me to tackle the main challenge he presented to me at the beginning of my Ph.D.: transcend disciplinary boundaries to seek meaningful answers that will improve the lives of the most needed. His scholarship guided me in that quest, and his sincerity, integrity, and humanity made him an academic and personal role model for me.

I thank my parents, Alba and Hugo, whose example and guidance have been paramount in my path. I am grateful to my siblings Juan, Federico, and Maria

Bethania for their moral support.

I owe thanks to Professor John D. Clapp, whose support and insight fostered my willingness to study complex social problems through the lens of dynamical systems.

I owe thanks to the members of my dissertation committee, Professors Andrea

Serrani and Atilla Eryilmaz for reviewing my work and providing inputs. Throughout the lectures and thought-provoking conversations with both, I have gained a new standard for doctoral pedagogy.

Finally, I give thanks to Oliva, the love of my life, for standing with me side by side at every step of the way. I am grateful to my son Rafael, for his kindness and selﬁshness in helping dad ﬁnish his Ph.D.

iv Vita

January 16, 1986 ...... Born - Asuncion, Paraguay

August 2011 ...... B.S., Elec. & and Mech. Engineering

2011,2013 ...... Senior Field Engineer, CIE S.A., Luque, Paraguay 2013–2015 ...... Fulbright Scholar, U.S. Department of State May 2015 ...... M.S., Elec. & Comp. Engineering

2018–present ...... Presidential Fellowship, The Ohio State University

Publications

Research Publications

H. Gonzalez Villasanti, K.M. Passino, J.D. Clapp, D.R. Madden A Control-Theoretic Assessment of Interventions During Drinking Events. IEEE Transactions on Cyber- netics, 49(2):604-615, Feb 2019.

H. Gonzalez Villasanti, L.F. Giraldo, K.M. Passino Feedback Control Engineering for Cooperative Community Development. IEEE Control Systems, 38(3):87-101, June 2018.

J.D. Clapp, D.R. Madden, H. Gonzalez Villasanti, L.F. Giraldo, K.M. Passino, M.B. Reed, I. Fernandez Puentes A System Dynamics Model of Drinking Events: Multi-Level Ecological Approach. Systems Research and Behavioral Science, 35(3):265-281, May 2018.

v H. Gonzalez Villasanti, K.M. Passino Feedback Controllers as Financial Advisors for Low-Income Individuals. IEEE Transactions on Control Systems Technology, 25(6):2194-2201, Nov 2017.

Fields of Study

Major Field: Electrical and Computer Engineering

vi Table of Contents

Page

Abstract...... ii

Dedication...... iii

Acknowledgments...... iv

Vita...... v

List of Figures...... ix

1. Introduction...... 1

2. Stability Preserving Mappings...... 4

2.1 Introduction...... 4 2.2 Notation...... 7 2.3 Formulation...... 10 2.4 Stability of Invariant Sets...... 14 2.5 Stability of Input/Output Systems...... 17 2.6 Stability Results...... 20 2.7 Stability of Interconnected Systems...... 26 2.8 Conclusion...... 28

3. Dynamics of Mood Disorders and Stability Analysis...... 29

3.1 Introduction...... 29 3.2 Mathematical Modeling of Mood Dynamics...... 35 3.2.1 Mood States, Trajectories, and Regions...... 35 3.2.2 Mood Dynamics and Equilibria...... 40 3.2.3 Basins of Attraction for Mood...... 43

vii 3.2.4 Adjusting Mood Dynamics and Including Inputs...... 46 3.2.5 Mood Trajectories for Various Mood Disorders...... 48 3.3 Nonlinear and Computational Analysis of Mood Dynamics..... 49 3.3.1 Stabilization to Euthymia...... 50 3.3.2 Pharmacotherapy: Triggering Mania...... 54 3.4 Conclusion...... 54

4. Conclusions and Future Directions...... 56

4.1 Conclusions...... 56 4.2 Future Directions...... 58

Bibliography...... 61

viii List of Figures

Figure Page

3.1 (a) One- and two-dimensional representations of mood; (b) The mood state is in a euthymic region (black circle centered at the green dot); (c) Mood trajectory example with mood starting in a manic/mixed state and decreasing to euthymia with mood lability; and (d) Regions on the mood plane of maximal mood variations for each of the bipolar spectrum disorders [2, 29]...... 36

3.2 (a) Unforced manic mood dynamics functions, and (b) Basins of at-

traction for equilibria via integration of Sm(0,M, 0) and interpretation of each attractor...... 42

3.3 Vector ﬁeld diagram (grid of red arrows) and trajectories (in blue) for (a) BD-II, (b) Cyclothymia, and (c) MDD...... 49

3.4 (a) Vector ﬁeld diagram and trajectories (in blue) for BD-I, showing 4 basins of attraction at normal, depression, mania and mixed states. (b) Mood episode shift from depression to mania after treatment with antidepressant...... 55

ix Chapter 1: INTRODUCTION

Solving some of the most pressing challenges that humanity faces nowadays re- quires interdisciplinary work [52, 79], which in turn demands a synthesis of the different languages, methodologies, and metrics employed by each field. A quest that is ubiquitous in the formal, basic, and applied sciences is that of providing qualitative equivalences between domains by employing structure or property-preserving relations. Very broadly, one can argue that any simplified description of a phenomenon, a model, is a property-preserving relation between a real or abstract domain and a codomain, where properties about unknown elements of the domain can be revealed, or are easier to be obtained. Despite being a common thread in research, scientific disciplines have adopted different methodologies to construct these property-preserving relations, among which the most common are the statistical methods used in pre- dictive modeling, computational methods employed in simulations, and analytical methods using mathematical models.

For the ﬁeld of dynamical systems and control theory, the stability of a set of trajectories of the system is a central property; and among the methods developed to verify whether a system is stable, the approaches introduced by Lyapunov [56] and

Zames [89] remain as the foundation of modern stability methods [75]. The notion of qualitative equivalence is present in the dynamical systems literature, usually through

1 coordinate transformations [50]. However, the stability-preserving mappings in [61] oﬀer a broader range of tools to establish qualitative equivalence between systems and includes most of Lyapunov results as a special case.

Motivated by the need to provide stability guarantees in control systems that integrate human and technology elements, this dissertation extends the applicability of stability preserving mappings to dynamical systems whose evolutionary processes depend explicitly on external input/output trajectories, thus bringing it closer to approaches based on Zames’ work [89]. The generality obtained by the model allows us to explore other areas of inquiry beyond those typically encountered in the applications of control and stability theory. This lead to the second main contribution of this dissertation: a nonlinear dynamical system model to integrate scientiﬁc ﬁndings on mood disorders, predict features of mood via nonlinear and computational analyses, and connect these to psychotherapeutic practice.

Chapter2 presents properties of a class of dynamical systems whose trajectory elements are subsets of metric spaces, and the input/output maps are binary relations.

We deﬁne the concept of stability of invariant sets and stability of input/output systems. The main result generalizes the stability preserving mappings deﬁned in

[61] to obtain conditions under which the input/output qualitative properties are preserved. We apply the results to the case of stability of interconnected systems.

In Chapter3, we explore an application of stability analysis to complex biopsychosocial conditions called mood disorders. An extensive overview of mood disorders literature in psychology and neuroscience, as well as dynamical systems models of mood disorders is provided. We propose a nonlinear model that captures the essential interplay between drivers of depressive and manic mood. The result of simulations

2 shows that the model captures the complexity of relevant symptoms of the disorders.

A stability analysis, using Lyapunov theory, allowed us to achieve a conclusion known in neuroscience literature to be fundamental to mood disorders.

We provide concluding remarks in Chapter4, where we also discuss further applications for the theory developed in Chapter2, as well as extensions of the model in

Chapter3.

3 Chapter 2: STABILITY PRESERVING MAPPINGS

2.1 Introduction

Any nontrivial dynamical systems appearing in science and engineering cannot be characterized with a set of equations whose solutions admit a closed form, or if such an expression exists, the behavior is too complex to make meaningful predictions.

The social and biological interactions between agents, the behavior of subatomic electrically charged particles, and the weather pattern across the globe are examples of complex systems that lack an exact and accurate characterization. Nevertheless, significant advances in the form of new theories and technologies have taken place by understanding the ”approximate” or qualitative properties of complex dynamical systems. Nowadays, textbooks on classical and evolutionary game theory, quantum electrodynamics, and numerical weather predictions feature those advances for the cited systems above. In the wake of greater integration between technology and society in the form of cyber-physical systems, identifying qualitative properties, such as stability and robustness, becomes an interdisciplinary task, usually among fields with different methodologies to pursue guarantees of those properties.

Stability is concerned with the behavior of a dynamical system under perturbations. The literature features diﬀerent deﬁnitions depending on the properties of the

4 perturbations and their eﬀect on the system. In Lyapunov stability, perturbations for initial conditions of trajectories close to invariant sets are considered, while Lagrange stability is concerned with the boundedness and ultimate boundedness of trajectories of a dynamical system, with respect to an arbitrary point in the underlying space.

Input-output stability characterizes the behavior of dynamical systems relative to perturbations in the domain of (multivalued) operators [89], and a unifying stability concept was introduced in [74], termed input-to-state stability (ISS). Other types of stability deﬁnitions include orbital stability [35], stability in terms of two measures

[83], and stability with respect to perturbations on time scales [58].

Methods based on Lyapunov’s work on the qualitative properties of solutions of ordinary differential equations [56] have been applied to establish qualitative properties of systems whose evolutionary process is described by ordinary differential and difference equations, ordinary differential and difference inequalities, functional differential equations, integrodifferential equations, and certain classes of partial differential equations [61]. These methods employ a suitable real-valued function, and conditions imposed on its derivative along trajectories of the system reveal qualitative properties such as stability, attractivity, and boundedness. For interconnected systems, test ma- trix methods exploit the structure of the interconnections to simplify the application of Lyapunov methods [63].

Conic sectors, introduced in [89] to establish the stability of input-output systems, feature well known small-gain theorem (SGT) for the stability of interconnected systems. Further generalization for input-output stability was provided by Safonov, in the form of the separation theorem, which features a distance functional to ensure topological separation of the graphs of multi-valued operators (binary relations) [73].

5 For an exposition on the application of the separation theorem to represent stability results, such as the small-gain theorem, Nyquist stability criterion, and stable coprime fractions, see [81].

Qualitative methods for nonlinear systems with inputs include passivity theorems

[84], and ISS-Lyapunov theorems [54]. See [59] for additional connections between passivity and conic sectors. The small-gain theorem was extended to nonlinear state space systems in [41], and [42] introduced an ISS-Lyapunov formulation of the SGT.

An extension of ISS-Lyapunov theorems to systems modeled by linear C0-semigroups is found in [13]. The advent of cyber-physical systems prompted a revision of stability methods in order to apply them to interconnections of topologically distinct systems, including Lyapunov methods for discrete event systems [66], and hybrid systems [9, 87] as well as non-Lyapunov methods [70].

Stability preserving mappings (SPM) have been employed to obtain qualitative equivalences between dynamical systems. Formally introduced in [60] for the qualitative study of discrete event systems, its application on a large class of dynamical systems whose trajectories lie on metric spaces offers a unified and efficient tool to study qualitative properties. Well known results, such as scalar and vector Lyapunov theorems, as well as some results on invariance theory and converse Lyapunov theorems can be formulated using SPM, and practical applications range from ordinary differential and difference equations, stochastic differential equations, to differential inclusions [38, 61].

This chapter extends the applicability of stability preserving mappings to include the qualitative analysis of systems with external behavior. To achieve this, we introduce multi-valued operators that link elements of the internal (state) and external

6 (input, output) behavior, maintaining the model in [61] as a particular case. We

extend the connections made for a previous formulation between Lyapunov theory

and input-output stability theory via the use of stability preserving mappings. In

Section 2.2 we introduce the notation employed in the rest of the chapter, partic-

ularly concerning binary relations. Section 2.3 details the formulation of a general

class of dynamical systems, as well as their input/output behavior, and in Section 2.4

we provide stability deﬁnitions for the mentioned systems. The main results of the

chapter are in Section 2.6 and applications to stability interconnected systems are

presented in 2.7. The chapter ends with conclusions in Section 2.8.

