Systems with Infinite-Dimensional State Space: The Hilbert Space Approach

Abstract-The contrd of infmitedimensional systems has received of them. Ideally, a complete theory would have several layers. much attention from engineers and even mathematicians Realizability There should be a general theory giving the structure common although fmt considered m [4] has been ignored until recently. Ironi- to many situations and then there should be a branched hier- dyenough while state-spacesystems theory was developing in the edy 1960’s a mathematical study of scattering and of wn-selfadjoint archy of increasingly special conditions and increasingly spe- operptorsproduced a parallel theory which was infmitedimensional cific theorems. Such an extensive theory will, of course, take %om the beginning. When the dose relationship between the two sub a long time to develop. On the positive side the general theory jects became known time invariant infiitedimensional systems theory of infinite-dimensional A, B, C, D-type linearsystems now advanced quickly and at a general level it now seems reasonably com- seems reasonably complete and that is whatthis article pre- plete. This paper desaibes the connection between mathematical scat- tering and systems Itthen gives a thaough treatment of infite sents. We describe a realistic setting for many infinite-dimen- dimensionnlrime invariant continuous time systems The last section sional problems to which the main properties of general finite- lists recent scattering results whi& might be of engineering intexest. dimensional systems adapt suitably. Although the world abounds in important infinite- I.INTRODUCTION dimensional systems, of what benefit is a theory about them? Certainly the abstract considerations to be presented here are ERY ROUGHLY SPEAKING, a situation which re- not of immediate interest to the practicing engineer. However, quires infinitelymany parameters to specify will be they may ultimately be influential. For example, theone V modeled by asystem with infinite-dimensional state problem common to mostinfinite-dimensional design situa- space. Very early in the come of a design, one restricts atten- tions is: How does one select a finitely parameterized approxi- tion to a few parameters which he believes are the most im- mation and how does one determine its accuracy?Problems portant. However, the resulting heavily proscribedsystems of this type are mathematically speaking infinite dimensional will still have infinite-dimensional state space. For example, a and one would have little hope of developing a decent theory variable coefficienttransmission linehas an infinitely of finite-dimensional approximationswithout first having a parameterized space of capacitance, inductance, etc., distribu- good theory of infinite-dimensionalsystems. Hopefully the tions and any system corresponding to a transmission line will approach taken here addresses that need and provides a foun- have infinitedimensional state space. Even thoughthere are dation for building the more specialized layers of theory which many ways to discard all but a finite numberof design parame- will follow. Another benefit of having an infinite-dimensional ters such as to consider only cascades of a prescribed number theory closely attuned to a classical mathematical physics ap- of uniform lines or to prescribe u priori the tapering of the proach (as this one is) is that developments in that area can be line, all such systemsstill have infinite-dimensional state adapted to (and encouraged by) engineering theory; some of space. In thissame vein semiconductorsadmit an infinite Section IV andthe entirety of Section V are examples. variety of doping intensities, of geometries, and of doping Whereas, it would be highly irresponsible to urge a practicing geometries and even the practical devices with heavily re- engineer to learn the abstract theory described herein it would stricted geometries and doping patterns have an infinite-dimen- be equallyfoolish for theoretical engineers to abandon such sional space of states. This applies to an enormous number of pursuits since unifieda andthorough understanding of devices and though finite dimensional systems suffice as mod- infinite-dimensional realizability should be valuable. els for basic circuits, many components of a modem circuit Now that we have motivated a study of infinite-dimensional when modeled carefully would be infinite dimensional. From systems, we give some idea of what the theory entails and how this viewpoint, infinitedimensional engineering problems pos- itconnects with other areas. The basic general principles sibly outnumber the intrinsicallyfinite-dimensional ones. To which have emerged in finite-dimensional systems theory are reemphasize this point we mention that besides the many in- the fact that two ‘minimal’ systems with the same frequency- finite-dimensional systems thought of more in connection with response function (FRF) are isomorphic, every rational func- control (cf. [43]) there are transmission lines, microwave cir- tion is the FRF forsome ‘minimal’ system, and the eigenvalues cuits, thin film devices, surface acoustic wave devices and in- of the ‘state operator’ for a minimal system give the pole loca- deed most semiconductor devices. tions of the FRF. During the early 1960’s,while this was de- In fact, there are so many examples of infinite-dimensional veioping a mathematicalcounterpart evolved independently. systems that any theory general enough to deal with most of The mathematical theory is infinite dimensional and by com- them would be too general to say much specific about any one bining it with’the systems approach precise analogs of these basic theorems are now available in infinite dimensions. Thus Manuscript received January 15, 1975;revised June 15, 1975. there has been a recent advance in systems theory initiated by This work was supported in part by the National Science Foundation. The author is withthe Department of Mathematics, University of the discovery of existing mathematical work and greatly expe- California at San Diego, La Jolla, CA. dited by its application. A surprising fact is that not only do 146 IEEE,PROCEEDINGS OF THE JANUARY 1976

the mathematical works provide techniques for the study of devoted his time to design of lumped circuits using the chain systems,the works themselves are reasonably close to being formalism (cf. the survey [ 16]), and his colleague Arov has studiesof systems, For example, the book by Nagy and pursued infinitedimensional problems [ 1 1. Foias [491, work which was done for purely mathematical rea- Amusingly enough a major problem which has troubled the sons, could be viewed after a translation of terms almost en- field since completion of the earlier work centers on deciding tirely as atreatise on infinitedimensional discrete-timesys- what one means by an infinite dimensional system. The type tems. If the results and methods of the book are restricted to of linear system treated in these works always has the form finitedimensions they agreewith thestandard engineering d ones for the case of‘lossles:’ discrete systems. A description - x(t) = Ax(t) + Bu(t) of the relationship appears in[ 281. dt The work of Lax and Phillips though slightly less similar in y(t)= Cx(t) +Du(r) form is moresimilar in spirit to systemstheory. This is a theory of scattering, which includes nonsymmetric scatterers but amusingly enough it is not at all clear what type of linear as opposed to the radially symmetric ones concentrated upon maps B and C should be. In fact this has been a major prob- in classical physics. The original work compiled in [37] treats lem troubling the field since the connection between systems lossless situationswhile [38] gives alossy scattering theory. andoperator theory was discovered.Fuhrmann andHelton This works puts great emphasis on the ‘state operator,’ but in- avoid the problem by treating mostly discrete time systems. put and output operators never appear and are replaced by a In treating the continuous time case Dewilde avoids B, C en- more general notion of “incoming and outgoing space.” De- tirely while Baras and Brockett choose them in a mathemati- spite these differences the author feels that Lax-Phillips scat- cally convenient way whicheliminates most standard exam- teringand infinite-dimensional systems theory arelinked so ples. Thus an essential task is to finda formulationfor closely in substanceand spirit that future development of continuoustime systems whichcontains the main examples these subjects should go hand in hand. Section 111 is devoted and to which the basic finite dimensional systems facts extend to describing the preciserelationship between the two sub- in a reasonable fashion. Ironically enough the techniques for jects. An immediateconsequence is thatone can seehow proving these facts have (for the last three yein) been much systems theory looks in the setting of classical scattering, how clearer than what the systems themselves should be. Even for for example the classical objects of scattering, wave operators, studying discrete time systems a slightly broader set up than andscattering operators, compare to objectsin systems the one presently used might be desirable since it is necessary theory. Quite possibly these more established objects will be to embrace some common circumstances (see Section11-B). easier to use for infinite dimensions than controllability and The major accomplishment of this paper is that it gives a observabilityoperators. Thearticle [281 connects Lax- theory of continuous time systemswhich is physically realistic Phillips scattering with systems in discrete time; while [311 as well as being complete at the general level. The theory also does it for nonlinear situations. fits together perfectly with Lax-Phillips scattering which guar- Now we describe some history. Among mathematicians the antees that it contains the physical situations they have stud- first work in this direction was done by Livsic [441, 1451 who ied and is consistent with their long experience in abstracting began a study of operators on Hilbert space which are not self- mathematics from physical situations. adjointin the early1950’s, cf. [9]. Thework of Lax and We also mentionthat the distributional approach of Phillips on scattering theory began in the early1960’s. Also Balakrishnan and more recently Aubin and Bensoussan is very about that time DeBrange and Rovnyak and Nagy and Foias general and contains many physical situations. The work does began a study of “nonunitary” operators which though similar not effectively introduce a Hilbert space structure and thus is to Livsic’s in general philosophy was basically very much dif- moreabstract than our approach. No isomorphism property ferent. Another closely related study was done by Helson. In or spectral results exist for distributional systems although the 1964, Adamajan and Arov [ 21 showed the Lax-Phillips and author does not know if these will come from additional work Nagy-Foias theories to be equivalent. In 1965, Livsic used his or if they require additional Banach or Hilbert space structure. operator theory to derive a type of systems theory. The book Our treatment of systems might be thought of as a combined received no attention in this country until connection of the distributional and Hilbert space approach. Hilbert spaces usu- above mathematical theories with systems was discovered here ally arise from energy or cost considerations. Energy consider- abd generated enough interest to produce a translation [46] in ations play a fundamental role in mathematical physics which 1973.The first work on infinite-dimensional systems was accounts for the prominence ofHilbert space there. done in control. In realizability the first work was Balakrish- Our notation is standard.If H is aHilbert space and nan’s [6] in 1966 which described the response function for a 1 < p < 00, then LP(a,b, H)denotes functions h(t) with values systemwith distributional entries. Little happened until re- in H so that $’ 11 h(t)llP dt< O0. A function inLz [-jm,jm, HI centlywhen the mathematicaltheory was seen to pertain; which is the Fourier transform of a function in Lz [O,m, HI Dewilde [ll], Fuhrmann [17]-[26], andHelton [27], [28]. has an analytic continuation to the right-half plane (RHP) of Also around that time systems theorists Brockett and his stu- complexnumbers; denote the space of thesefunctions by dent, Baras, became interested in the subject [ 51. There was H2(H).The corresponding space for the left-half plane (LHP) considerable subsequent work on this [ 61, [ 121, [ 181, [ 19 1, is z2(H).The space of all infinitely differentiable H valued [291,[311, [321 and related topics [131, [201-[261, [301. functions defined on [a, b] and vanishing near a and b is de- In quite a different spirit there is the algebraic approach taken noted CF(a, b, H). If H and K are Hilbert spaces QH, K) de- by Kamen [ 341 -[36] and the distribution theoretic approach notes the space of all bounded linear operators (ones whose ofAubin and Bensoussan [ 31. More recentRussian work norm known to the author is the book [45] describing new direc- tions in systems theory. Also the mathematician Potopov has HELTON: SYSTEMS WITH INFINITE-DIMENSIONAL STATE SPACE 147 is finite) from H to K. Note a linear operator is continuous if Thus a signal entering thetransmission line at x = 1 satisfies and only if it is bounded. The space of all uniformly bounded functions on [a, b] with L(H, K)values is Lm(u,b, H, K)while H"(H, K) [resp. F(H,K)] denotes those which have a uni- formlybounded analytic continuation tothe RHP [resp. while one leaving satisfies LHP] . A one parameter semigroup G(t)of linear operators on H will always satisfyadifferential equation of the form d C(t)/dt = AG(t) on a dense subspace 9 (A)of H on which the linear operator A is defied. The converse also holds. A com- plete reference on the subject is [33], an introductory refer- this describes the input and output to the line. The state-space ence is [ 561 and an intermediate reference is [ 151. A good equations for this setup are: general reference on functional analysis is [ 5 5 1. '(')=A(;)+Buat 11. THESETUP A. Continuous Time. For our purposes a system [A,B, C,D ] is defied by Y = c(:)