2.2 Notation

Let R be the set of real numbers and P(R) its power set. We endow R with the

usual total order relation “<” and norm |x|. Let R+ = [0, +∞) denote the set of

nonnegative real numbers, and let Z+ be the set of nonnegative integers. We denote a binary relation as the subset R ⊂ A × B, with set departure A and

codomain B. The domain of R is given by the set dom(R) = {a ∈ A :(a, b) ∈ R, b ∈ B},

its range is the set ran(R) = {b ∈ A :(a, b) ∈ R, a ∈ A}, and its inverse relation is

given by R−1 ⊂ B×A, where R−1 = {(b, a) ∈ B × A :(a, b) ∈ R}. The image of a subset H ⊂ A under the relation R is denoted with R (H) = {b ∈ B :(a, b) ∈ R, a ∈ H}.

If (a, b) ∈ R, we will sometimes write aRb. When we refer to R as a subset of the

cartesian product A × B, we will say we are referring to the graph of R. Similarly,

Qn i j an n-ary relation is a subset P ⊂ i=1 P , for n ≥ 2. The image of the set A where

7 Aj ⊂ P j under the n-ary relation P is deﬁned as

n j 1 j−1 j+1 n Y i 1 n j P A = (p , . . . , p , p , . . . p ) ∈ P :(p , . . . , p ) ∈ P, pj ∈ A (2.1) i=1,i6=j

Similarly, the image of two sets Aj ⊂ P j and Ak ⊂ P k under the n-ary relation P is deﬁned as P Aj,Ak = (P [Aj]) Ak.

A function or mapping is a left total and right unique binary relation, in which

case we will use the standard notation p : A → B. The family of all functions with

domain A and codomain B, will be denoted by BA. Given the functions pi ∈ (Bi)Ai , we consider their ordered n-tuple as

n n Y Y (p1, . . . pn) = Ai → Bi i=1 i=1 such that, for all j = 1, . . . , n we have that πj,a ◦ (p1, . . . , pn) = pj ◦ πj,b, where

j,a Qn i j j,b Qn i j π : i=1 A → A and π : i=1 B → B are standard projection functions. The

Qn i Ai j cartesian product i=1(B ) is the set of all n-tuples of functions. If the sets A are subsets of the real line, it is possible to propose alternative deﬁnitions of the product

mapping. A choice that takes into account the fact that there exists an order and

Qn i metric preserving bijection between i=1 A and a subset of the real line, and was considered in [87] is deﬁned as

n n [ Y (p1, . . . pn) = Ai → Bi (2.2) i=1 i=1

such that, for all j = 1, . . . , n we have that πj,a ◦ (p1, . . . , pn) =p ¯j, where ( pj(t), if t ∈ Aj p¯j(t) = (2.3) πj,b(x), if t∈ / Aj

Qn i for all j = 1, . . . , n, where x ∈ i=1 B is a ﬁxed point in the codomain of the n-tuple.

8 We label a metric space with the pair (X, d), where X is the underlying set of elements in the space and the function d : X × X → R is the associated metric. The distance from x ∈ X to a set Z ⊂ X is given by

d(x, Z) = inf d(x, z) (2.4) z∈M

Given two metric spaces (Xu, du) and (Xy, dy), unless otherwise stated, we will endow the cartesian product X = Xu × Xy with the metric d : X × X → R such that for any x, x0 ∈ X we have

d(x, x0) = max {du(πu(x), πu(x0)), dy(πy(x), πy(x0))} (2.5)

Let (X, d) be a metric space, T ⊂ R, and consider a mapping p : T → X and a set M ⊂ X. We deﬁne the distance from the mapping p to M as

d¯(p, M) = sup inf d(p(t), m) = sup d(p(t),M) (2.6) t∈T m∈M t∈T

We will employ the following comparison functions. A continuous functions φ :

[0, r] → R+ belongs to class N if ψ is nondreceasing on [0, r]. A continuous function

+ + η :[s0, ∞) → R belongs to class M if η is nonincreasing on [s0, ∞) where s0 ∈ R .

A continuous functions ψ : [0, r] → R+ belongs to class K if ψ(0) = 0 and if ψ is strictly increasing on [0, r]. A continuous functions γ : R+ → R+ belongs to class

+ K∞ if γ(0) = 0, γ is strictly increasing on R , and limr→∞ γ(r) = ∞. A continuous

+ function σ :[s1, ∞) → R belongs to class L if σ is strictly increasing on [s1, ∞) and

+ + if lims→∞ σ(s) = 0, where s1 ∈ R . A function β : [0, r1] × [s0, ∞) → R is said to be in class KL if for any ﬁxed s ∈ [s0, ∞) the mapping β(·, s) belongs to class K and if for any r ∈ [0, r1] the mapping β(r, ·) is decreasing on [s0, ∞) and lims→∞ β(r, s) = 0,

+ where s0 ∈ R .

9 2.3 Formulation

The concept of motions is central to the dynamical systems analysis. We deﬁne here the structure of a set of motions

Deﬁnition 2.3.1. (Dynamical System). Let (X, d) be a metric space, and let A ⊂ X

p p and T0 ⊂ R. For any ﬁxed ap ∈ A, τp ∈ T0, let T ⊂ R with min(T ) = τp, with a

p p mapping p(·, ap, τp): T → X, with T ⊂ Rτp being called a motion if p(τp, ap, τp) =

T p p ap. Let S be a family of motions, i.e S ⊂ ∪T p⊂RX , and let T = ∪p∈ST . Then

{T,T0, X, A, S} is a dynamical system.

This deﬁnition reduces to Deﬁnition 2.2.2 in [61] if we assume that the domain of

p a motion p(·, ap, τp) ∈ S is given by T = [τp, tf ) ∩ T for T ⊂ R. The above definition reduces to an embedding of a hybrid system, as defined in [87], if we add the additional requirements min(T ) = 0, and that the domain of each motion p(·, ap, τp) ∈ S can be extended to R+ via the transformation in Equation 2.3). The choice of not assuming a unique T ⊂ R for all motions allows the representation of the class of hybrid systems that do not have a fixed time step in its evolution, e.g. [25]. Furthermore, under the

(reasonable) requirement that the time domains of the motions to be closed subsets of the real line, we can employ the calculus in measure chains [6] to develop a comparison theory for the above examples. Besides the initial value problems involving differential or difference equations, Definition 2.3.1 can be used to model the discrete event systems in [65], while the model in [61], which is a special case of our definition, is employed to model systems governed by partial differential equations, integrodifferential equations, differential inclusions, and stochastic differential equations.

10 Since we are not making any assumptions about the continuity of a motion

p(·, ap, τp) ∈ S with respect to ap and τp, and in order to reduce the notation burden

in the following sections, we choose to avoid this extended notation and instead de-

note a motion p(·, ap, τp) ∈ S simply as p ∈ S whenever it is understood by context.

In the case that a restriction of the mapping p(·, ap, τp) is also a motion in S, we will

treat the restriction as a diﬀerent motion in S. Note that it is possible that r ∈ S,

r p r 6= p, with T = T and r(τr) = p(τp). For any ﬁxed τ0 ∈ T0, consider the set

q Sτ0 = {q ∈ S : min(T ) = τq = τ0}. Then, a partition of S is given by S = ∪τ0∈T0 Sτ0 .

This is indeed a partition, since every set Sτ0 is nonempty and Sτ0 ∩ Sτ1 = ∅, for

any τ1 ∈ T0, τ1 6= τ0. The topology induced by the partition can be used to deﬁne a

metric in S, provided other assumptions are made.

In order the extend the application to a broader class, we now deﬁne the in-

put/output dynamical systems. In the deﬁnition above, we will employ the sets

P (Rr0 ) = {D ∈ P (R) : min(D) = r0} − {∅}.

Deﬁnition 2.3.2. (Input/Output Dynamical Systems). Let (X, d), (Xu, du), (Xy, dy)

be metric spaces, and let T0 ⊂ R and A ⊂ X. For any ﬁxed τp ∈ T0 and ap ∈ A, a set of mappings

[ T x Sτp,ap ⊂ (X) x T ∈P(Rτp ) p is called a set of motions if for any p(·, ap, τp) ∈ Sτp,ap where p(·, ap, τp): T → X with

p T ∈ P Rτp , we have that p(τp, ap, τp) = ap. We call the mapping p(·, ap, τp) ∈ Sτp,ap

11 a motion. Consider the binary relations

u [ u T u Σ (τp, ap) ⊂ (X ) × Sτp,ap u T ∈P(Rτp ) y [ y T y Σ (τp, ap) ⊂ Sτp,ap × (X ) y T ∈P(Rτp )

A motion p(·, ap, τp) ∈ Sτp,ap is called a controlled motion if for some mapping up(·, τp):

p,u u p,u u T → X with T ∈ P Rτp , we have that (up(·, τp), p(τp, ap, τp)) ∈ Σ (τp, ap). A

motion p(·, ap, τp) ∈ Sτp,ap is called a measured motion if for some mapping yp(·, τp):

p,y y p,y y T → X with T ∈ P Rτp we have that (p(τp, ap, τp), yp(·, τp)) ∈ Σ (τp, ap). Then, an input/output dynamical system is denoted by the tuple {S, Σu, Σy} where

[ S = Sτp,ap

(τp,ap)∈T0×A u [ u Σ = Σ (τp, ap)

(τp,ap)∈T0×A y [ u Σ = Σ (τp, ap)

(τp,ap)∈T0×A with S is a family of motions, Σu and Σy are called input dynamical system and

u u y y output dynamical system respectively, with F = dom Σ , and F = ran Σ , Su =

u y p dom Σ , Sy = ran Σ , and T = ∪p∈ST . Finally, the input-to-output dynamical is given by Σ = Σy ◦ Σu, where

y u n u y u yo Σ ◦ Σ = (up, yp) ∈ F × F : ∃ p ∈ S :(up, p) ∈ Σ ∧ (p, yp) ∈ Σ

Following the discussion on the construction of a partition for the family of motions

S defined in Definition 2.3.1, and a claim after Definition 2.2.2 in [61], it is clear that the family of motions S in Definition 2.3.2 holds the same level of generality than the one in Definition 2.3.1, despite the different construction. As discussed after

12 Deﬁnition 2.3.1, we will avoid the notation p(·, ap, τp, up, yp) for a motion related to

Σ whenever possible.

It is important to note that since the theory of stability preserving mappings considered in this work did not require additional topological structure to the set

T ⊂ R+ in the deﬁnition of dynamical systems 2.3.1, we did not seek to deﬁne a particular choice for T , since there are several non-equivalent alternatives. The reader should consult[25] for a review on time sets, and [62] for a description of a class of time sets that are order and metric isomorphic to subsets of the real line.

Furthermore, the cartesian product of motions can account for diﬀerent choices for each particular binary relation.

In the above definition, the concept of “state” can be interpreted as the motions in S that describe the internal behavior of the system, subject to disturbances represented in F u. The mappings in F y are usually employed as the external representation of the internal states of the system. Clearly, this particular decomposition may not be unique, but modifications to the definition to accommodate the concept of system’s terminals employed by Willems in [85] appear possible. Furthermore, studying whether S ⊂ ranΣu or S ⊂ domΣy can produce similar concepts as those studied under Kalman’s canonical decomposition [48]. This work will be focused on stability, leaving controllability and observability for future research.

Note that we can provide a deﬁnition for the internal behavior as the relation

0 0 0 0 Σ ⊂ S × S, where S is a set of mappings of the form p : {τ0} → {a0}. In [35], this notion was employed with the motion p0 ∈ S0 representing the initial function vector in delay differential equations (differential difference equations). Since A ⊂ X is an arbitrary metric space, we do not need to make any further generalizations.

13 2.4 Stability of Invariant Sets

Here, we provide the stability deﬁnitions for invariant sets. We ﬁrst review the

stability deﬁnitions for dynamical systems in Deﬁnition 2.3.1.

Deﬁnition 2.4.1. (Invariant Set). Let S be a dynamical system. A set M ⊂ A is

said to be invariant with respect to S, denoted by (S,M), if ap ∈ M implies that

p p(t, ap, τp) ∈ M, for all t ∈ T , all τp ∈ T0 and all p(·, ap, τp) ∈ S.

When referring to the asymptotic behavior of motions, we will assume that for

any p ∈ S, sup T p = ∞. Equivalent (if and only if) deﬁnitions [35] using comparison

functions will be provided after the qualitative deﬁnitions using − δ arguments.

Deﬁnition 2.4.2. (Stability). (S,M) is stable if for any s > 0 and any τp ∈ T0,

p there exists δs = δs(s, τp) > 0 such that d(p(t, ap, τp),M) < s for all t ∈ T and all

p(·, ap, τp) ∈ S, whenever d(ap,M) < δs.

(S,M) is uniformly stable if Deﬁnition 2.4.2 holds with δs(s) independent of τp.

With comparison functions, (S,M) is stable if for any τp ∈ T0, there exists αs ∈ K which might depend on τp, and ρ > 0 such that d(p(t, ap, τp),M) ≤ αs (d(ap,M)) for

p all p(·, ap, τp) ∈ S and all t ∈ T whenever d(ap,M) < ρ.(S,M) is uniformly stable if the function αs is independent of τp.

Deﬁnition 2.4.3. (Attractivity). (S,M) is attractive if for any a > 0, any τp ∈

T0, there exists δa = δa(τp) > 0 and there exists Ta = Ta(a, δa, τp, p) > 0 such that

p d(p(t, ap, τp),M) < a for all t ≥ τp + Ta, t ∈ T , and all p(·, ap, τp) ∈ S, whenever d(ap,M) < δa.

(S,M) is uniformly attractive if Deﬁnition 2.4.3 holds with Ta(a, δa) independent of p(·, ap, τp). With comparison functions, f for any τp ∈ T0, there exists ηa = ηa(τp) >

14 0 such that for any p(·, ap, τp) ∈ S, there exists σa ∈ L, which might depend on τp

p and p(·, ap, τp), such that d(p(t, ap, τp),M) < σa (t − τp) for all t ∈ T whenever d(ap,M) < ηa.(S,M) is uniformly attractive if the function σa is independent of τp and p(·, ap, τp).

The following deﬁnitions of boundedness do not assume the invariance of the set

M, which in this case is an arbitrary bounded subset of X.

Deﬁnition 2.4.4 (Boundedness). A dynamical system S is bounded if for any τp ∈ T0 and any δb > 0 there exists b = b(δb, τp) > 0 such that d(p(t, ap, τp),M) < b for all

p t ∈ T and all p(·, ap, τp ∈ S, whenever d(ap,M) < δb.

S is uniformly bounded if Deﬁnition 2.4.4 holds with b(δb) independent of τp.

With comparison functions, S is bounded around M if there exists a function φb ∈ N

p deﬁned on [0, rb], rb > 0, such that d(p(t, ap, τp),M) ≤ φb (d(ap,M)) for all t ∈ T

and for all p(·, ap, τp) ∈ S for which d(ap,M) < rb. S is uniformly bounded around

M if the function φb is independent of τp.

Deﬁnition 2.4.5 (Ultimate Boundedness). A dynamical system S is ultimately

bounded if there exists m > 0 and for any δm > 0, there exists Tm = Tm(m, δm) > 0

p such that d(p(t, ap, τp),M) < m for all t ≥ τp + Tm, t ∈ T and all p(·, ap, τp) ∈ S

whenever d(ap,M) < δm.

S is uniformly ultimately bounded if Deﬁnition 2.4.5 holds with Tm(m, δm) in-

dependent of p. With comparison functions, S is ultimately bounded if for any

p(·, ap, τp) ∈ S for which there exists rm > 0 and a function σm ∈ M, such that

p d(p(t, ap, τp),M) < σm (t − τp) for all t ∈ T whenever d(ap,M) < rm. S is uniformly

ultimately bounded if the function σm is independent of p(·, ap, τp).

15 Combinations of the above deﬁnitions allows us to deﬁne additional stability con-

cepts. We say that (S,M) is asymptotically stable if it is stable and attractive and is uniformly asymptotically stable if it is uniformly stable and uniformly attractive.

Using comparison functions, (S,M) is asymptotically stable for each τp ∈ T0, there

exists φ ∈ K deﬁned on [0, ra], ra > 0, and there exists a function σ ∈ L such that if d(ap,M) < ra, then

d(p(t, ap, τp),M) ≤ φ (d(ap,M)) σ(t − τp)

for all t ∈ T p.(S,M) is uniformly asymptotically stable (UAS) if φ is independent of

τp and σ is independent of τp and p ∈ S. Furthermore, (S,M) is globally uniformly asymptotically stable (UGAS) if, in addition to the condition for UAS, we have that ra > 0 can be arbitrarily chosen and φ ∈ K∞.

We now introduce the stability-preserving mapping for systems with no inputs

nor outputs.

Deﬁnition 2.4.6. (Stability-Preserving Mapping). For dynamical systems S1 and

S2, a mapping V : T × X1 → X2 is a stability preserving mapping if:

1. it induces a mapping V : S1 → S2, with n S2 = V (S1) = q(·, aq, τq): q(t, aq, τq) = V (t, p(t, ap, τp)) , p ∈ S1,

q po aq = V (τp, ap),T = T

2. M2 = V (T × M1) = x2 ∈ X2 : ∃ x1 ∈ M1, t ∈ T : x2 = V (t, x1) ;

3. the invariance of (S1,M1) is equivalent (if and only if) the invariance of (S2,M2)

4. the stability, uniform stability, attractivity, and uniform attractivity of (S1,M1)

are preserved in the corresponding properties for (S2,M2).

16 Deﬁnition 2.4.6 coincides with the concept of strongly stability preserving mapping

in [61].

2.5 Stability of Input/Output Systems

In this section we provide stability deﬁnitions that include elements from the

external behavior. Generalizations of the concept of invariant sets are presented

next.

Deﬁnition 2.5.1. (Zu-Invariant Set). Consider a set of input mappings Zu ⊂ F u.

u u u A set M ⊂ A is Z -invariant with respect to Σ , denoted by (Σ ,M), if ap ∈ M

u p and up ∈ Z implies that p(t, ap, τp, up) ∈ M for all t ∈ T , all τp ∈ T0, and all

p(·, ap, τp, up) ∈ Su.

Deﬁnition 2.5.2. (Zy-Invariant Set). Consider a set of output mappings Zy ⊂ F y.

y y y A set M ⊂ A is Z -invariant with respect to Σ , denoted by (Σ ,M) if ap ∈ M

y p and yp ∈ Z implies that p(t, ap, τp, yp) ∈ M for all t ∈ T , all τp ∈ T0, and all

p(·, ap, τp, yp) ∈ Sy.

y y y n,y As above, we will refer to the set N = {yn ∈ S : yn(t) ∈ M ∀ t ∈ T } and

N = {n ∈ domΣy : n(t) ∈ M ∀ t ∈ T n}.

Throughout this work, we will assume that for j = u, y, the following sets are

closed:

j j j j j z,j M = {x ∈ X : ∃ jz ∈ Z : x = jz(t), t ∈ T }

We will employ the distance in Equation 2.6 in the following stability deﬁnitions.

Deﬁnition 2.5.3. (Input-to-State Stability). (Σu,M) is input-to-state stable if for

a u u any τp ∈ T0 there exists ρ > 0, ρ > 0, and α, γ ∈ K, such that for each

17 a ¯u u u p(·, ap, τp, up) ∈ Su, there exists σ ∈ L such that if d(ap,M) < ρ and d (up,M ) < ρ , then

u ¯u u d(p(t, ap, τp, up),M) ≤ max{α (d(ap,M)) σ(t − τp), γ d (up,M ) }

for all t ∈ T p.

(Σu,M) is uniformly input-to-state stable if the function σ is independent of

u τp, up, p(·, ap, τp, up) and the functions α, γ are independent of τp. The deﬁnition

of uniform input-to-state stability reduces to the deﬁnition of local-input-to-state

stability employed in [76] if we assume our metric spaces are subsets of euclidean

u u spaces and Σ is a linear operator that assigns to each ap ∈ A, τp ∈ T0, and up ∈ S

u the corresponding motion p(·, ap, τp, up) ∈ Su. Furthermore we say that (Σ ,M) is

u uniformly globally input-to-state stable if there exists β ∈ KL, γ ∈ K∞ such that

u for any τp ∈ T0, any ap ∈ A, any up ∈ F

u ¯u u d(p(t, ap, τp, up),M) ≤ max β (d(ap,M), t − τp) , γ d (up,M )

p for all p(·, ap, τp, up) ∈ Su, for all t ∈ T .

Deﬁnition 2.5.4. (Output-to-State Stability). (Σy,M) is output-to-state stable if

a y y for any τp ∈ T0 there exists ρ > 0, ρ > 0, and α, γ ∈ K, such that for each

a ¯y y y p(·, ap, τp, yp) ∈ Sy, there exists σ ∈ L such that if d(ap,M) < ρ and d (yp,M ) < ρ ,

then

y ¯y y d(p(t, ap, τp, yp),M) ≤ max{α (d(ap,M)) σ(t − τp), γ d (yp,M ) }

for all t ∈ T p.

18 (Σy,M) is uniformly output-to-state stable if the function σ is independent of

y τp, yp, p(·, ap, τp, yp) and the functions α, γ are independent of τp. The deﬁnition of uniform output-to-state stability reduces to the deﬁnition of local-output-to-state stability employed in [77] if we assume our metric spaces are subsets of euclidean

u spaces and Σ is a memoryless map that assigns to each ap ∈ A, τp ∈ T0, and

y p(·, ap, τp) ∈ Sy the corresponding output mapping yp ∈ F . A similar deﬁnition that includes the input was deﬁned in [81] under the name “boundedness observability”.

Similarly, we say that (Σy,M) is uniformly globally output-to-state stable if there

y y exists β ∈ KL, γ ∈ K∞ such that for any τp ∈ T0, any ap ∈ A, any yp ∈ F

y ¯y y d(p(t, ap, τp, yp),M) ≤ max β (d(ap,M), t − τp) , γ d (yp,M )

p for all p(·, ap, τp, yp) ∈ Sy, for all t ∈ T .

Note that the associated concepts of “practical stability” [81] are already included, since, for any metric space (X, d) and scalar c > 0, the function d(x, y) + c is also metric in X.

The following deﬁnition constitutes the main object of inquiry of this chapter.

Deﬁnition 2.5.5. (I/O Stability Preserving Mapping). For I/O dynamical systems

u y {Si, Σi , Σi }, i = 1, 2, a mapping V : T × X1 → X2 is an I/O stability preserving mapping if:

1. it induces mappings Vj : Sj,1 → Sj,2, for j = u, y with

n Sj,2 = Vj (Sj,1) = q(·, aq, τq): q(t, aq, τq, jq) = V (t, p(t, ap, τp, jp)) , p ∈ Sj,1

q p j jo aq = V (τp, ap),T = T , jp ∈ F1 , jq ∈ F2

2. M2 = V (T × M1) = x2 ∈ X2 : ∃ x1 ∈ M1, t ∈ T : x2 = V (t, x1) ;

19 j j 3. the Z -invariance of M1 with respect to Σ1 is equivalent (if and only if) the

j j Z -invariance of M2 with respect to Σ2, for j = u, y

u 4. the input-to-state stability of (Σ1 ,M1) and the output-to-state stability of

y u y (Σ1,M1) are preserved in the corresponding properties for (Σ2 ,M2) and (Σ2,M2) respectively.

2.6 Stability Results

Results for stability of the invariant set (S,M) are found in [61] and extensions are provided next for input/output dynamical systems.

u y Theorem 2.6.1. Let {Si, Σi , Σi }, i = 1, 2, be I/O dynamical systems and let Mi ⊂

Ai, be closed sets, for i = 1, 2 . Assume that there exists a mapping V : T × X1 → X2 such that:

1) Sj,2 = Vj (Sj,1) for j = u, y;

+ 2) There exists ψ1, ψ2 ∈ K, deﬁned on R such that ∀x ∈ X1, t ∈ T ;

ψ1 (d1(x, M1)) ≤ d2 (V (t, x),M2) ≤ ψ2 (d1(x, M1)) (2.7)

j j + j 3) There exists γ1, γ2 ∈ K deﬁned on R such that for any q = Vj(p), jp ∈ F1 and

j jq ∈ F2 for j = u, y, we have

j ¯j j j ¯j j j ¯j j ψ1 ◦ γ1 d1(jp,M1 ) ≤ γ2 d2(jq,M2 ) ≤ ψ2 ◦ γ1 d1(jp,M1 ) (2.8)

Then V is a I/O stability-preserving mapping.

Proof. The task is to oﬀer proof for points2)-4) in Deﬁnition 2.5.5. For point2), from Equation 2.7), it is true that x ∈ M1 if and only if V (t, x) ∈ M2 for any t ∈ T .