d*) d*) = Ax( t) + Bu( t) where A is the unbounded operator onHE defined by dt y(t)= Cx(t) + Du(t) where A, B,C, D, are linear operators and x(t), u(t), and y(t) are vector-valued functions. If the vector spaces involved are not finite dimensional, then precisely where the operators B and C mapfrom and to is adelicate matter. One naturally acting on domain wants to specify them in a nice way but it turns out that the simplest definitions exclude many interestingexamples. For example, one seemingly reasonable set up would be to have Hilbert spaces U, Y, X input, output, and state spaces, finite i(1) - 41) = 0 and 40)= 0 , respectively. The operator A is densely defined on X,and is the infinitesimal generator of a strongly continuous semigroup } eAt on X, the operator B maps U to X and C maps X to Y. Ba = 6(x - 1) (- :)a, and Ck) = 41) + i(1). Despite thenatural appearance of this structureit is very *restrictive. As Baras and Brockett [ 51 observe (see also [31) As another example, consider this same transmission line in 'the impulse response function C 8'Bu applied to input vector the impedanceformalism. Theinput is then i(1) while the u is continuous in t. This property does not holdeven for loss- output is 41). The state-space equations are again (2.1), but less transmission lines; to cite the simplest example a uniform this time transmission line with c = I = 1, unit length, and with one end short-circuitedhas impulse response equalthe diracdelta 6(t - 2) when treated in the scattering formalism (reference line c = I = 1). Thus a more elaborate definition of system is required in order to avoid vacuousnw of the theory. Before Ba=6(x - 1) (:)a, and C(i)= v(1). presenting our definition of "system" we give as an example the statespace description of a lossless transmission line. This Let us make an observation about the examples; while A is illustrates the difficulties just described and gives one some- an unbounded operator on X as we expected, B is a distribu- thing concrete to hold onto during abstract discussions. The tion and C is an unbounded operator. Thus the situation is example is like the one in[ 321. partly operator theoretic and partly distributional. This type Considera transmission linewith capacitance c(x), induc- of behavior occurs frequently and to deal with it we shall use a tance I(x), nonvanishing and resistance dx), conductanceg(x) "rigged Hilbert space."' Such spaces should be familiar to fol- at the point x in [ 1 1. The energy of a current-voltage dis- 0, lowers of the Lions school; a general reference is [25]. tribution in the line is (i) If H and H1 are two Hilbert spaces with H 3 H1, then a third Hilbert space H' can be defined as the set of all continu- ous linear functionals on H1. Since to each x in H we may as- sign the linear function I on H1 defined by I(y)= (y,x) for and HE will denotethe Hilbertspace of all (i) withfinite all y in H1,the space H is regarded as being imbedded in H'. energy. The line is short-circuited at x = 0. To study this line If M is an operator onH whoce adjoint M* maps H1 into itself, we connect the x = 1 end to a 'lossless reference line having thzn it induces an operator M on H' according to the formulz c = I = 1 and send signals down it toward x = 1. A signal mov- [MI] (y)= I(M*y) for each I E H' and y E H1. The operatorM ing left [resp. right] is known to have a spatial distribution at each instantof time 'An alterqativeand equivalent setting involves bilinear forms on U X H,. We could use thisformalism throuaout withthe advantage that bilinear forms are more fashionable in current mathematical phys- ics (see [ 55, ch. 81) than rigged Hilbert space. On the other hand, the rigged Hilbert space seemedless encumbered. 148 IEEE,PROCEEDINGS OF THE JANUARY 1976 is just the continuous extension of M to H,because Gl,(y) = line, X is a space of functions 3 (C) is a subspace of fairly (M*y, x) = (y,Mx) = l~~(y)and so it will be denoted M. If B smooth functions and Bu for each u in U is a "rough func- maps Hilbert space U into H',then its adjoint maps H1 to U tion." Moreover, the operator (A - z0)-' is an integral opera- and is given by (u, B*x) = [Bu] (x). Likewise if C: H1+ U its tor which hasa smoothing effect and (A - zo)-'Bu will be adjoint sends U to H' and is given by [C*u] (x) = (u, Cx). Note smooth enough to lie in 9 (0. This guarantees that B** = B and C** = C. C(A - z0)-' Bu and C(A - z0)-' e(A-zo)tBu exist and conse- Let X,U, Y be Hilbert spaces. The closed operator A maps quently that the output y(t)= Cx(t) + D u(t)is well defied at a dense subspace g(A) of X to X and generates a uniformly least for anydifferentiable inputfunction u(t). This isthe boundedone parametersemigroup denoted eAr. The motivation behind the definition of compatible system. expression There are further properties which systems frequently have. These properties of systems have been isolated by Lax-Phillips (1,Y)A = (x, Y)+ (Ax,AY) as common to all systems they study. is meaningful for pairs of vectors x, y in 9(A)and gives an in- Meromorphic property: ner producton 9 (A) whichmakes it aHilbert space. Let $(A)' be the dual of 9 (A) and we have a rigged structure The operator valued function (zl- A)-' is mero- %(A)'3 X 3 %(A). The same can be said for %(A*). Since A morphic in thewhole complex plane @. generates a nice semigroup A* does, and this implies that if A> 0 the map (A - A)*-' sends X continuously into 9 (A*) A stronger propertyis with I1 IIA* topology. Thus (A - A)-' is well defined on 9 (A*)' Compactness property: and for w in 9 (A*)' the linear functional [(A - A)-' w](x) = w((A - A)*-'x) is continuous in the X topology. By the Riesz There is a number T > 0 so that eTA (Zl- A I-'is a compact operator forRe z > 0. representationtheorem it equals (x, wo)with Wn in X. In other words (A - A)-' w = wo and so (A - A)-' 9 (A*)' C x. The A, B, C, 0 from the transmission line example is a compat- Also eAIt $(A*) C 9(~*)(cf. 115, ch. VI11 1.5, lemma 7(b)l ible system having T = 0 compactness. Note, compactness is and is uniformly [-]A: boundedand so eAr is defined on related to stability in thatany asymptotically stable system 9(A*)' and is uniformly bounded there. with thecompactness property is exponentially stable. A system [A, B, C, Dl will consist of maps, A as above, The basic objects one uses to study asystem are control- B: U+$(A*)', C: !??(C)+ Y, and D:U+ Y where the domain lability, observability, andeither input-output, impuise- of C satisfies %(A)C 9(C) C D(A*)', the operator B is con- response, or frequency-response maps. Inthis context they tinuous, and C is continuous on % (A) in the (, )A topology. are defined as follows: A compatible system satisfies (zol- A)-'B C 9(C) for some The controllability map e: L' (U) + 9 (A*)' is zo with Re zo > 0. Most systems in the author's experience are compatible. Let us pause for a moment to see what type of objects one eu = lm&'B u(t) dt can have for solutions x(t), y(t)to the system equations. For- mally the state x(t) arising from input functionu(t) is where u is the function u(t) in L' (U).

The observability map 9 : L' + 9 (A)is x(t) = eArxo + I' B u(s) ds. (2.2) (Y) If x. E !??(A*)'and u(t) is a continuous U-valued function, qy = [Cdi'] * y(t)dt. then #'xo is in 9 (A*)', Bu(s) is a continuous 9 (A*)' valued function as is eA(t-S)Bu(s) and its integral is in 9 (A*)'; thus The FRF is x(t) is a continuous function with values in !??(A*)'.So much for solutions tothe state equation. To obtainoutputs y(t) D + C(ZI- A)-' B from the states one needs to apply C to x(t). This for systems for Re z > 0. as we have defined them is impossible because of their extreme Thelater object requires muchexplanation. It is well dc generality: C simply cannot be applied to an arbitrary element Tiedfor a compatible system. To check thisnote in 9 (A*)'. The situation is not really so grim, for set x. = O C(zol- A)-'B is well defined and since the operator on the and suppose that u(s) is differentiable. Thenintegration by right-hand side of the "resolvent identity" parts in the formula forx(t) gives for any zo with Re zo > 0 (zl- /I)-'- (zol- A)- =(zo - z) (zol- A)-' (zl- A)--' (2.3) maps D (A*) into 9 (A) we can apply C from the left and B from the right to get a natural definition of C(zl- A)-'B. The FRF is not necessarily defined for an arbitrary system. The The integral is in %(A)C 9(C) and so, the only obstruction to FRF is, however, given in an intuitive sense up to a constant having Cx(t) defined lines in the last two terms. These vanish by whenever support u C (0,t) and as will be discussed later this D + (zO - Z) C(z01- A)-' (zZ- A)-'B. (2.4) insures that the "Hankel" operator for the system exists. In most practical situations,for example, see the transmission For many purposes the indeterminacy is unimportant. This is HELTON: SYSTEMS WITH INFINITE-DIMENSIONAL STATE SPACE 149 because the Hankel operator for a system plays a dominant exactobservability, etc. will be defined as above with 2 re- role in realizability theory and according to some thought is placing C. A one parameter semigroup eAr of operators on X about the only thing oneusually measures anyway; the Hankel is called asymptotically stableif eA'x + 0 for each x in X. operator for a system is determined by its FRF but is inde- Remark 2.1. In the defmition of system we assumed that A pendent of any constant added to the FRF. generated a uniformly bounded semigroup eAr. One could re- Now we discuss Hankel operators. If F( jy) takes values in place this by the assumption that thesemigroup is bounded by oe(U, Y) and is uniformly bounded, then for u in H2(v) the ll&'ll Q ept with p > 0, since a simple scale change A - p con- function F( jy) u(jy)is in L2(-", 00, Y) and can be written verts this to auniformly bounded semigroup. TheFRF (at uniquely as the sum a + b of functions a in H2(Y)and b in least (2.4)) will be defined on the half-plane Re z > p rather p(Y). Define HankelF u 2 a. Note that if F is constant, then than the RHP, and everything else works with analogous modi- HankelF = 0 so two functions differing by a constant have the fication. In fact the main reason for taking p = 0 was to avoid same Hankel operators. If the function F given by (2.4) is uni- carrying the subscript p through the whole paper. formly bounded on RHP, then the Hankel operatorfor the Remark 2.2. A more general but natural definition of sys- system is HankelF. Now let's lookat HankelF in the time tem is possible. Namely, the operators A and D remain as is, domain. Suppose for the moment that the Fourier transform the operator B maps U into % (An*)', while C: 9 (Am)+ Y. 8 of F which in general must be regarded as a distribution &) With this definition everything in the section after a straight- 1s in fact a well-behaved function.Then the Fourier trans- forwardmodification still goes through.To make sense of formed Hankel operator x: L2 [-", 0, U] + L2(0,-, Y)is most formulas one simply requires what in the time domain amountsto repeated integration by parts. For example, in (2.5) just integrateby parts n + m - 2 more times. [Xu] (s) = imE(t + s) u(-'s) ds. B. DiscreteTime If fi is a distribution this definition extends naturallyto give a What is the correct notion of a discrete-time system? Let us map X: c;(-=,0, V)+ ~~(0,m, Y). The formally associ- consider a typical situation-discretizing the transmission line ated with a system is from subsection a. The method we choose is a usual one (cf. [ 10, Appendix C] namely, send in pulses of duration T and read values or averages of the output at timeintervals of length [xu](s) = CeA('+')B u(t) dt T. The state propagation equation for the discrete version of (2.l)is which rigorously is defined on CF[-m, 0, v) through integra- w(n + 1) = &v(n) + &(n) tion by parts where A" = 8" and d2 [xu](r) = C(A - p)-2 e(A-P)('+')B - u(t))dt 1 dt2 B" = I eAq B dq. (2.5 1 and does not require F to be uniformly bounded. The input, output, and statespace are the same as before. For each a(t) put into the continuous time system, the resulting The adjoint system to [A,B, C, Dl is [A*, C*, B*, D*] . The controllability [resp. observability] maps forthe adjoint are state at time T is f eA(T-q)Ba(q) dnand its energy and con- the same as the observability [resp.controllability] maps sequently 11 11; is no greater than .jT la(t)12 dt the energy of a(t). If a(t)is a pulse of height a, then the state is precisely forthe original. Notethat 2 : 9 (A)+L2(0, m, Y) takes x E $(A) into the functionCpx and similarly for e. Thus h;thus B" is a bounded operator. Since A is dissipative eA" has norm Q 1. Now we turn to C. There are several reason- able ways to define it. A common one is set 2; = C. Thp 2; is 2*Cu = C&('+')B u(t)dt an unbounded operator on HE. Another is to have E(:) give the average value of i - u over some small interval (0, 1). If T 1- is small and 0 is chosen appropriately this can be approximated which up to u(r) + u(-r) equals the Hankel Operator for the in practice by taking a time average-of $1, t)- u( 1, t) over a system. sampleinterval of length T. Here C is a boundedoperator. While the controllability map sends L' (O,m, v) to 9 (A*)' The first methodthough seemingly simple is incompatible integration by parts shows that functions m CF(0, m, v) map with energy considerations, while the second is consistent with into X. If the set of vectors so obtained is dense in X the sys- energy considerations. tem is called approximately controllable or often just control- The second type of discretesystem is treated in (281 and lable. If in addition e is continuous it is called continuoudy works of Fuhrmann.The first can becompletely analyzed controllable. If maps some subspace of L2 onto all of X, using the same techniques adapted to a setting like we use in then it is exactly controllable. By the open mapping theorem this paper. a pseudoa inverse of anycontinuously exactly control- lablesystem is a boundedoperator. If R is asubspace of 111. LAX-PHILLIPSSCATTERING AND SYSTEMS L2(0, m, v) for which range CIR = range e,then for most pur- poses we can use CIR instead of e. We call such a subspace A. The Lax-Phillips Model controlling for e. Reachability is also used by some authors This section begins with a summary of Lax-Phillips structure instead of the term controllability. Henceforth observability, adaptedsomewhat toour purposes. Chapter 1 of [37] is a 150 PROCEEDINGSJANUARY OF THE IEEE, 1976