20 u u For the proof of point3), assume M1 is Z1 -invariant with respect to Σ1 . For any pair

u u (uq, q) ∈ Σ2 such that aq ∈ M2, and uq ∈ Z2 it follows from assumption1) that there

p exists p ∈ Su,1 such that for all t ∈ T , V (t, p(t)) = q(t), with V (τp, ap) = aq, which

u q,u implies that ap ∈ M1. Since uq(t) ∈ M2 for all t ∈ T , then by Equation 2.8), we

u p,u u u have that up(t) ∈ M1 for all t ∈ T , which implies that up ∈ Z1 . By Z1 -invariance

u p of M1 with respect to Σ1 , we have that p(t) ∈ M1 for all t ∈ T . By Equation 2.7),

q u this implies that q(t) ∈ M2 for all t ∈ T , hence M2 is Z2 -invariant with respect to

u Σ2 .

u u u Now assume M2 is Z2 -invariant with respect to Σ2 . For any pair (up, p) ∈ Σ1 such

u that ap ∈ M1, and up ∈ Z1 it follows from assumption1) that there exists q ∈ Su,2

p such that for all t ∈ T , V (t, p(t)) = q(t), with V (τp, ap) = aq, which implies that

u p,u aq ∈ M2. Since up(t) ∈ M1 for all t ∈ T , then by Equation 2.8), we have that

u q,u u u uq(t) ∈ M2 for all t ∈ T , which implies that uq ∈ Z2 . By Z2 -invariance of M2 with

u q respect to Σ2 , we have that q(t) ∈ M2 for all t ∈ T . By Equation 2.7), this implies

p u u that p(t) ∈ M1 for all t ∈ T , hence M1 is Z1 -invariant with respect to Σ1 .

y We now proof equivalences in output invariance. Assume M1 is Z1 -invariant with

y y y respect to Σ1. For any pair (q, yq) ∈ Σ2 such that aq ∈ M2, and yq ∈ Z2 it follows from

p assumption1) that there exists p ∈ Sy,1 such that for all t ∈ T , V (t, p(t)) = q(t),

y q,y with V (τp, ap) = aq, which implies that ap ∈ M1. Since yq(t) ∈ M2 for all t ∈ T ,

y p,y then by Equation 2.8), we have that yp(t) ∈ M1 for all t ∈ T , which implies that

y y y yp ∈ Z1 . By Z1 -invariance of M1 with respect to Σ1, we have that p(t) ∈ M1 for all

p q t ∈ T . By Equation 2.7), this implies that q(t) ∈ M2 for all t ∈ T , hence M2 is

y y Z2 -invariant with respect to Σ2.

21 y y Conversely, assume M2 is Z2 -invariant with respect to Σ2. For any pair (p, yp) ∈

y y Σ1 such that ap ∈ M1, and yp ∈ Z1 it follows from assumption1) that there exists

p q ∈ Sy,2 such that for all t ∈ T , V (t, p(t)) = q(t), with V (τp, ap) = aq, which implies

y p,y that aq ∈ M2. Since yp(t) ∈ M1 for all t ∈ T , then by Equation 2.8), we have that

y q,y y y yq(t) ∈ M2 for all t ∈ T , which implies that yq ∈ Z2 . By Z2 -invariance of M2 with

y q respect to Σ2, we have that q(t) ∈ M2 for all t ∈ T . By Equation 2.7), this implies

p y y that p(t) ∈ M1 for all t ∈ T , hence M1 is Z1 -invariant with respect to Σ1.

u u For the following proofs, we assume that for (up, p) ∈ Σ1 , then (uq, q) ∈ Σ2 with q = Vu(p). For input-to-state stability equivalence, we will divide the proof into the

“stability” part, which we call “input-stability”, working with α and γu, and later

u prove the attractivity part. Assume (Σ1 ,M1) is input-stable, with some α1 ∈ K and

u a a a γ1 from Equation 2.8). Let ρ = ψ1(ρ1) and assume d(aq,M2) < ρ . By Equation 2.7) we have

−1 −1 a a d1(ap,M1) ≤ ψ1 (d2 (aq,M2)) < ψ1 (ρ ) = ρ1 (2.9)

u u−1 u u ¯u u u Also, let ρ = γ2 ◦ ψ1 ◦ γ1 (ρ1 ) and assume d (uq,M2 ) < ρ . By Equation 2.8) we have

¯u u u−1 −1 u ¯u u u−1 −1 u u u d1 (up,M1 ) ≤ γ1 ◦ ψ1 ◦ γ2 d2 (uq,M2 ) < γ1 ◦ ψ1 ◦ γ2 (ρ ) = ρ1 (2.10)

u q By input-stability of (Σ1 ,M1) and Equation 2.7) we have for all t ∈ T

n u ¯u u o d2 (q(t),M2) ≤ ψ2 max α1 (d1(ap,M1)) , γ1 d1 (up,M1 ) n o u ¯u u ≤ max ψ2 ◦ α1 (d1(ap,M1)) , ψ2 ◦ γ1 d1 (up,M1 )

−1 u u u−1 −1 u Using Equations 2.9) and 2.10), with α = ψ2◦α1◦ψ1 and γ = ψ2◦γ1 ◦γ1 ◦ψ1 ◦γ2 =

−1 u ψ2 ◦ ψ1 ◦ γ2 we obtain

u ¯u u d2 (q(t),M2) ≤ max α (d2(aq,M2)) , γ d2 (uq,M2 )

22 a ¯u u u u whenever d2 (aq,M2) < ρ and d2 (uq,M2 ) < ρ . Then, (Σ2 ,M2) is input-stable.

u Uniform input-stability follows if we choose α1, and γ1 to be independent of τp.

u u Conversely, assume (Σ2 ,M2) is input-stable, with some α2 ∈ K and γ2 from

a −1 a a Equation 2.8). Let ρ = ψ2 (ρ2) and assume d(ap,M1) < ρ . By Equation 2.7) we have

a a d2(aq,M2) ≤ ψ2 (d1 (ap,M1)) < ψ2 (ρ ) = ρ2 (2.11)

u u−1 −1 u u ¯u u u Let ρ = γ1 ◦ ψ2 ◦ γ2 (ρ2 ) and assume d (up,M1 ) < ρ . By Equation 2.8) we have

¯u u u−1 u ¯u u u−1 u u u d2 (uq,M2 ) ≤ γ2 ◦ ψ2 ◦ γ1 d1 (up,M1 ) < γ2 ◦ ψ2 ◦ γ1 (ρ ) = ρ2 (2.12)

u p By input-stability of (Σ2 ,M2) and Equation 2.7) we have for all t ∈ T

n o −1 u ¯u u d1 (p(t),M1) ≤ ψ1 max α2 (d2(aq,M2)) , γ2 d2 (uq,M2 )

n −1 −1 u ¯u u o ≤ max ψ1 ◦ α2 (d2(aq,M2)) , ψ1 ◦ γ2 d2 (uq,M2 )

−1 u −1 u u−1 Using Equations 2.11) and 2.12), with α = ψ1 ◦ α2 ◦ ψ2 and γ = ψ1 ◦ γ2 ◦ γ2 ◦

u −1 u ψ2 ◦ γ1 = ψ1 ◦ ψ2 ◦ γ1 we obtain

u ¯u u d1 (p(t),M1) ≤ max α (d1(ap,M1)) , γ d1 (up,M1 )

a ¯u u u u whenever d1 (ap,M1) < ρ and d1 (up,M1 ) < ρ . Then, (Σ1 ,M1) is input-stable.

u Uniform input-stability follows if we choose α2, and γ2 to be independent of τp.

Now for the attractivity part, it suffices to show that attractivity of (S1,M1) is equivalent to the attractivity of (S2,M2). Therefore, we will only prove the attractivity component for the case when (S2,M2), since the proof for (S1,M1) follows symmetrically. In this proof, we will employ the equivalent definition of attractivity in Definition 2.4.3. Assume (S2,M2) is attractive, i.e., for any τq ∈ T0, there exists

δ2 = δ2(τq) > 0 such that for any 2 > 0, there exist T2 = T2(2, δ2, τq, q) > 0 such that

23 q if d2(aq,M2) < δ2, then d2(q(t),M2) < 2 for all t ≥ τq + T2, t ∈ T . For any τp ∈ T0

−1 and for any a > 0, let 2 = ψ1(a) and δa = (ψ2) (δ2). Let p ∈ S1 and V(p) = q. If d1(ap,M1) < δa, then by Equation 2.7) with t = τp

d2(aq,M2) ≤ ψ2 (d1 (ap,M1)) < ψ2 (δs) = δ2

Thus, by attractivity of (S2,M2), we have that d2(q(t),M2) < 2 for all t ≥ τq + T2,

q t ∈ T . From Equation 2.7) we obtain that for all p ∈ S1

−1 d1 (p(t),M1) ≤ (ψ1) (d2 (q(t),M2))

−1 < (ψ1) (2) = a

p for all t ≥ τp + T2, t ∈ T whenever d1 (ap,M1) < δa. Then, (S1,M1) is attractive.

Uniform attractivity follows if we choose δ2, δa, and T2 to be independent of τq and q.

y y For the following proofs, we assume that for (p, yp) ∈ Σ1, then (q, yq) ∈ Σ2 with q = Vy(p). For output-to-state stability equivalence, since the attractivity and stability estimates with respect to ap of input-to-state stability are identical to the attractivity and stability estimates of output-to-state stability, we will focus on prov-

y ing the “output-stability” part. Assume (Σ1,M1) is output-stable, with some α1 ∈ K

y−1 y y−1 y y ¯y y y and γ1 from Equation 2.8). Let ρ = γ2 ◦ ψ1 ◦ γ1 (ρ1) and assume d (yq,M2 ) < ρ . By Equation 2.8) we have

¯y y y−1 −1 y ¯y y y−1 −1 y y y d1(yp,M1 ) ≤ γ1 ◦ ψ1 ◦ γ2 d2 (yq,M2 ) < γ1 ◦ ψ1 ◦ γ2 (ρ ) = ρ1 (2.13)

y q By output-stability of (Σ1,M1) and Equation 2.7) we have for all t ∈ T

y ¯y y d2 (q(t),M2) ≤ ψ2 γ1 d1(yp,M1 )

y ¯y y ≤ ψ2 ◦ γ1 d1(yp,M1 )

24 y y y−1 −1 y −1 y Using Equation 2.13), with γ = ψ2 ◦ γ1 ◦ γ1 ◦ ψ1 ◦ γ2 = ψ2 ◦ ψ1 ◦ γ2 we obtain

y ¯y y d2 (q(t),M2) ≤ γ d2(yq,M2 )

¯y y y y whenever d2(yq,M2 ) < ρ . Then, (Σ2,M2) is output-to-state-stable. Uniform output-

y to-state stability follows if we choose γ1 to be independent of τp. The proof for the

y case when (Σ2,M2) is output-to-state stable follows by symmetry.

Remark

• With respect to inputs, assumption1) in the statement of the theorem implies

u−1 that if a motion p ∈ Su,1 is a controlled motion, controlled by up ∈ Σ1 (p),

then it must be mapped to another controlled motion q ∈ Su,2 controlled by

u−1 u u uq ∈ Σ2 (q), such that the “eﬀects” of up and uq, estimated via γ1 and γ2 ,

u u are preserved by the relation Vu ◦ Σ ⊂ S1 × S2 using Equation2 .8) which

u u results from their composition of γ1 and γ2 with the mappings in Equation 2.7).

u Altogether, this could be interpreted as Vu ◦Σ1 being similar to a controllability- preserving mapping, although more research is required to employ the term

controllability rigurously.

• With respect to outputs, the assumptions in part1) maintain the same tone as in

the previous remark: that if a motion p ∈ Sy,1 is a measured motion, measured

y by yp ∈ Σ1(p), then it must be mapped to another measured motion q ∈ Sy,2

y measured by yq ∈ Σ2(q), such that the “readings” of yp and yq, estimated via

y y y−1 y γ1 and γ2 , are preserved by the relation Vy ◦ Σ ⊂ F1 × S2 using Equation

y y 2.8) which results from the composition of γ1 and γ2 with the mappings in

25 y−1 Equation 2.7). This could also be interpreted as Vy ◦ Σ1 being similar to a measurability-preserving mapping.