boonto aphysical understanding of thisformalism-rather HEATER dryly presented here. INCONING I Let U,Y, X be Hilbert spaces. We define H to be the Hilbert L TRANSNISSION HOT ROD space LINE OUTGO ING H=L2(-=,0,U)@X@L2(0,=,Y) -ll -I TPANSNISSION and T(t) to be a strongly continuous uniformly bounded one LINE parameter semigroup of operators on H which describes how TENPERATURE information in H changes as time goes by. Let T-(t) [resp. HEASURENENT y+(r)] denote translation by t units to the right on L2(-=, PROVERBIAL 0, v) [resp. L2(0, w, Y)]. Denote by S[-,,b] the subspace BLACK BOX L2(-a, 0, U)@X@L2(0,b, Y) of H; by S[X, the sub- Fig. 1. space X @L2(0,-, Y), by s[~,-] the subspace L-J (0, =, Y), etc. Let b] be the orthogonal projection onto S[-a, bl, Pi-,, Ba = ah(r) etc. The semigroup T(t)has the following properties. cx = x(+) (i) T(t)g = T+(t)gon ~'(0,=, Y) T(r)*g= T-(t)*g on L'(-=, 0, v). D = 0. (ii) For a > 0 define Z(t)= P[-,,,] T(t)P[,, -1. Then It is compatible and satisfies the compactness property. To Z(t) and Z(t)* are asymptotically stable. fit the system into a Lax-Phillips model, connect an infinitely (iii) For a > 0 longtransmission line to the heater and another one io the measuring device. Energy is supplied to the heater by sending an incoming electric signal down the input wire, and the mea- surement is converted into an outgoing electric signal which travels out the output wire (see Fig. 1). The space of possible incoming [resp. outgoing] signals is The number a is not critical and in practice a is chosen in an arbitrary fashion. The choice of a effects the scattering matrix but in a trivial fashion, namely, Sb(z)= e(b-4zS,(z). {(-f) with f in L2(-=, 0, R ') A further restriction [38, eq. (1.2)1 which all Lax-Phillips 1 models satisfy is that of inertness: resp.{(:)withfinL2(0,-,R') 1. T+(t)S[o,-l = T(t)SIo,-1 is orthogonal to range T(t)S[,,a Thus the incoming and outgoing spaces are naturally identifi- T-(t)*S[-,, 0~ = T(rF S[--, 0~ is orthogonal to range able withL2 spacesandXisL2(-=,0,R')@X@L2(0,=,R'). At aparticular time in theoperation of thiscontraption a power signal will be coming in, something will be going on in An inert modelwhich satisfies (ii) and (iii) for aparticular X,and a measurement signal will be going out, in other words, a > 0 will satisfy them for any a > 0. We call the setup just one would observe avector in X. Later t timeunits one described a Lax-Phillips model (with reference a). would observe a vector h'. The map sending h to h' defined on all X by this process is T(t)the time evolutionsemigroup. We B.Example could write it down explicitly but do not toavoid redundancy The transmission line example fits nicely in the Lax-Phillips with the next section which gives a formula for doing this in framework and in fact is the typeof situation which motivated general. The Lax-Phillips model here is inert. Clearly any in- their theory. By reading [37, ch. 11 one can see how to do coming signal or state when propagated for t time units has no this example in the form of Section 111-A. However, it is more influence on L2 [ t, =, R '), thus the first condition is satisfied. instructive to do a typical systems example which is somewhat Intuitively the condition says that state and incoming waves bizarre from the Lax-Phillips viewpoint since this underscores cannot have influence in the outgoing space which propagates what is happening ingeneral. faster than theprevailing signal speed. Consider a rod of unit length and temperature distribution x(r, t) which is fixed, x(0, t)= x( 1, r) = 0, at the ends. There C. Getting a Lax-Phillips Model froma System is one heater in the rodwhich upon receipt of input a supplies We just saw how a Lax-Phillips model could be associated to heat in a smooth distribution &(r), and one measuring instru- one particular system; now we show how to dothis in general. ment which reads off the temperature atr = 4. The associated It is done with an abstraction of the preceding construction system is which we call the imbedding construction. Proposition 3.1: If [A, B,C, D] is a system the imbedding construction associates with it a Lax-Phillips model with ar- bitrary reference if and only if [A,B, C, D ] (a) is continuously controllable and observable, Af =-d2f on 9 (A) dx2 (b) has uniformly bounded FRF, (c) has d" and Prasymptotically stable. To describe the imbedding constructionwe need definitions. Let x[o,tl be the function on R' which equals 1 on the intet- HE LTON: SYSTEMS HELTON: WITH INFINITE-DIMENSIONAL STATE SPACE 151

Val [ 0, t] and 0 off of that interval. Define R, : L2(0,00, H)+- following identities L2(0, 00, H) by [Rrf](r) = x[~,rj(r)f(t - T) and denote by m y+(t)K(W) + K(t)Pw=K(f + W) (3.2a) the reversal operator u(r) --f u(-r). The space X in the Lax- Phillips model is obviously LZ(-=,0, U)@ X @L2(0,=, Y) d'rp(w) +P(t) = P(t + w) (3.2b) and we must define operators T(t) on X. Since X is naturally decomposed into three subspaces it is natural to express T(f) Y+(t)p(W) K(t)P(w) p(t) 3-(W) = P(t W) (3.2~) as an operator entried 3 X 3 matrix: which we must verify. Lets check 1 ; its right side is

y+(t)R,Cd" + R,Cd'('+ w). The first term is C&(w- r+ 'I, for t Q r Q w + c and 0 for all

other r, while the second is Cd'('- + w, for 0 < r < t and so bothterms sum to Cd'(f*w-r) for O

Properties of the systemcorrespond toproperties of T(t) as follows: plus System Lax-Phillips model

(a) + (b)+ eA ' uniformly bounded T(r)uniformly bounded restriction (i) no restriction (C) (ii) plus (a) + (4 (iii) restriction inertnessno restriction t-r CJ, e~(r-h)Bf(-X-w) dA, for r E 10, tl. as we now demonstrate. Since IlRtll = 1 = Ilmll, we have IIP(t)ll < IIeII and Ilu(t)ll < (I 2*11 = 119 II for all t. Conversely A change of variables makes the last integral llRrfllLl --f llfll~l as t +- +- and so max, IIK(f)II = 119 11. More- over Ut > range Rt is dense in L2(0, =, V), so max, IIP(t)II = #(r-r-k+w) B f(-X) dX. IICII. Since p(t) is Rr applied tothe input-output map Jwr-r+w max, Ilp(t)ll equals the norm of the input-output map which in turn equals maxRez > 0 IIS(z)II where S(z) is the FRF. This This plus the middle integral is establishes the equivalence surrounding boundedness.

Lax-Phillips property (i) and inertness are built intothe &(t+w-r-A) Bf(-X) dX, for r E [0, t] c.onstruction. Property (ii) is the same as (c).The second cI1+w-* ,.art of property (iii) says pfPxh +P(t)P[-,,olh + 0 and

3-(t)Pi--, 01h --f 0. The last convergence is automatic and the eAr convergence follows from (c). Let us treat P(t)u for which added to the first integral extendsthe formula to [ 0, t + w]. However, this is R,+ J', 8('-')B f(-h) dX as fixed u in Lz[--, 0, VI. Suppose E> 0, pick N so that required. To verify (c) when D not 0, all we must do is IIP[-m,~~u ll~l N is check that 3+(t)RwD +.R,D 3Jw) = R,+ wD,. The opera- tor D factors out to the left, m y-(w) = y+(w)*rn and so the P(t)u = I"+ 6&(f-r)B u(-r) dr problem reduces to verifying y+(t)Rw +Rr3+(w)*=Rt+w and note that the second integral is dominated by IIClle. The fist integral is P('-")x, where x = $: &(N-r)Bu(-r) dr and which is easy. The imbedding construction can be carried out with many so goes to zero as t +- -. Thus continuous controllability plus (i) gives the second part of property (ii): the first part follows variations. One we shall use takes similarly from continuous observability. ;IC=LZ(-m,-Q,U)8X,dLZ(U,m,Y) We still mustshow that T(t) is a semigroup. Upon multi- p&ing T(t)T(w) and equating it to T(t + w) we obtain the with Q > 0; we call it the a-imbedding. 152 PROCEEDINGSJANUARY OF THE IEEE, 1976