2.7 Stability of Interconnected Systems

In this section, we present an application of the above formulation and stability results for the case of interconnected systems. Consider two input/output dynamical

u y systems {Si, Σi , Σi }, i = 1, 2. We deﬁne the dynamical systems composed of motions resulting from their input-output interconnection as follows:

Deﬁnition 2.7.1. Consider the feedback interconnection of two input-output dy-

u y u y namical systems {S1, Σ1 , Σ1} and {S2, Σ2 , Σ2}, given by the relation

−1 Σ = Σ2,1 ∩ (Σ1,2) ⊂ S2 × S1 (2.14)

with

u y n y u y uo Σ2,1 ⊂ Σ1 ◦Σ2 = (r, p) ∈ S2×S1 : ∃yr = up ∈ F2 ∩F1 :(r, yr) ∈ Σ2∧(up, p) ∈ Σ1

u y n y u y uo Σ1,2 ⊂ Σ2 ◦Σ1 = (s, q) ∈ S1×S2 : ∃ys = uq ∈ F1 ∩F2 :(s, ys) ∈ Σ1∧(uq, q) ∈ Σ2

We say that the interconnection is well deﬁned if Σ 6= ∅. Then the interconnected ¯ dynamical system S ⊂ S2 × S1 is given by the graph of Σ, with T0 = T0,2 ∩ T0,1 and

A = {(ar, ap) ∈ A2 × A1 : ∃ (r, p) ∈ Σ}, with some product metric in X = X1 × X2,

and appropriate time embedding, e.g. Equation 2.3)

The following result applies the small gain theorem to our class of dynamical

systems

26 ¯ Theorem 2.7.1. Assume that the feedback dynamical system S ⊂ S2 × S1 is well

u u deﬁned, with M = M1 × M2 ⊂ A, and that (Σ2 ,M2) and (Σ1 ,M1) are uniformly

u u globally input-to-state stable with input gains γ2 , γ1 ∈ K∞. Also assume that the

y y output-dynamical systems Σ1 and Σ2 are equal to the (diagonal) identity relations.

u u ¯ Assume that it holds that γ1 ◦ γ2 (b) < b for any b > 0. Then S is globally asymptotically stable.

u Proof. By uniformly global input-to-state-stability of (Σ1 ,M1), we have that there

u exists β1 ∈ KL for any τp ∈ T0,1, any ap ∈ A1, any up ∈ F1 such that

u ¯u u d1(p(t, ap, τp, up),M1) ≤ max β1 (d1(ap,M1), t − τp) , γ1 d1 (up,M1 )

p u for all t ∈ T . Similarly, by uniformly global input-to-state-stability of (Σ2 ,M2), we

u have that there exists β2 ∈ KL for any τr ∈ T0,2, any ar ∈ A2, any ur ∈ F2 such that

u ¯u u d2(r(t, ar, τr, ur),M2) ≤ max β2 (d2(ar,M2), t − τp) , γ2 d2 (ur,M2 ) for all t ∈ T r. For any (r, p) ∈ Σ, we have that

u ¯ d1(p(t, ap, τp),M1) ≤max {β1 (d1(ap,M1), t − τp) , γ1 d2(r, M2)

u ¯ d2(r(t, ar, τr),M2) ≤ max{β2 (d2(ar,M2), t − τp) , γ2 d1(p, M1) }

For any x, y ∈ X = X2 × X1, adopt the metric

2 2 1 1 d(x, y) = max{d2(π (x), π (y)), d1(π (x), π (y))}

For any (r, p) ∈ S¯, adopt the embedding in Equation 2.3), with x ∈ M. Employ the

u u same embedding for β1, β2, p, r. Using the fact that γ1 ◦ γ2 (b) < b for any b > 0, we

27 have that for any t ∈ T r ∪ T p

n o d((r, p)(t),M) ≤ max d2(r(t),M2), d1(p(t),M1) n ≤ max β1 (d1(ap,M1), t − τp) , β2 (d2(ar,M2), t − τp) ,

u u o γ1 ◦ β2 (d2(ar,M2), t − τp) , γ2 ◦ β1 (d1(ap,M1), t − τp)

≤ β (d ((ar, ap),M) , t − τp)

2.8 Conclusion

We have extended the applicability domain of the SPM for qualitative analysis of dynamical systems by including a class of input/output dynamical systems modeled with multi-valued operators, or binary relations. We first introduced a general model of dynamical systems, as well as the input/output processes modeling the external behavior of the dynamical systems. Stability definitions were provided for invariant sets of dynamical systems, for sets of motions in input/output dynamical systems, as well as mixed definitions such as input-to-state stability. Theorem 2.6.1 generalizes existing Lyapunov results that employ real-valued functions, and provides insights on controller synthesis methods, such as control-Lyapunov functions. In the following section, we employed our model and SPMs to analyze the stability of interconnected systems, drawing connections with classical methods such as the small-gain theorem.

These results provide a basis for the qualitative analysis of input/output systems.

28 Chapter 3: DYNAMICS OF MOOD DISORDERS AND STABILITY ANALYSIS

3.1 Introduction

The framework developed in Chapter2 is considered in this chapter, where we present an input/output dynamical systems model that integrates biopsychosocial scientiﬁc ﬁndings on mood disorders that enables the prediction of features of mood via nonlinear and computational analyses.

Mood disorders are prevalent and disabling illnesses that have received significant attention from clinical (e.g., psycho/pharmaco therapy), scientific (e.g., neurotrans- mitters or RCTs), and technological (e.g., rTMS or ECT) perspectives [29, 30]. Be- hind this work lies the areas of mathematics (e.g., statistics or electromagnetic theory for the axon) and engineering (e.g., electrical engineering for rTMS/ECT devices or biomedical engineering for new medicines). Here, we use a mathematical (nonlinear) dynamical system model to integrate scientific findings about mood disorders, predict features of mood via nonlinear and computational analyses, and connect these to psychotherapeutic practice. It is natural to take a dynamical systems approach to psychodynamics, especially for mood disorders, considering, for example, mood

“swings” in bipolar disorders or recurrent depression. Our approach is a type of

29 “meta-analysis” in that we do not conduct new experiments, but use results of multiple existing experimental studies, and link these together to obtain a dynamical systems-level representation and conclusions. We build on, then move past, the re- ductionistic paradigm.

The mathematical and computational modeling and analysis of mood has been studied using a number of different approaches. In [72] the bipolarity of mood is characterized using a thermodynamics perspective, and fixed, periodic, and chaotic attractors are discussed; however, no mathematical models are used. Such dynamics, however, are partially supported by the self-report data from bipolar patients analyzed in [31], which shows that mood swings are not truly cyclic, but chaotic. Studies that employed specifically-designed experiments to validate their models are mostly limited to drift-diffusion models on binary choice (“Flanker task”), studying the role of rumination, attention, and executive functions in mood disorders [17, 67]. How- ever, these studies are constrained to only specific features of mood. On the other hand, stochastic nonlinear models have been constructed to represent several dynamic features of mood disorders, with different levels of connection to neurobiological and psychological determinants. The effects of noise on bifurcations and episode sensiti- zation in mood disorders have been described with linear oscillator models [39, 40].

In [27, 28] bipolarity is envisioned as arising from two possible types of regulation of a bistable system that result in mood oscillations characterized by two variables, depression and mania, with some connections to mood disorder determinants and therapy. Analysis in [21] considered a nonlinear limit cycle model of mood variations based on biochemical reaction equations. In [80] a model of the behavioral activation system, linked to bipolar disorder episodes, is constructed via a nonlinear stochastic

30 model, and mono- and bi-stability of mood oscillations are studied. Also, the eﬀects of noise and changes in nonlinearities are studied, by comparing simulations with empirical and observational data. Mood regulation is analyzed in [49] with an inverted pendulum model, and therapy interventions represented as feedback controllers. The mixed state is accounted for in the oscillator model in [34]. While these studies focus on a particular subset of depressive disorder determinants using up to two state variables, integrative dynamical models accounting for biological and psychological determinants of depressive disorder were proposed in [86], with an emphasis on psychosocial states, and in [5], with an emphasis on neurobiological factors. Additional models considering depression are in [15] where a ﬁnite-state machine is used, and

[82] where a stochastic model of random aspects of mood is employed. Also, there are a number of general computational/mathematical models used in psychiatry and psychology (e.g., see [7, 67, 71]) that are relevant to the above models. All these studies, however, are limited in terms of experimental model validation, computational and mathematical analysis, and results obtained from analysis.

Bipolar I disorder (BD-I) and major depressive disorder (MDD) have large mood variations and clear relations to other mood disorders; hence, these two will take a prominent role in this work. Mood dynamics of bipolar II disorder (BD-II) and cyclothymia are considered here to be a special case of those of BD-I, and dysthymia mood dynamics, a special case of MDD (in range of mood variation, not necessarily depressed mood persistence). This is partly justiﬁed by the close relations between mood symptoms of these disorders in DSM-5 (e.g., compare BD-I to BD-II or cyclothymia, BD-I to MDD, or MDD to dysthymia) [2]. Yet, mood disorders on the

31 “bipolar spectrum” [29] have different symptoms, severity, and experienced-time- length requirements for diagnosis (e.g., consider the differences between unipolar and bipolar depression [12]). Also, there are some symptoms that are significantly im- pairing and cannot be ignored by a clinician, but are essentially ignored here (e.g., expansive/racing thoughts and risk-taking in mania, suicidal ideation in depression, or sleep/weight disruption for several mood disorders). Colloquially, and in some scientific/clinical literature, mania and depression are sometimes described as opposite poles of a single vertical axis with mania at the top (“up”) and depression at the bottom (“down”) (but, see [44]). Yet, in the bipolar “mixed state” [57] there are both manic and depressed symptoms at the same time [2]. The mixed state is relatively common [29]. Also, in [10] the authors say that (i) “Goodwin and Jamison (1990) found that symptoms of depression and irritability, not simply elation, occur in 70%-

80% of patients with mania” (see [29]) and (ii) “Goldberg et al. (2009), in ﬁndings from the STEP-BD study (n=1380), found a majority of bipolar patients with a full depressive episode have clinically relevant manic symptoms” (see [26]). Furthermore, see also pp. 144-147 in [88] where dimensional and categorical approaches to diagnosis based on depression and mania variables are used, with the mixed state corresponding to diﬀerent combinations of severity levels for depression and mania (Figure 8.1 on p. 145 of [88] is related to Figure 3.1d below). Also, the existence of a mixed

(mood) state is logically consistent with the literature on aﬀective dimensions that identify mixed aﬀect/emotion [8]. The mixed state cannot be understood in terms of a single axis with a variable that indicates whether someone is up or down; here, we include a second axis/variable so that there is one for mania severity, and another for depression severity. This does not imply that only two variables need to be used in

32 order to represent mood dynamics. Indeed, below, there will be many other variables that inﬂuence the evolution of the mania and depression variables, and the dynamics of these.

In this paper, unlike the above past research, the mathematical model represents a wider range of features: (i) euthymia, mania, depression, the mixed state, anhedonia, hedonia, and flat or blunted affect (all on a continuum of numeric values that represent the extent to which someone possesses a mood characteristic, or its “severity level,” as in “this person is fully manic, is close to euthymia, or is severely depressed”); (ii) mood dynamics for smooth variations between mood states (e.g., mania to euthymia to depression, and back, and severity levels for these as mood changes continuously); and (iii) “attractors” (“traps”) that mood can fall in to and get stuck (e.g., being stuck in a severe depressed state), ones that are parameterized in terms of a person’s diagnosis (e.g., BD-I or MDD), and other characteristics of a person (defined below); some justification for such an attractors (basins of attraction) approach is given in

[45] where such basins were found experimentally for a group depressed of patients.