D. Getting a system from a Lax-Phillips model a/2): Observe using inertness that K(t)[P[-,,XI x + ,I x] To a Lax-Phillips model for each number a > 0, we associate is orthogonal to range T+(t+ 0/2), in other words, it is identi- the compatible system: cally zero on [t+a/2, -1. Thus K(t)x = ~(t)p+P[~,,]xon [ t + a/2, =o] and if x E $(A) the argument used to imply con- A the infinitesimalgenerator of the semigroup P[-,,,] tinuity of x € 2(A*) at -a implies here that p+P[~,~lx is a TWP[-,,-1, left continuousfunction at a. Thus [K(f)X] (T) = [K(t)p+ * Eu is the linear functional on 9 (A*) satisfying [Eu](x) = P[o,a~~l(~)=Rr~+(r)P[o,~~x),,butlim,tt T+(t-r)PIo,.1x= (u, [P[-,,01x1 (-a)), [P[O,,]xl(a) = cx. Cx [P[o,,l lx(a), for x in ii? (A) + SIII,u~~which we call the Since p, 0, adjoint have the same basic form as K,K, the argu- emitted system. ment above implies that p= p,. Now we turn to p and p,. Theirdifference 6 satisfies (3.3) with g(s) = T-(s). Inertness The first order of business is to clarify these defiitions. In implies that the action of T(t) a function in Lz -a, U) particular we need to establish that E and C are defined for on (-00, is not felt in Lz(a, Y)for 2u time units; thus p(t)= 0 if those x cited. After completing that we prove theemitting 00, and imbedding construction are inverse to each other. t < 2.u. This is also true of ps(t). Therefore, A(t) = 0 for t 2.u and for a t we have f A(t)(r) = 0. the Proposition3.2: The a-imbedding construction applied to < < < 2u lim, r On other hand,inertness implies that 3+(t)*A(t)= 0 which to- the system emitted at a from a Lax-Phillips model gives the gether with (3.3) gives A(t) = 3(t)*A(r s). That is A(t s) original model and conversely. + + . To prove that E is well defined we need to show for x in (r + s) = A(t)(s); so lim, t t A(t)(r)is independent of t and from the above observation it is 0. 9 (A*) the function g Pi-,, ol x is right continuous at -a. Let 4 At this point, we mention an additional example. The trans- PO, p- be positive smooth functions on [-a, 01 with po + p- = mission line example effectively demonstrates the system one 1 and p- 0 outside of [-a, -a/4], The vector x is in ii?(d*) if would use to model many typical one-dimensional scattering and only if P[+,,I [ (T(t)*- I)/t]x converges. Consider situations. While the state operator A varies with the trans- P[-,,,I A(t)p-g =P[-u,n~(A(t)p-)g +P[-,,al (T-(~)*P-)Nt)g mission line one is testing the operators E and C do not; they are intrinsic to the reference line. For scattering in odd dimen- where \(f) = I)/tl. The first term converges to [(T-(t)* - sions the same is true and the input-output operatorsE and C P[-,,,]p-g. Inertness implies that P[-,,,] (IT-(t)*p-)[T(t)* - can be written very explicitly by using directly the formulas I/t] P[~,,lx= 0 for small t, thusthe second term equals of [ 38, sect. 71. Instead of writing out the setup and formulas P[-,,,I (T-(t)*p-)[ T(t)*- Z/t] x which converges since x E explicitly we just give the recipe. Here = Y = Lz(S"-') %(A*). We have shown that limr+.oP[-a,ulA(t)p-g exists U where S"-' = {(x1, * * * , X") : Ixl Iz + * + lxnf = 1). To de- in Lz [-a, 0, Y]. In other words p-g is differentiable in a - fine Eu plug h(x * w, w)= 6-,(x w)u(w) into (7.17). To certain sense. It is well known (and easy to prove, cf. [53, - define C one evaluates jd of (7.11) ata; note (7.1 1) is given in Theorem 1.1 51 ) that this type of differentiability implies terms of (7.8) and (7.4). p-g is continuous. Thus PI-,,o~x is continuous near -a. A similar argument holds for x € %(A)and suffices to show that E. Finer Structure C is well defined. Proofof Proposition3.2: The space Jc arising from the Now that we have shown how systems theory and scattering a-variation on the imbedding constructionfrom one system theory correspond it is natural to examine how basic objects [A, E, C] is ~=L'(-=o,-a, U)@X,@Lz(a,00, Y)and we such as the FRF the controllabilityand observability operators denote by Os, K,, and ps theconstituents of the semigroup compare to basic objects in scattering theory. There are three TJt) in thismodel. The space in the original Lax-Phillips such objects in scattering: the scattering matrix, the forward model can be identified with Jc in the apparent way. (X, % and the backward wave operators.Since systems correspond Lz (-a, 0, U)@ X @ L2(0, a, Y),etc.) and the semigroup T(t) to Lax-Phillips models and any Lax-Phillips model has wave can be written as a 3 X 3 matrix with critical entries denoted and scattering operators these objects can be associated with p, K,p. We shall show that p = Os, etc.; the main tool is any system. The bulk of this section is devoted to comput- Lemma 3.3: Suppose that Z is a vector space and g(s) is a ing them explicitly for a system; we find that they are closely one parameter semigroup of linear operators on Z. If for each related to C, 0, and the FRF. In [281 this business was thor- t, s 2 0 the map 6(t): Z + L2(O,m, Y)satisfies oughly worked out for thediscrete-time case. Before beginning the computation we mention another di- T+(t)6(s) G(t)g(s) = 6(t +s) (3.3) - rection. Certain functions called 'distorted plane waves' play and if for eachfuted z, t the Lz function 6(t)z satisfies a fundamental role in classical scattering theory. Helton and lim, t b [6(t)z](r) = 0,then S(b)z is identically zero on [0,b]. Ralston [32] showed that for the system [A,E, C] naturally Proof: For fixed t,s the function fT+(t)S(s) z is identically associated with the lossless transmission line (as in Section I!) zero on [0, t) and so on that interval 6(t + s)z = -6(t)g(s)z. the state-space valued function (z - A)-' Eu[resp. (z - A)-' Thus 1-im, t [6(t+ s)z] (r) exists and is 0; this is independent C*y] actuallyequals the classical physics incoming[resp. of s so the lemma follows. outgoing] distorted plane wave. (Here u, y must be chosen in Consider 6 = K, - K, since both K, and K satisfy (3.2a), acertain basis.) The (z - d)-'E objects are used repeatedly their difference 6 does also; note(3.2a) is a special case of in studying systems and the author feels it is quite interesting (3.3). For x in ii? (A) the lim, t b [K,(b)X] (r) exists and is cx. that they correspond to such important objects in scattering To complete the proof we need only show that this limit on theory. It seems reasonable clear that this correspondence will K equals C because the result subsequentlyfollows from carry to most situations. One should be justified in thinking Lemma 3.3. To do thislet po,p+ be smoothfunctions on of (z - A)-'&, etc., as asuperposition of distorted plane IO, a] which satisfy po + p+ = 1 and po E 0 outside of [O, waves. HELTON:SYSTEMS WITH INFINITE-DIMENSIONALSTATE SPACE 153

Nowwe computethe wave andscattering operatorsfor a Bg(f')d[ + D. The lower left entry is given system.The basic philosophybehind scattering is to study a situation by comparing it to one where nothing hap- pens,freea situation. Given a Lax-Phillips model as in P(t)pl(t)*g=CR, B:P:('-'))C:R,g(r) dr. Section 111-A the appropriate free model has evolution opera- tor denoted S(t) which is the shift by t units to the right on Uponwriting R, explicitly, making the change of variables L(-m, 00, U). Forthe sake of convenience we shall always t - r -+ f', and sending t +- we obtain P(t)pl(t)*g -+ c /; * take U= Y which is no real restriction since the smaller space B:eA;(*-[)C:g(f') dt which we abbreviate a:. Here s1 is can be enlarged to give two equidimensional spaces which are the transfer operator for [Al, B1,C1, O,] . Finally, the top unitarilyequivalent. If the Lax-Phillips modelhas reference comer entry is a, then let be a map of LZ(-a,a, U)into LZ(-a,0, U)@X @ Lz(O,a, U). Let p denote the map of LZ(-m,m, U)into JC which is the identity on L2(-w, -a, U)and Lz(a, QO,U) and p" on LZ(-a,(I, U). The backward wave operator is the map W- of Lz (-00, m, U)to X defined as

This we now compute. Let [A1, B1, C1, 01 be the system Change variables twice, namely t - s -+ 5 and then t - h + q to whose Lax-Phillips model with reference Q is the free system. obtain Then T(t)p S(t)* written in matrix form is: : ). Pl(t)* Pl(t)* 5--(t)* Thelimitast+ais5,Sl*g. These expressions can be seen in a very simple way because

As t goes to infinity y+(t)*,K (t),and &If go to zero (in the theaction of and 5 11 = 5 :are trivial. Theoperator W- strongoperator topology [55, ch. 6, sect. 11.Thus we see when written with respect to the basis immediately that p has no effect on Furthermore, since W-. LZ(-m, 0, U)@LZ(O, m, U) the Lax-Phillips model for T(t) is inert we have p(t) y-(t)*, P(t)- 3-(t)*,3-(t)pl(t)*, and y-(t)pl(t)*equal zero. What -+L2(-=, 0, U)$1 @LZ(O,=, Y) remains is is

t+- 0 w-=(! i). The main influence of Q is in the middle row of S(t)* and we The classical forward wave operatormakes no sense for most have seen that this row disappears in the limit. The influence lossy situationsbut natural generalizations lim, + 0 T(t)* * of Q on the rest of the calculation is merely in adjusting ranges S(t)* = W+ orits adjoint Wz= lim, + s(t)TO) domake of integration. Suchbookkeeping merely confusesotherwise sense, cf. [ 291 or [38]. The same procedure as above yields illuminatingand straightforward computations.Therefore, that W, in the basis Lz(-m, 0, U)@L2 (0, m, U)+ LZ(-=, we shall henceforth take Q = 0. 0, U)@X @L'(o, m, V)is Now we computethe entries of the matrix. Thebottom right one is W+= (:0 3Q . The classical scattering operator is W,*W- and from ourcom- Above it we have putations is (;*e 9). The entry [Q*Cgl (Q= C I!.. eA6-')Bg(q) dq while the inte- gral expression for 5 has already been given. When these are substituted into the matrix onefinds that the scattering opera- tor is [I: C eACt-')B g(g) dt + D g(f')1 . That is, the scattering operator is g. This completes the com- putation of wave and scattering operators. 154 PROCEEDINGS OF THE IEEE, JANUARY 1976