Also, a computational analysis of mood attractors is given for multiple cases on the bipolar spectrum (e.g., for BD-I, BD-II, cyclothymia, and MDD, and for each, with attractors for euthymia, depression, mania, and the mixed state). This shows how, in a sense, the characterization of mood disorders here can be related to a “dimensional diagnosis” (i.e., when mood stays in certain regions, or visits a region for some length of time, a certain type of disorder is indicated). Also unlike the above work, our mathematical model and analysis provides conditions for when a person will return to euthymia no matter how their mood is perturbed (i.e., when euthymia is a “global” mood attractor). It is explained how these conditions have clinical implications, in

33 psycho and pharmacotherapy, for how to stabilize a person to euthymia. Also, our analysis provides a novel explanation of the mechanism underlying the mood stabilizer

(e.g., Lithium Carbonate, Li2CO3), that illustrates how it changes from being an antimanic to anti-depressant agent, and also how it operates in the mixed state, serving simultaneously as an anti-manic and anti-depressant agent. These results resolve the statement where researchers identify the “...paradoxical eﬀects of Lithium as both an antidepressant and antimanic agent” [29]. Next, it is shown that for some depressed persons, if they are given an anti-depressant, the medicine could result in a mood trajectory that moves to the fully manic state (e.g., someone who was diagnosed with MDD, was not on a mood-stabilizer, and was driven fully manic by the antidepressant–indicating that they might actually have BD-I).

To the limit scope of our work, we ignore: (i) details of emotion regulation and

“fast” dynamics of emotions that occur on a time scale of less than 2-5 seconds

[32] (but do consider the longer-term inﬂuence of emotions on mood as in [47]); (ii) eﬀects of stress (e.g., see [24]); (iii) mood-congruent attention (see, e.g., [29, 47, 68]);

(iv) positive/negative rumination and inter-episode features (e.g., see [33, 55]); (v) behavioral activation/inhibition system (BAS/BIS) sensitivity (e.g., see [1, 46]); (vi) goal pursuit (e.g., [23]) and goal dysregulation [43]; and (vii) “slow” dynamics that occur on a time scale of multiple years (e.g., kindling or seasonal inﬂuences [29,

69]). Also, we largely do not reach down to model the neural level, but the model nonlinearities are informed by it via referenced scientiﬁc studies. The objective here is to uncover principles of mood dynamics that are common across the spectrum, and only consider other features in how they inﬂuence mood, and not via an explicit model of their own dynamics (e.g., a dynamical model of emotion regulation dysfunction or

34 the coupling between mania and sleep). The ability to consider cross-spectrum issues arises here, in part, due to the use of a “dimensional approach,” as opposed to the

“categorical” one in [2] (a comparative analysis these two approaches is beyond the scope of this paper).

3.2 Mathematical Modeling of Mood Dynamics

3.2.1 Mood States, Trajectories, and Regions

Mood is represented in two dimensions; however, consider brieﬂy the one-dimensional case where the mood state is simply a scalar. Let D(t) ≥ 0 and M(t) ≥ 0 represent the severity of depression (respectively, mania) at time t ≥ 0. Consider Figure 3.1a, but ignore everything except the red/blue vertical line. Place both D(t) and M(t) on this line (one dimension). Thinking of depression and mania as opposites, assume

D(t) is measured on the blue vertical axis, and M(t) on the red vertical axis. The

“×” represents normal/euthymic. If D(t) starts at normal and its severity increases, mood decreases in the downward direction along the blue line (toward a “pole”). If

M(t) starts at normal and its severity increases, the mood point increases in the up- ward direction along the red line (toward the other pole). Representing mood in this manner allows for the colloquial manner of discussing “mood swings” going “up and down” that ignores the mixed state (i.e., only symptoms of depression or mania can be present at the same time, not both). For instance, the mood (black dot) shown on the vertical blue line, along with the vector pointing up, represents someone who is depressed, but with depression that is lifting (decreasing) in the direction of normal.

35 (a) Mood plane/states. (b) Mood in euthymic region.

Figure 3.1: (a) One- and two-dimensional representations of mood; (b) The mood state is in a euthymic region (black circle centered at the green dot); (c) Mood trajectory example with mood starting in a manic/mixed state and decreasing to euthymia with mood lability; and (d) Regions on the mood plane of maximal mood variations for each of the bipolar spectrum disorders [2, 29].

36 The “mood state” [29] is represented here via two independent variables at time

t ≥ 0, depressive mood D(t) and manic mood M(t). Numerical values and scales result from standard instruments for measuring depression (e.g., BDI or HAM-D; see

[30]), mania (e.g., YMRS; see [29]), or the mixed state (e.g., see [10, 37]). Here, we assume that the numeric results from such instruments are aggregated and/or scaled so that they take on values between zero and one. Then. the variables D(t)

and M(t) can be represented on a standard cartesian plane (two-dimensional plot),

with the horizontal being depression, D(t), and the vertical, mania M(t). Then, all

combinations of severity levels for depression and mania can be represented via D(t)

and M(t), with values between zero and one (i.e., D(t) ∈ [0, 1] and M(t) ∈ [0, 1])

representing, in general, the mixed state when the two variables are nonzero (i.e.,

D(t) > 0 and M(t) > 0). In Figure 3.1, we rotate the standard two-dimensional

axis by −45◦ to obtain a “mood plane” representing the combinations of D(t) and

M(t) values. There are three advantages to this rotation: (i) the resulting mood

plane generally corresponds to traditional descriptions of mood disorders, like BD-

I, where mania corresponds to “up” and depression corresponds to “down;” (ii) the

mood plane highlights not only “bipolarity” of BD-I, but also the mixed state and

ﬂat aﬀect poles; and (iii) it conceptualizes euthymia as a type of mixed state where

there is a normal mix of emotions that create that state.

Next, consider the mood plane in Figure 3.1a. Diﬀerent mood states are rep-

resented with black dots. Mood change directions are represented by the arrows

(vectors) For example, if the mood state represents low depression, and hypomania

(upper-left dot), the vector represents that depression is staying constant, but mania

is increasing. The black dot in the center, bottom, represents that someone is in

37 a mixed state with more depression than mania, and the vector pointing up means

that depression is decreasing at the same rate as mania is increasing. In Figure 3.1b,

the mood state is in a euthymic region (black circle centered at the green dot), with

“normal” deﬁned for a person or population (see below). A “mood trajectory” is a

time-sequence of mood states. Figure 3.1c shows a mood trajectory example, with

mood starting in a manic/mixed state (upper-right) and decreasing to euthymia, but

with mood lability (see, e.g., [36] for a discussion on this case); many other mood

trajectories will be considered below.

Figure 3.1d shows regions on the mood plane of maximal mood variations for each of the bipolar spectrum disorders [2, 29]. The general description of disorders on the mood spectrum as vertical lines representing mood variations has, e.g., a longer vertical line for BD-I than, e.g., cyclothymia, as mood swings over a wider range for

BD-I (e.g., see pp. 22-23, and Figure 1-1, of [29]). When considering the mood plane, the mood variation region for one disorder (e.g., cyclothymia) is a subset of another disorder (BD-I). An “ordering” of the illnesses on the spectrum analogous to the one in [29] holds, but now in terms of subsets. The dimensional characterization of mood disorders in Figure 3.1d is related to the one used in Figure 8.1 on p. 145 of [88].

Let the ﬁxed constants nd ∈ [0, 1] and nm ∈ [0, 1] represent a point in the mood

plane. Referencing this point, other mood features can be added to the mood plane

cases in Figure 3.1, such as anhedonia, hedonia, ﬂat aﬀect, and hypomania:

1. Euthymia: For nd ∈ [0, 1] and nm ∈ [0, 1], we assume that if D(t) = nd and

M(t) = nm this describes “normal” or “euthymic” mood [29, 30]. The values of

nd and nm could be speciﬁed for an individual via assessment or a population by

38 averaging individual assessments. As an example, in Figure 3.1 nd = nm = 0.1

is used to represent the center of a euthymic region;

2. Extreme mood states: D(t) = 1 or M(t) = 1 represents maximally severe depres-

sion (respectively, mania), and if D(t) = M(t) = 1 this represents a maximally

severe mixed state [2];

3. Mixed states: Intermediate values of D(t) and M(t) represent mixed mood

states. If nd = nm = 0.1, mixed state examples include: (a) D(t) = 0.75

and M(t) = 0.1 representing very depressed but no mania as compared to

normal (e.g., an MDD state); (b) D(t) = 0.2 and M(t) = 0.1 representing light

depression but no mania as compared to normal (e.g., a dysphoric state); (c)

D(t) = 0.25 and M(t) = 0.25 representing moderate depression as compared to

normal and moderate mania as compared to normal (e.g., as in a mixed state

in cyclothymia); (d) D(t) = 0.1 and M(t) = 0.4 representing no depression as

compared to normal but hypomania (e.g., as in BD-II); and (e) D(t) = 0.75

and M(t) = 0.75 representing a mixed state (e.g., in BD-I).

4. Absence of depression and/or mania: D(t) = 0 (M(t) = 0) represents the

total absence of depression (respectively, mania), and if D(t) = M(t) = 0 this

represents “ﬂat aﬀect” [2]. D(t) ≥ 0 with M(t) = 0 represents “anhedonia.”

D(t) = 0 with M(t) ≥ 0 represents “hedonia” [29];

The time units adopted in this work are days. We are considering adults who do not experience full-range mood swings within 24 hours [29]. We are not considering children who sometimes can have very fast mood swings, on the scale of minutes or tens of minutes [22]. The model will distinguish between “very rapid cycling” (e.g.

39 over an hours time scale) and the mixed state, even though these can be confused in

some cases [57]. Even though mood can vary with time, for simplicity we frequently

drop the notation for time dependency and simply use D and M (similarly, for other

variables).

3.2.2 Mood Dynamics and Equilibria

Mood dynamics are represented by the diﬀerential equations

dD = S (D, M, u ) dt d d dM = S (D, M, u ) (3.1) dt m m

with nonlinear functions Sd(D, M, ud) and Sm(D, M, um) specifying the rates of change

dD dM of mood (derivatives, dt and dt ), where ud(t) and um(t) are the internal/external inputs to the depressive and manic dynamics, respectively, e.g., from stimuli psycho-

physiological response systems.

Consider the “unforced” mania mood dynamics, that is, without the inﬂuence of

˙ dM depressive mood or other external inputs and outputs so that M = dt = Sm(0,M, 0). Deﬁne

2 2 ˙ am (M − nm + cm) fm (M − nm − gm) M = bm 2 − dm 2 −hm (M − nm) am (M − nm + cm) + 1 fm (M − nm − gm) + 1 | {z } | {z } Linear decay Double sigmoid (composed of two sigmoids) (3.2) ˙ In the depressive case, D = Sd(D, 0, 0) is deﬁned in an analogous manner. Solutions to

these diﬀerential equations exist, and are unique, since Sm and Sd are continuous and

satisfy Lipschitz conditions. To illustrate why the shape of the nonlinear function

Sm(0,M, 0) represents key features of mood dynamics, consider an example. Let

nm = 0.5, bm = 0.07, dm = −bm, cm = 0.34, gm = −cm, am = 29, fm = am, and

40 hm = 0.19. Figure 3.2a shows the linear decay line (blue, diagonal, see third term in Equation 3.2 vs. M), the sum of the rational functions that compose the double- ˙ sigmoid (red, see ﬁrst two terms in Equation 3.2) vs. M), and M = Sm(0,M, 0) vs.

M (magenta, right-hand-side of Equation 3.2)). For the linear decay (blue) plot, for a given value of M ≥ nm (M < nm) on the horizontal axis there is a negative

(positive) value moving mood down (respectively, up); that is, −hm (M − nm) tries to stabilize mood to normal. The first two terms in Equation 3.2) vs. M, the red line, have (i) no influence at M = nm; (ii) an increasing positive (negative) influence on mood change as M moves to intermediate values above (below) M = nm representing ˙ destabilizing effects on mood (e.g., making M positive when M > nm so it increases further); and (iii) a lower positive (negative) influence on mood change as M moves above the peaks in the red line, representing destabilizing effects on mood that are ˙ weaker for high values of ±M. The M = Sm(0,M, 0) vs. M case, the magenta line, is the sum of all three right-hand side terms in Equation 3.2, which are the red line and the negative of the blue line. Notice that by plotting the linear decay vs. M we can see the intersection points in Figure 3.2a that are the five points on the magenta ˙ line that cross zero. These zero points identify M values where M = Sm(0,M, 0) = 0, that is, where there is no change in mood, up or down (these are “equilibria”). For ˙ instance, at M = nm, M = Sm(0,M, 0) = 0, so that when the person is at normal, and there are no influences from depression or internal/external inputs, then mood will stay at normal.

41 0

0.1 0.5 0.9

(a) Functions that deﬁne mania mood change.

Euthymia

Anhedonia Euphoria

0.1 0.5 0.9

(b) Basins of attraction for mania.

Figure 3.2: (a) Unforced manic mood dynamics functions, and (b) Basins of attraction for equilibria via integration of Sm(0,M, 0) and interpretation of each attractor.

42 3.2.3 Basins of Attraction for Mood

To visualize the dynamics in the vicinity of the ﬁve equilibria, imagine drawing

arrows on the horizontal axis of Figure 3.2a, with the directions indicating how M will change, as speciﬁed by the sign of M˙ , for each value of M ∈ [0, 1]. For example,

for M values just above (below) nm = 0.5, the arrow will point to the left (right) since ˙ ˙ M = Sm(0,M, 0) < 0 (M = Sm(0,M, 0) > 0, respectively) and this shows graphically

that nm = 0.5 is an asymptotically stable equilibrium point. Since such arrows will

move away from the point to the right of nm = 0.5 where the bottom of the valley

exists, it is called an unstable equilibrium point (if the M value is to the left of the

bottom of that valley, M will decrease towards nm but if it is to the right of the

bottom of the valley, M will increase, moving away from the valley bottom). Similar

analyses works for the other three cases where Sm(0,M, 0) = 0, and a full analysis of

stability is given below.

For another way to view the dynamics and equilibria consider Equations 3.1)

and 3.2), and taking a continuous-time gradient optimization perspective on the ma-

nia dynamics, we let, for any M(0) ∈ [0, 1] and all t ≥ 0,

dM(t) ∂J(M) = −α = Sm(0,M(t), 0) dt ∂M M=M(t)

where α > 0 is a constant “step size” and J(M) is the function to be minimized, one

that must be chosen so that the dynamics speciﬁed by Equation 3.1) are matched

by this equation. Independent of time t ≥ 0, for any M ∈ [0, 1], integrating the two

right-hand-side terms of this equation we get

Z M ∂J(λ) 1 Z M J(M) = dλ = − Sm(0, λ, 0)dλ 0 ∂λ α 0

43 For α = 1, the plot of J(M) is shown in Figure 3.2 where there are “basins of attraction” (valleys) for euthymia, anhedonia, and euphoria (in the depression case, there are basins for euthymia, hedonia, and dysphoria, with some justiﬁcation for this in [45]). The gradient optimization perspective says that if mood is perturbed from the bottom of one of these basins, it will move to go “down hill” until it reaches the bottom of the basin. Hence, the bottoms of these basis “attract” the mood trajectory

M(t) if it is in the vicinity of the bottom of the basin. The peaks on the two hills represent “unstable” points where if M is perturbed even slightly to the left or right, mood will tend to move to the left or right more, and hence M will move away from the peak–the tendency is always to move down the J(M) function if it is not at a peak.

Mood shifts between basins in Figure 3.2 has been linked to external and internal stimuli such as stressors, sleep and seasonal patterns, and the response to these stimuli by other internal processes like physiological arousal or behavioral activation/inhibition systems. For instance, in Figure 3.2 if mood M starts at euthymia, it is important to know if other variables (e.g., stress and sleep deﬁcits) can move it out of out of the euthymia basin, to the right, over the hill, then down/farther to the right to end up at euphoria. Alternatively, in the analogous diagram to Figure 3.2 for depression, if mood M starts at dysphoria, it is important to know if other variables

(e.g., an anti-depressant) can inﬂuence it to move it out of the dysphoria basin, to the right, over the hill, then down/farther to the right to end up at euthymia. In [19], two main processes are identiﬁed in the dynamics of depression: neurobiological processes responsible for mood-congruent cognitive biases in attention, processing, rumination, and self-referential schemes, and attenuated cognitive control to correct these biases.

44 Along those lines, we argue that the size of a basin is mostly aﬀected by the degree of

biased processing of relevant stimuli. For example, increased processing of negative

stimuli by limbic structures in major depressive disorder, which contributes to a re-

duced stressor tolerance and a reduced threshold for mood switching, is represented

by a narrow normal equilibrium basin and wide basin for dysphoria, in the depressive

mood case. Also, a basin’s depth could be correlated to the reinforcing elements of

mood disorders that maintains mood in the basin, increasing the duration of abnormal

episodes. In this case, biased self-reference schemas, mood congruent attention and

maladaptive strategies, such as rumination, could increase the depth of the abnormal

equilibria’s basin.

Decreasing the parameters am and fm has a larger effect in increasing the depth ratio between the depth of the abnormal and the normal basins. Therefore, low values of these parameters indicates a higher effect of mood-congruency on the cognitive biases in the abnormal basins. This could increase the duration of episodes of abnormal mood, compared to the duration in normal levels. Decreasing cm and gm has a greater effect in displacing the unstable equilibria, decreasing the euthymic basin’s size. Thus, low values of cm and gm reflect biased processing and low switching thresh- olds to abnormal mood. An increase in bm and dm results in an increase in both the depth and size of the abnormal basins, thus high values in these parameters represent high severity of the cognitive biases and self-referential schemas that attract mood to the abnormal equilibria, leading to long duration or even chronic episodes due to the difficulty of escaping the basin. Low hm can be associated with attenuated regulatory processes in the prefrontal cortex that regulate mood [19].

45 3.2.4 Adjusting Mood Dynamics and Including Inputs

Generally, we are interested in the euthymic equilibrium point at M = nm, where we obtain 2 2 amcm fmgm Sm(0, nm, 0) = bm 2 − dm 2 = 0 (3.3) amcm + 1 fmgm + 1 which is an under-determined algebraic equation, as we have only one equation and six variables. To reduce the number of variables, the parameter choice in the symmetric case in Figure 3.2a represents a feasible solution of Equation 3.3), with bm = dm, fm = am, and gm = cm. However, some mental disorders are not symmetric, notably, the lack of an euphoria basin in major depressive disorder. By further analyzing

Equation 3.2), some simplifications can be made to obtain an equilibrium point at nm with parameters that allow asymmetry in the manic mood basins, and at the same time provide tractability for a qualitative analysis of mood disorders. First, by converting the functions in Equation 3.2) into a single rational function, we note that the polynomial’s least common multiple, which is the denominator of the resulting rational function, is always positive and continuous, thus the function’s domain is the entire real line. Also, the degree of the numerator of the rational function is greater than the degree of the denominator due to the linear decay term, avoiding the undesirable effects of horizontal asymptotes in M˙ . Finally, our main interest is to study the behavior of the system close to the zeros of the rational function, which are the zeros of its numerator. With these insights, the behavior of the rational function resulting from Equation 3.2 in the domain of interest can be approximated by its numerator, properly scaled to account for the values of the denominator in this domain. An adequate choice for such a scaling factor for M ∈ [0, 1] is the value of the denominator at M = nm, which is confirmed in simulations. The resulting

46 polynomial function is of ﬁfth order with non zero coeﬃcients, while the symmetric

choice of parameters given in Figure 3.2a results in a generic polynomial of ﬁfth

order with only odd exponentials, which is suitable for stability analysis. It is seen

in simulations that introducing a quadratic term with a small coeﬃcient is suﬃcient

to eliminate a stable equilibrium at abnormal manic mood, while maintaining the

normal equilibrium at nm. Hence, we relax the constraint on the rate of change of

the sigmoids to fm − am = m only on the quadratic term of the generic polynomial.

Thus, the above interpretation of the model parameters in terms of mood disorders remains largely unchanged, with m > 0 representing the bias towards euphoria, and

m < 0 the bias towards anhedonia. With these considerations, the resulting map for the unforced and uncoupled dynamics of M˙ is

˙ 5 3 2 M = −pm1 (M − nm) − pm2 (M − nm) − pm3 (M − nm) + pm4 (M − nm)(3.4)

with

2 fmhm pm1 = 2 2 ≥ 0 (fmgm + 1) 2 2fmhm (fmgm − 1) pm2 = 2 2 (fmgm + 1) m (2gmhm − dm) pm3 = 2 2 (fmgm + 1) 3 hm + fmgm (hmfmgm + 2hmgm − 4dm) pm4 = 2 2 (fmgm + 1)

It can be seen that pm1 is the only parameter whose sign is determined by the basic assumptions made for Equation 3.1). As the main determinants of the characteristics of the basins are spread out between the p-parameters, we will refer to the insights made above on fm, gm, and hm in discussions in the next section. The exception is

47 parameter pm3, which by itself represents the bias between either side of the normal equilibrium.

We can now consider the coupling eﬀect between depression and mania, as well as the forcing inputs ud and um. The revised equation for the mood dynamics given in Equation 3.1 is

˙ 5 3 2 D = −pd1 (D − nd) + pd2 (D − nd) + pd3 (D − nd) − pd4 (D − nd) (3.5)

+qd (M − nm) + Bd (D, ud)

˙ 5 3 2 M = −pm1 (M − nm) + pm2 (M − nm) + pm3 (M − nm) − pm4 (M − nm)

+qm (D − nd) + Bm (M, um)

The coupling parameters qd ∈ R and qm ∈ R provide the existence of a mixed state, as well as a way to model the cyclic nature of bipolar disorders. If big enough, these parameters can introduce oscillations, analogous to what is seen in rapid cycling. The functions Bd (D, ud) and Bm (M, um) are the eﬀects of external and internal inputs on depressive and manic mood respectively at time t.

3.2.5 Mood Trajectories for Various Mood Disorders

The diﬀerential equation model in Equation 3.5) is ﬂexible enough to represent several of the most important mood disorders in the DSM-5. In Figure 3.3, the vector

field diagrams for BD-II, cyclothymia, and MDD are presented. By altering the size and depth of the basins of attraction for the equilibria, for both the depressive and manic dimensions (as in Figure 3.2), we can create equilibria at asymmetric locations in the mood plane. The vector field diagram of BD-II features an equilibrium at a hypomania and major depression, while the cyclothymia vector field depicts equilibria at hypomania, and mild depression, as well as in the mixed state, but with increased

48 Vector field diagram: BD-II Vector field diagram: Cyclothymia 1 1 Manic Manic 0.5 0.5

0.5 1 0.5 1 Depressive Depressive (a) (b)

Vector field diagram: MDD 0.7

0.5 Manic

0.5 1 Depressive

(c)

Figure 3.3: Vector ﬁeld diagram (grid of red arrows) and trajectories (in blue) for (a) BD-II, (b) Cyclothymia, and (c) MDD.

basin depths to reﬂect the extended episodic durations in these disorders. MDD features only two equilibria in the normal and major depression episodes.

3.3 Nonlinear and Computational Analysis of Mood Dynam- ics

This section features the main technical result of the chapter, as well as a computational analysis for mood dynamics under pharmacotherapy with mood stabilizers.

49 3.3.1 Stabilization to Euthymia

Here, we present conditions for global exponential stability of the manic mood

equilibrium that corresponds to euthymia:

Theorem 3.3.1. Stability of Euthymia: Consider the manic mood dynamics given

in Equation 3.4, and assume that

2 pm2 < 4pm1pm4, pm4 > 0. (3.6)

If pm3 is such that q 2 2 2 3 2 pm3 < (12pm1pm4 + pm2) − pm2 36pm1pm4 − pm2 (3.7) 27pm1 ¯ then the manic mood equilibrium point at M = nm is globally exponentially stable.

ˆ Proof. First, we make the change of coordinates M = M − nm, such that the new ¯ˆ coordinate of the equilibrium point at nm is translated to M = 0. Consider the ˆ 1 ˆ 2 continuously diﬀerentiable, radially unbounded, positive function V M = 2 M as a Lyapunov function candidate. Its derivative along the trajectories given by

Equation 3.4 is

˙ ˆ ˆ ˆ 5 ˆ 3 ˆ 2 ˆ V M = M −pm1M + pm2M + pm3M − pm4M ˆ 2 ˆ 4 ˆ 2 ˆ = −M pm1M − pm2M − pm3M + pm4

Our goal is to prove that V˙ Mˆ < −αkMˆ kβ, for α > 0 and β > 0, which is equivalent to proving that the last polynomial expression in the parenthesis is positive, noting that the parameters pm2 and pm3 are sign indeterminate. To simplify notation, we restate the polynomial in the parentheses as

α = P Mˆ = Mˆ 4 + qMˆ 2 + rMˆ + s (3.8)

50 where q = −pm2/pm1, r = −pm3/pm1, and s = pm4/pm1. In this form, the polynomial is called a “depressed quartic function” (there is no cubic term).