Now we give a brief scattering to systems dictionary. Lemma 4.2: Projx, 9 (A*) iscontained and is dense in %A:). H erein Lax-Phillips Herein Proof: Write dl*' as a 2 X 2 operator entried matrix with respect to thedecomposition X, @X? of X, applyit to a Df vector (g) in X, and integrate D! 3 (t)or Uo(t) 3(-t)or uo(-t) ''9 - 1 Pf" and P!+' X, TO),20) = P translation representationwith It is straightforward to check that f(x,s) is in D(A*). Clearly U=Y=N Projx- 9(A*)C $(A:) and so p(x,s) P Projx, f(x,s) is in imaginary, jw axis real axis !D(A:). Since e x is norm continuous with limit x at t = 0, we get p(x,s)/s -P x, moreover, if x E $(A,?), then A: p(x,s)/ s + A,?x. Thus p(x,S)/S +X in the 1) ll~,*topology. Iv. THE BASICPROPERTIES OF SYSTEMS The first lemma guarantees that E, once defined on Projx, There are three basic theorems which hold for finite dimen- Y)(A*) is defined on adense subspace of $(A *) while the sional systems: there is a controllable and observable system second says that E, has a continuous extension to $(A:). with a given FRF, two such systems are equivalent in a certain The various properties asystem might have are preserved sense, every 'reasonable' function is the FRF for some system. under his cut down operation. Onecan check ina straight- These threeproperties when suitably interpretedhold in forward way that since eArXc C X,, the operator (z - A,)-' infinite dimensions as is described in the first three parts of equals (z - A)-' restricted to moreover, (z - A,)-' E, = this section.The fourth part shows thatfor meromorphic X,; (z - A)-' E for Re z 0. This is enough to show that the systems the poles of the FRF correspond to eigenvalues of A > function - A)-' (s - A)-' E, which determines the Hankel as aconsequence of Lax-Phillips theory. It also describes C(z operatorfor the original systemequals C,(z - A,)-' (s eigenvalue behavior foranother natural class of systems. - Throughout this section we work only with continuously con- A,)-* B,; thusthe two Hankel operators areequal. If the original system is compatible (z A)-' E is contained in trollable and observable systems. - $(C) as well as X,; thus it is contained in D(C,) = 9(C) n X, and A. Canonical Decomposition of a System the new system is compatible. The two FRF's are clearly the same. If an operator-valued functionis meromorphic or We shall see that iny systemcan be 'cut down' to acon- compact valued its restrictionto an invariant subspace also has trollable observable one having the same Hankel operator (or these properties;thus the meromorphic and compactness FRF if the original system is compatible). Here is the proce- property are preserved. dure. Let X, = cl range c and X0 = cl range 9 for the system The same type of process works withobservability and [A,E, C, Dl. Let X,- and Xb denote the orthogonal comple- projecting [A,E, C,Dl down to X,. One proof relies on the ment of X, and X,, respectively. fact that Q is the controllability map for the adjoint system Now we describe how to restrict the system [A,E, C,Dl to [A*, C*,E*, D*]; the preceding construction applies. Then X, and obtain a system [A,, E,, C,, D,] which acts on X,. one can take adjoints to obtain [A0, CO,Eo, Dl , an observable Since eAf X, C X,, its restriction toX, is a semigroup and con- system. If one combines thesetwo processes, onegets (by sequently has an infinitesimal generatordenoted A, whose first decomposing the system [A,E, C, Dl into controllable domain Y) (A,) is contained in Y) (A). Set D, = D. The operator and uncontrollable parts and then decomposing these systems C, is defined on Y) (C) n X, by C,x = Cx. The operator C, is into obserable and unobservable parts). 11 [la, continuous. B, is first defined as a linear functional on Theorem 4.3: There is a closed subspace X,, of the state projx, 9 (A*)by space X of a system [A, B, C, Dl so that the system [A,,, [Bcul (w)= [Bul (w) E,,, C,,, D]gotten by restricting and projecting the original system to X, (as described above) is controllable and observ- and then extended to w in ?(A:) which is shown to be possi- able and has the same Hankel operator as the original systerk ble by the next twolemmas. The system is compatible if the original one is and the FRF is Lemma 4.1: If u E U,there is a constant K(u), so that the the same forboth. The meromorphic and compactness linear functional Eu satisfies properties are ako preserved. I[Bul (w)l QK(u)IIProjx, wll~ Note: WeE, and C, mightbe regarded formally as restrictionsfollowed by projections A,, is definedby the for each w in Y) (A9. relationship = Projx,, d"tx,, andthis is notthe same Proof: For p < 0 set x&) =C(X[~,~Ie-pf u); then (x&), thing. x>=$,' (Be-"' u(t), &*f x) dt. Since A* generates a uniformly bounded semigroup range A * - p =X,see [33, Theorem E. Two Systems with the Same ERE 2.3.21. If we set x = (A* - ply, the integral becomes [Eu] Thissection describes theextent to whicha system is ({&*-Ps - I)w). Since P)~WII~~-to we get lirn,,, . determined by its Hankel operator or if the system is compati- (xp(s), x)= - [Eu] (w). Thus xp(s) converges weakly to a ble with itsFRF. First it is necessary to discuss pseudo-inverses vector x(=) which is in X, because each xp(s)is. We have the of an operator. Suppose thatM is an unbounded operator with estimate I[Bu] (w)l< Ilx(=)ll IIProjx, (A* - p)wll from which domain 3 fromone Hilbertspace H1 to another Hz, and the lemmafollows. suppose 63 is a subspace of 3, which does not intersect thenull HELTON:SYSTEMS WITH INFINITE-DIMENSIONAL STATE SPACE 155

space of M. We definethe 63 pseudo-inverse of M, written From (4.1), we see that Mi', to be the operator G with range equal to 61 so that GM M eAr = eArM (resp. MG) is the identity operator on63 (resp. MG). The main theorem here is on range andadditionally Theorem 4.4: Suppose that two.co?trollable and observable MA systems [A,B, C,Dl and [A,B, C, Dl have the same Hankel =AM operator. If the two systems are continuously controllable and on SA. Furthermore, Me= e. Thus if u E U, the vectors observable, thenthere is a possibly unbounded closeable operator M with dense range suchthat the 'intertwining'

equations d'f Bu dt 4 b(u, s), PrBu dt 4 b(u, s) S I,' MA=AM MB=B CM=C D=D in range and e satisfyMb(u, s) = d(u, s). Now (x, Mu, s)) + each holdon adense set. One suchoperator M is gotten [Bul (x)Fd (x, b(u, s)) + [hul (x) and so [MBu] (x)= [Bu] * explicitly by (M*x) = [Bu] (x) for x .in SA*. To see that M intertwines C ~=(9;~)*2=&;l and C observe that C eAsMx = 2 *Mx = 2 *x = C eAsx for x E SA. Set s equal to 0 to obtain the result. defmedon range C whenever -63 and 6: are controlling and The 'note' and last line of the theorem are straightforward observing spaces for e, and 2,2,respectively. Note that to prove (cf. [ 28 p. 291 1. Remark 4.5: If thetwo systems in Theorem 4.4 areem- di= [nullC=nulltIlandL= [null2 =null91' bedded in a Lax-Phillips model, then T(f)and T(t)are related by = are one set of spaces for which the theorem applies. Clearly GT(r) T(t)C where C in matrix form is if bothsystems are continuously exactly controllable or observable, then M is bounded and invertible. Proof: The continuous operator 9 * ehf applied to u in Cr[O, -, U] gives the function C dlseAr 1; eArB u(t) dt of s. Since unlimited integration by parts is possible, there is no problemin interchanging operatorswith integration to get C. System Realizability 2 eArCu(s) = (- 1)" C (A p)" B(ePru(r))(") dt. * Jt dits"") - Thissubsection describes how to construct systema When [O,gl containsthe support of u we may substitute with preassigned FRF. An operator-valued function F(z) = u(g t)for u then change variables and get

c(f) - analyticon the RHP is said to converge to 0 inthe

positive direction if lim, -,+* max yllF(x +jy)ll is 0, it con- eA(s+r+g-r) verges to F, if F(z)- F, converges to 0. The function F is Q* eArC C(s> = (- 1)" I,"+s+r unitary inner if F(jy) is a unitary operator for almost all y. Note that if F is analytic on Re z >p and is the FRF of a (A - p)-" B(eP(g-') u(t))(")dt. compatible system C(z - A)-'B, then the modification A -+ This is just the response of the system [A,B, C, Dl to input A - p - E of the system changes the region of analyticity for ep(8-r)u for values >g. Thus by hypothesis we have FRF to the RHP or greater. Thus as far as the realizability problem is concerned we might as well assume that the (pro- Q* @re = g*,Ar e spective) FRF is well behaved on theRHP. Theorem 4.5: If F(z) is a uniformly bounded 6:( U, Y)-valued for u Cr and by continuity it holds for all u L2 [ 0, 00, v] E E . analytic function on the RHP with a limit F(-) in the positive From this, one gets if 6? and are controlling and observing 6: direction, then there is a compatible continuously controllable, for and 9, respectively, then e, 9, and observable system satisfying Il@'II < 1 with F as its FRF, (9-1~)2 * eAr x- - gr e-5-1ax which (i) is exactlycontrollable and observable if andonly if $r x E range e. By setting K = 0, we see that HankelF has closed range. Note suchsystems are always (C&l)* 9 * = (4.1) asymptotically stable. (ii) has 8' and d" asymptoticallystable if F(jy) has holds on range =ea, regardless of which 6: and fl are used. either the form V(jy)G(jy) or G(jy)U(jy) where Uis Define M to be this operator on range e. The construction also a unitaryinner function on the RHP and G is the works for the adjoint system and yields an operator M+ de- boundary value of an analytic and uniformly bounded fied on range 4); the two operatorssatisfy (Mx,y) = (x,M'y) function in theLHP. for x,y in their respectivedomains. Thus M has adensely (iii) has controllabilityand observability operatorswith defied adjoint and by 159, ch. VII, sect. 2, Th. 31 is closable. isometric adjointsif F is unitary inner. Let SA = {oE g(A) :a=& for a uEC"(O,-, U) with Proof: We begin informally and give an intuitive idea of tuin g(A)}. Since any u in Cr(0, -, U) will produce sucha the construction. Assume that F goes to zero in the positive vector, SA is dense. (even inthe II lh topology) in $(A)n direction. Let X(t) denote the shift by t units to the left on range : .likewise MSA is dense. By (4.1) the map M takes SA L2(z-, -, Y). Its infinitesimal generator A" is dldx acting on into g(A) and so M as an unbounded operator on I)(A) is $(A)= {uEL2(--,=, Y): the derivative of u is in L2(0, closable. Likewise M* restricted to a dense subspace SA* of m, Y)}. The adjoint of A" is -d/dx acting on the same domain. $(A*) is closable, etc. The inverse Fourier transform of Fu is not in general a func- 156 PROCEEDINGS OF THE IEEE, JANUARY 1976