Although providing an analytic solution for this quartic polynomial is cumber- some, in [51] it is shown that the positivity of the depressed quartic polynomial in Equation 3.8) is guaranteed by showing that: (i) the discriminant of P Mˆ is strictly positive, and (ii) s > q2/4. The assumption in Equation 3.6) fulﬁlls (ii), thus we concentrate on the discriminant of the quartic polynomial, which is

4 3 2 2 2 2 4 3 ∆4 = 16q s − 4q r − 128q s + 144qr s − 27r + 256s

The discriminant is the product of the squares of the diﬀerences of the roots of P Mˆ .

We are interested in the eﬀect of the parameter r in this discriminant, noting that

2 4 ∆4 can be factored into a quadratic polynomial in r . However, the coeﬃcient for r

2 is negative, and ∆4 → ∞ as r → ∞. When r = 0, the assumption in Equation 3.6) allows ∆4 > 0 for all values of q and s, therefore the polynomial in r has only two real solutions, that are symmetric. We seek to constrain the value of r2 in between these roots, which are given by q − (−4q3 + 144sq) ± (−4q3 + 144sq)2 + 4(−27)16s(q4 − 8q2s + 256s3) r2 = (3.9) 0 −2(27)

Algebraic simpliﬁcation results in

2 q r2 = (12s + q2)3 + q 36s − q2 (3.10) 0 27 2 2 ˆ We see that the constraint r < r0 is a suﬃcient condition for positivity of P M . We express this inequality in terms of the parameters in Equation 3.4) as

q 2 2 2 3 2 pm3 < (12pm1pm4 + pm2) − pm2 36pm1pm4 − pm2 27pm1 51 which holds by the assumption in Equation 3.7). Also, by the assumption in Equa-

tion 3.6), this bound is strictly positive. Therefore, since this condition holds, α = P Mˆ > 0, and the system in Equation 3.4) is globally exponentially stable.

Note the assumption in Equation 3.6) is equivalent to stating a lower bound on

the regulation rate hm, given by

dm hm > gm

which is independent of fm. This shows that the maximum slope of the sigmoids in the

equilibrium corresponding to euthymia is given by dm/gm. Hence, this is analogous

to saying that when the mood regulation rate is large enough that it can regulate the

cognitive biases, then the multi-stability (e.g. equilibria at euphoria and anhedonia)

disappears. Larger hm can be obtained via psychotherapy that promotes awareness of

cognitive biases and tools to regulate emotions and mood, as well as pharmacother-

apy that targets biological determinants of the PFC’s activity, like serotonin levels.

Furthermore, as detailed in the proof, when r = 0, which implies that pm3 = 0, we always have that the discriminant ∆4 > 0, therefore, the unbiased system with

pm3 = 0 is also globally exponentially stable. The bound in Equation 3.7, within the

region where assumption in Equation 3.6) is fulﬁlled, is larger with larger values of

pm2 which increases in fm. Here, fm determines the depth of the basins. Larger fm

represents shallow abnormal basins, which gives room for the imbalance pm3. The

value of proving this bound goes beyond this result, as it provides a bound on any

other external or internal input that introduces quadratic or linear terms in Equation

3.4.

This analysis is particularly instructive as it provides natural parametric condi-

tions that, when they hold, a person will return to euthymia in the presence of any

52 mood perturbation (e.g., arising from life events). For the case of mania (depression is analogous):

1. Let hm > 0 be the “mood regulation rate,” which is the rate at which manic

mood is returned to euthymia (when that is possible).

2. Let dm > 0 be a “mood reinforcement parameter,” which represents the posi-

tive reinforcing eﬀect to mood variations produced by cognitive and behavioral

biases.

3. Let gm > 0 be a “mood switch parameter,” which represents “distress tolerance”

[53].

The meaning and eﬀects of these parameters is discussed in [14, 18]. A person will return to euthymia no matter how their mood is perturbed if

dm hm > gm

This condition shows if the mood regulation rate is fast enough (i.e., a strong enough tendency to stabilize mood), a person will return to euthymia no matter where their mood starts at (e.g., fully manic, depressed, or mixed). Here “fast enough” is quan- tiﬁed by the fraction dm/gm, so that if it is large, then the mood regulation rate hm must be even higher, but if it is small, then the hm can be lower. The relative sizes of dm and gm determine the size of dm/gm: for a ﬁxed dm, gm being small means that dm/gm will be large, and gm being large means that dm/gm will be small. The same type of idea holds for holding gm constant, and varying dm.

53 3.3.2 Pharmacotherapy: Triggering Mania

Simulations are provided here with depressive mood and manic mood as state variables. Figure 3.4a shows the vector field diagram for BD-I, with red arrows indicating the direction of mood change for different points in the mood plane. The blue lines represent trajectories with circles at their initial conditions and squares as their final state at time T = 10 days. There are four stable equilibria, namely, normal/euthymia, depression, mania, and mixed states where the trajectories may converge.

Note that the rate of convergence matches the onset times given in the literature, which are, on average, three days for mania and seven days for depression [29]. Fig- ure 3.4b shows the time trajectories for depressive mood in blue, and manic mood in red when there is a misdiagnosis of major depressive disorder and the use of an antidepressant. The initial mood state corresponds to the depression episode equilibrium, and at the second day, an antidepressant is administered for two days, modeled here as a constant negative input in depressive mood and a smaller, positive input in manic mood. Even after stopping the medication, manic and depressive mood are attracted to the equilibrium representig mania, which, according to the DSM-5, constitutes suﬃcient evidence for a BD-I diagnosis.

3.4 Conclusion

A review of scientiﬁc ﬁndings concerning the dynamics of mood in the bipolar and depression spectra revealed the heterogeneity of the subject and the need of interdisciplinary work to improve our understanding of the main biopsychosocial mechanisms

54 Vector field diagram: BD-I Mood shift in BD after AD 1 Depressive 0.8 Manic MU

Manic 0.6 0.5

0.4 0.5 1 0 2 4 6 8 10 Depressive Time (days) (a) (b)

Figure 3.4: (a) Vector ﬁeld diagram and trajectories (in blue) for BD-I, showing 4 basins of attraction at normal, depression, mania and mixed states. (b) Mood episode shift from depression to mania after treatment with antidepressant.

behind manic and depressive episodes. This chapter extends this literature by providing a novel mathematical model that characterized an expanded set of features of mood disorders. We offered qualitative descriptions of the primary mood states: euthymia, mania, depression, mixed states, anhedonia, hedonia, and flat affect, as well as the dynamics for smooth variations between these states. Computational simulations portrayed these mood states as attractors, and a stability analysis provided a mathematical expression that supports the hypothesis given in [18] of conditions for global stability of euthymia. Finally, simulations of our model provided a new explanation of the mechanism underlying mood stabilizers.

55 Chapter 4: CONCLUSIONS AND FUTURE DIRECTIONS

4.1 Conclusions

The challenges stemming from a deeper integration of technology and society, fueled by the fast pace of technological development and data availability, call for a reciprocal integration between scientiﬁc and engineering disciplines in order to provide

”no-harm” guarantees for systems operating under humans-in-the-loop architectures.

Stability and robustness transcend scientific boundaries and represent essential properties that the systems mentioned above should possess. However, there is not in place a unified methodology for the definition and verification of qualitative properties of systems that can be understood and implemented by both social scientists and engineers designing these systems.

This dissertation explores qualitative equivalence via stability preserving mappings (SPMs) [61], in the quest of proposing a framework for the analysis of socio- technological systems. Chapter2 extends the applicability of SPMs to study systems with input/output behavior. To achieve this, we introduced multi-valued operators that link elements of the internal and external behavior. Hence, our model can represent systems whose evolutionary process are dependent on external motions, as well as the endogenous motions considered in [61]. Examples of those evolutionary processes

56 include sets of ordinary differential and difference equations, stochastic differential equations and autoregressive models, functional differential equations, differential inclusions, and multivalued operators. The model enables the definition of input- to-output and input-to-state stability, as well as other mixed definitions. Our main result in this chapter reveals that SPMs are suitable mathematical tools to establish qualitative equivalence between input/output dynamical systems, and these results are applied in the stability robustness analysis of interconnected systems, offering a generalization of known Lyapunov/small-gain theorems to our class of systems.

In Chapter3 we applied our qualitative methodology in a novel mathematical model of mood dynamics in the bipolar and depressive spectra. The model integrates scientiﬁc ﬁndings of mood disorders, predict features of mood via nonlinear and computational analyses, and connect these to psychotherapeutic interventions.

Unlike other mathematical representations of the dynamics generated by the interplay of the main determinants of mood disorders, our model represents a broader range of features. These include characterization of attractors corresponding to euthymia, mania, depression, the mixed state, anhedonia, hedonia, and flat or blunted affect, and the transitions among these attractors caused by external elements, like stress and medication. A stability analysis provided a mathematical expression that supports the hypothesis given in [18] of conditions for global stability of euthymia, as well as offering additional conditions for stability of euthymia under a particular class of feedback inputs.

57 4.2 Future Directions

There are several avenues to extend the results from Chapter2 of this dissertation,

of which extending the current local results to semi-global and global results. A more

uniﬁed deﬁnition of dynamical systems could merge the concepts of internal and

external independent binary relations, i.e., S0 and Su, usually interpreted as the set

of initial conditions and external inputs. A suitable metric in the product of the

codomains of initial conditions and inputs U = A × Xu can be found, for example, by using metric preserving functions [20]. This approach can be used to study the connection between tools employed in stability analysis of interconnected systems, like M-matrices [63] and the small-gain theorem. However, more research is needed to answer whether the conceptual simplicity obtained by treating these codomains as a single object will match the freedom of assigning stability or attractivity properties to only those codomains, or if the results obtained in metric spaces can be carried to inﬁnite and ﬁnite dimensional normed spaces.

A more detailed analysis of the composition and intersection of binary relations that result from the interconnection of dynamical systems, as shown in Section 2.7, reveals that arguing for global stability results will not be a trivial exercise, due to the lack of totality of the relations. This problem, in turn, could open the possibil- ity of characterizing further qualitative properties, controllability, and observability, for instance, through the study of images and preimages of the binary relations that represent the external behavior of dynamical systems. Future research should focus on extending the deﬁnition of SPMs, by introducing mappings linking the graphs of binary relations representing the input/output behavior to achieve such results. Fi- nally, the extensive use of binary relations in this work paves the way to employ other

58 mathematical tools such as those found in category theory [3] to generalize deﬁnitions and results. Recent dynamical systems models [4, 78] oﬀer exciting research avenues in that area.

The theory presented in Chapter2 can be directly applied to input/output systems whose trajectories do not lie in inﬁnite or ﬁnite-dimensional normed spaces.

Examples of such systems are encountered in domain theory, game theory, temporal logic, linguistics, and signal processing to cite a few. Further applications might be discovered by reviewing the extensive list of metrics, metric spaces, and their domain of application found in [16].

Research eﬀorts seeking to expand the work presented in Chapter3 should focus on model validation, and integration of other features of mood disorders, which include the internal mechanisms for goal seeking and emotion regulation, behavioral activation/inhibition systems, as well as input/output stability analysis featuring stress responses and mood-congruent attention. The literature on models for cognitive control by the prefrontal cortex is vast, but one that stands out for its similarity with existing models of resource allocation in engineering is the supervisory attentional system (SAS) in [64]. Here, each possible stimulus response is a value, similar to a suitability function. A fast allocator is responsible for selecting stimulus-driven responses, while the SAS can override the previous allocator by promoting goal-oriented responses. This view of attention as the main allocator of resources for regulation is supported in [11]. Furthermore, results for interconnected dynamical systems of

Chapter2 can be employed to model and analyze networks of social support with individuals in the bipolar and depression spectra.

59 Concrete technological applications of the theoretical development in Chapter3 could improve solutions in psychoeducation (PE) for patients in the bipolar spectrum. As seen in Chapter3, mood disorders feature complex dynamics and diﬀerent reinforcing and feedback eﬀects between variables. Psychoeducation interventions are evidence-based treatments for MD that educate the patient about these dynamics and raise awareness about possible leverage points for intervention. These applications could acquire somatic signals from electrocardiogram sensors and behavioral and cognitive signals from the interaction of the patient with the technology (e.g., a computer game), and employ multisensory stimulation via video and audio, as well as NTSB, regulated through feedback control. The controller must account for intra-individual variabilities to adjust the stimulus variables. The theory developed in Chapter2 can be employed to obtain stability guarantees for the closed-loop (controller-patient) system.

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