tion, but it can be naturally viewed as a linear functional on By construction c = Hank+, so the system [A,B, C, Dl is g(z*)= $(A); namely, if 3 dezotesthe ordinary Fourier exactly controllable if and only if range HankelF is closed. As transform on L2 functions, set [Bul (f)=.E,, F(j[) Sf([) d[ mentionednear the end of SectionI the operator 9*Cm is where f E g(A*) and B is the generalized Fourier transform of unitarily equivalent to HankelF. Thus 119*xll >e Ilxll; conse- F. Note that the integral exists because Sf(t)/(l + [[I2) is in quently, range 9 isclosed and wehave exactobservability. L2(-m, m, Y). A simple estimate (cf. [53, Th. 7.251) shows Conversely, forany continuously exactly controllable and that evey fun_ctionin $@) is continuous, thus we may de:ine observable system 9 * c has closed range; thus, the associated amapC:$(A)+YbyCh=h(O)forhina(&.Thei,B,e Hankel operator has closedrange. An argument like that in constitute a system. If the Fourier transform of F were a con- [28, Remark4.21 demonstratesthat continuous exact con- tinuous function, then [ Z(t)FF]- (x) = 3F(t + x) and so the trollability[resp. observabilityl forces 8' [resp. &'] to be impulseresponse function ?eA' au = [SFu] (t) is 3F, as asymptotically stable. required. To prove (ii), suppose thatthe first factorization holds. Now we turn to aprocedure which works in general and Then range Hankelu 3 HankelF and so if x belong to therange rigorously.Everything is set in the frequency domain. First of the latter it can be written x = Hankeluw for some w in **-d Fourier transform A,B, C: L2(-=, 0, Y). Now E(t)*x = Hankelug(t) where g(t) = PL'[--, 0, y] Z(f)*w. Since g(t)+ 0 as t +m, we had 3-'(z- &)=z-j[ E(t)*x + 0, that is T(t)*x + 0. Note T(t)x = PL'(~,-, y)

m E(f)x automaticallygoes to 0. If the reverse factorization holds, then a similar argument with the adjoint system proves [S-' (g) = SF(jC;)ug([) d[ &I 1, stability. If F(j[) is a unitary operator then it is straightforward to . rM show that HankelF is apartial isometry. An argumentlike that on [28, p. 321 Finishes the theorem. D. Poles and Eigenvalues while this is a system it is not necessarily compatible so we When X is finite dimensional the polesof the FRF fora con- extend C to be defined on g(C) = {g :R--f YI 3-'g has finitely trollable and observable system are located at the eigenvalues many poles in the RHP and [3-'g(E) goes to zero in the posi- of A, in other words, input and output information completely tive direction}.This system is compatible because 3-' (z - determinethe eigenvalues of A. Theimportance ofpole &-'& = (z - jt)-' F(j0 is in 39(C) and its FRF is location and of this fact in fiiite dimensions is well known. For infinitedimensional situations the author is more familiar withscattering theoretic implications than engineering ones (possibly because of background and possibly because efforts in the first areaare moreadvanced). The eigenvaluesof A This is one boundary of integration over the contour {z : lzl = correspond to modeswhich are decaying in the inside of M,Re z > 0) U {z : -M

Toapply their [38, Th. 5.51, we require in additionto required bythe theorem but with W replacing F. Thusit themeromorphic property a hypothesison 'incoming and remains to show that the singularities of W equal those of F. outgoing vectors'. Actually an immediate consequence of this Clearly any singularity of F is a singularity of W. Note that in additionalhypothesis [38, Lemma5.31 suffices. In systems the nonpseudomeromorphic case W is not unitary inner and theory language, it says that if x is an eigenvector of A [resp. a refined version of the Moeller theorem gives the last state- A*],then Cx # 0 [resp. B*x # 01. The statement about C is ment in the theorem. a weaker requirement then observability, while the one on B Now we show that every singularity of W is also a singularity follows from controllability. The conclusion of their theorem of F. Obviously it will depend heavily on the fact that W-' says that the poles of the s-matrix are located at precisely the and G have no common (left) factor whichcould cause the eigenvalues of A. The FRF for a system is just the s-matrixof cancelation of asingularity. This indivisibility property has the Lax-Phillips model in which it is imbedded and so the geometric inteIpretation that thesum S of the subspaces Theorem 4.7 follows. W-'?(Y) and GH2(U)is dense inHZ(Y), because CISbeing A compatible meromorphic system has a meromorphic FRF. (backwards) translation invariantcan bythe Lax-Beurling A more general class of FRF is mathematically reasonable to theorem be written cl S = 82(Y) for 8 a conjugate an_alytic consider,namely, the pseudomerornorphic F(z) whichare inner function. Since S contains both W-' Hz(Y) and GHZ(V) qose having a factorization F(jy)= G(jy) V(jy)or equiva- thefunction 8 divides both W-' and G from-the left and, fently V1 (jy) G1 (jy)with the G's analytically continuable to therefore, 0 = constant or equivalently cl S = H2 (y). Since a bounded function analytic in the LHP and the V's unitary S consists of vector functions of the form inneron the RHP. Theseoccurred in Theorem 4.6, part n = W-'c+Gd (4.2) b. When U and Y are finite dimensional another definition of pseudomeromorphic is possible [ 131.There is a function where cEHZ(U) and d€H2(Y) the space 61& {c+Fd} is H(z) meromorphic and of bounded type in the LHP so that dense in 61' 4 Wgz(Y).We shall show that this can happen lim,,o H(x + jy) = F(jy) exists for almost all y. Thus F is only if W has no more singularities than F. meromorphic except on the jy-axis where it could have bad For the sake of a clear presentation let us do the dim Y = 1 discontinuities. It is in these finitedimensional circumstances case first; then W and G are scalar valued. Let us suppose that where pseudomeromorphic FRF's are particularly interesting W has a singularity at zo while F does not. Note that since the since they turn out to be precisely those function for which singularity set for F is a closed subset of { z : (z(2 l}, zo is at there is anycorrespondence between properties of A and some positive distance from any singularity of F and so F is properties of theFRF. Thus they are thebiggest class of uniformly boundedin some region about zo. A classical functions that one would probably want to use in connection factorization theorem due to Buerling (see [26]) implies that with classical A, B, C, D type systems theory. This point is w = m * p where both m and p are inner, m has no singularity well illustrated by the type of system described in Section IV-A in common with F and p is bounded in a neighborhood of zo. stten byrestricting the shift to acertain subspace. Such a Since W-'F is in Hz(@)and m and F have nocommon restricted shift system, as Fuhrmann calls it, has a structure singularities p-' F is in [email protected] from (4.2) every rnn is in which can be immediately analyzed by standard mathematical Hz(@)which is impossible since the n aredense and m is a theory arising from Moeller's theorem I261, [371, [491. nonconstant function in Hz(@). Theorem 4.8: The singularities of a pseudomeromorphic After setting things up properly this same type of reasoning FRF for a restricted shift system with U and Y finite dimen- will apply to the dim Y = K > 1 case. Suppose there is a zo as sional are equal precisely to the spectrum of A. The spectrum before. Then there will besome entry, say w of the matrix of A with Re z < 0 consists of isolated eigenvalues. function W which is singular at zo . The functions w '1 factor as If the FRF is not pseudomeromorphic, then the spectrumof before into mlpl. Now d"= (2 : (21 < 1). Proof: It suffices to work on the unit disk rather than the RHP since a standard transformation (Caley) connects the two. If F is a unitary inner function then the theorem is essentially Moeller's theorem(cf. [37, ch. 3, sect. 31 or [49,ch. VI, is contained in xh. 4.1 I) and so our proof consists of reducing Theorem $3 tothe unitary inner case. Supposethat F is pseudo- meromorphic.Then rangeHankelF is asubspace of H2(Y) whose orthogonal complement 7 is invariant under multiplica- tion by z. By the Lax-Beurling theoremthere is an 'inner' which implies that for an entry fir ofF the function m;'m;' function W such that 7 = WHz (Y). The pseudomeromorphic * * * mi'p;' . * pk'fll is in B2(Q)and consequently p;' * * * property forces W to have V as a factor and so W is unitary pk'fll is in p(Q).Since 61; is dense in 611, one has w11 = inner; this is equivalent to F being pseudomeromorphic. Set n pl E clR ;. However, thisforces ml pi' * . * pi' to be in G(eie)= W(eie) *F(eie).Observe if fis any function inH2(U), p(Q)which is impossible, since ml is singular at zo and the then Ff E 7' + pz(Y)and so W*Ff is in H2(Y). This forces p;' cannot cancel this singularity. The proof is complete. G to belong to H"(U, Y). We have F = WG where by construc- As an aside it might be informative to observe how Theorem tion cl range HankelF = range Hankelw. This latter property 4.7follows from Theorem 4.8for U, Y finite dimensional implies that there is no nonconstant inner function W1 which (and otherwisewith slightly more work).Let [A,B, C, Dl divides both W* and G fromthe left, i.e., WIK= W* and satisfy the hypothesis of Theorem 4.7; with FRF denoted F. WIL= G for some analytic K and L. By construction eAr is Let A arise from the restricted shift. realization of F. By just the shift restricted to HankelF = Hankelw. The Moeller Section IV-B, there is M with AM =MA. Since the restricted theorem says thatthe spectrum of A behaves precisely as shift system is exactly controllable the inverse of its control- 158 PROCEEDINGSJANUARY OF THE IEEE, 1976 lability operator and therefore M is a bounded operator (by the formula for M). If Ax = k,then AMx = XMx so X isan eigenvalue of A. Thus every pole of F is an eigenvalue of A. The reverse directionfollows in an elementary fashion from (z - A)-’ meromorphic together with the (X - A)w = 0 * Cw f 0 and (x - A *) w’ = 0 *B*w‘ = 0 property. The subsectionconcludes with some historical remarks. A correspondence between the spectrum of A and poles of the FRF was widely observed once the connection between sys- tems theory and the work of Lax-Phillips, Nagy-Foias, Helson, etc. was understood and provided many new techniques, [ 121, [ 181, [ 271.The first thoroughpresentation for the non- Fig. 2. boundary singularities is [ 121 . The pseudomeromorphic func- tions were studied first by mathematicians [ 141. An equiva- where all coefficients are smooth and nonnegative-A(x) is a lent notion called ‘roomy’ was introduced independently by positive definite matrix-and we assume Lu = Au, for Ix I > R. Dewilde who asserted strongly that they were the proper class The objective is to study the coefficients in terms of the scat; for the study of systems [ 111. He also asserted that certain tering matrix. The onedimensional case includes the losslesd ‘roomy’ functions were appropriatefor the electrical design transmission line, e.g., b = 0, u = (l/c), A 1/2. Very recently technique called Darlington synthesis. In [ 13] the equivalence rudimentary Lax-Phillips scattering has been applied to sound of ‘roomy’ with [ 141 was established and they were shown to impinging on an obstacle with a ‘springy’ elastic surface [8]. be precisely the class for which Darlington synthesis is appro- In one dimension this corresponds to a singly infinite constant priate; the result was independently done in Russia by Arov coefficient transmission line connected to a circuit with one [ 1I . Meanwhile Fuhrmann in his work on systems theory de- resister, capacitor and inductor at the finite end. We reiterate voted quite a bit of attention to pseudomeromorphic functions thatmany of the resultswhich follow deal withor extend [22], [23]. Inparticular he concentrated on a subclass of easily to mixtures of the first and second situation (the third functions which are precisely those which can be the FRF of has not been studied yet). In fact for many purposes an ob- an exactlycontrollable and observable system. It was ob- stacle problem can be thought of as a limit of a variable coef- served by Clark and Helton [28, Th. 3c.l plusRemark ficient problem as appropriate coefficients get infiiitely large. 3c.21 that in the scalar discrete-time case this class has a very Also many of the results will most likely have an analog for nice characterization and Fuhrmann extend this to the multi- lossy situations.The preoccupation with losslessness derives variable and certain real time cases [ 191 , [ 2 1I. In the author’s fromthe fact that in classical physics most phenomena are opinion the next necessary step in this direction to fiidlossy energy conserving; a general ‘lossy’ scattering theory did not exactlycontrollable and observable systemswhich arise emerge until 1973 [381. Another point is that although the physically. one dimensional case will be emphasized through the expos: tion this is entirely for concreteness. Practically all structure v. MISCELLANEOUSRESULTS IN SCATTERING THEQRY is greatlysimpler for dimension one than for higherdimen- This section has a different spirit than the rest of the paper sions. Thus it is regarded as trivial and frequently ignored in and is not related to systems per se. It lists some qualitative the works to be described; three-dimensional structure, how- results obtained in scattering theory over the last five years. ever, is a good guide to behavior in all odd dimensions. The hope is that this will benefit engineers primarily by sug- One general class of results concerns how close poles of a gesting structure which they did not know. Most surely the scattering matrix can come to the imaginary axis. results as they stand will not apply without modification to 1) According to Beale [7], suppose that a sequence of ob- someone’s particular problem. However, practically all of the stacles 0, gets closer and closer to enclosing a region R (see results are flexible andnobody knows their ultimategenerality. Fig. 2) and that the Laplacian on R with zero boundary con- The mathematicians involved have not bothered to do rela- ditions has eigenvalues e 1, ez , * * * . Then each corresponding tively straightforward generalizations because motivation was scattering matrix s,(z) has a sequence of poles p:, p:, * * * lacking. Thus if a particular fact listed herein catches a readers such that pi + j e&. This result will almost surely apply to tha fancy he is urged to consult the source article for a precise variable coefficient case, say for example to atransmissioh statement of the result and if the context in thatarticle is not line withvariable capacitance having a huge well. Very crudely quite right he should not dispair since the result will probably speaking it describes an effect that pockets of very high signal hold in many physically reasonable contexts. We emphasize speed have on poles near the imaginary axis. that the listing here is not intended to be complete (orprecise) 2) For which situations does the scattering matrix have all andconcentrates on work done primarilyby people in the of its poles in Re z < -6 < 0 for some6? A stronger question Lax-Phillips branch of scattering theory. for the corresonding system is when is eAr exponentially We begin bydescribing thecommonly studied situations. stable, i.e., 11 < e-“? Beforedescribing general theorems, Thereisobsrucle scattering for the wave equation in 1,3, 5, . * we list some special cases where exponential stability holds: dimensions. Here one has an obstacle and bounces waves off when the obstacle is starlike (anypoint inside the obstacle of the obstacle usually in order to determine its shape. (Di- can be joined to the origin with a line lying insidethe obstacle), mensions 2,4, e - - are highly anomalous, see [41I, and will not when the variable coefficientsrestricted to anyray through be discussed here.) Variable coefficientscattering usually the origin decrease with distance from theorigin. The principle treats an equation behind this can be thought of in terms of geometrical optics. There are high frequency solutions of (5.1) with b = 0 which un=Lu =u(x)V.A(x)Vu - b(x)u (5.1) canbe thought of as propagating along rays-called bichar- HELTON: SYSTEMS WITH INFINITE-DIMENSIONAL STATE SPACE 159 acteristic curves of L. They are the rigorous version of ‘light A is constant near a52 then if ?(x) never equals 1 on a52, the rays’ and bend according to the laws which govern light rays system is observable. Inthe one-dimensional case this says going through a changing medium.For the general variable that if the transmission line in Section I1 is terminated in resis- coefficient case one has exponential decay provided that any tance R and has coefficients constrained as above (more than bicharacteristic curve leaves any finiteregion in a finite amount necessary), then the system associated to it in Section I1 is not of time, [ 37, Appendix 21, [41]. This bicharacteristic condi- observable if and only if R is the characteristic impedance of tion also implies [401 that the poles of s(z) are in the region the line.

Rez<-(a+blogIzI) ACKNOWLEDGMENT The author would like to thank J. V. Ralston for his con- for some a 0. In dimension one rays can never be trapped in > siderable help with Section V. a finite region and so one always has exponential decay. For the obstacle case one fixesa sphere around the obstacle 52, draws a line to the obstacle, optically reflects it, follows it to REFERENCES the obstacleagain, optically reflects it,etc. until the ray D. Z. Arov, “Darlington’s method for dissipative systems,” Sov. crosses the sphere (on its way out). Let be the length of the PhyS., Dok&dy,vol. 16, pp. 954-956, 1972. I A. Adamajan and D. Z. Arov, “On unitary couplings of semiuni- which youtake to be 00 if the path requiresinfinitely tary operators,” Trans. Math. Monogr. AMs,vol. 95, pp. 75-129, many reflections. Let L(n)be the maximum of all lengths I 1970. J. P. Aubin and A. Bensoussan, “Models of representation of possibly obtained thisway. The conjecture (137, p. 155)l is linear time invariant systems in continuous time,” MRSTech. that (a) if L(52) < m, then eAt decays exponentially, (b) if Rep. 1286,Sept. 1973. L(52) = 00, then I(eAr 11 = 1 for all t. In other words one sus- A. V. Balakrishnan, “On the statespace theory of linear systems,” J. Math. AM^. Appl., vol. 14,pp. 371-391, 1966. pects the same principle is behind both the variable coefficient J. Baras and R. Brockett, “H functions and infinite dimensional and the obstacle case. A main difficulty with :he present state realization theory,” SIAM J. Control, J. Baras, R. Brockett, and P. Fuhrmann, “State-space models for of the art is that rays hitting the obstacle tangentially can pro- infinite dimensional systems,” IEEE Trans. Automat..Contr.,vol. duce someincredible mathematicalpathology. Ralston [SO] AC-19, pp. 693-700, Dec. 1974. verified (b) while (a) has been proved for a large class of ob- T. Beale, “Scatteringfrequencies of resonators,” to appear in Commun. Pure Appl. Math. stacles, most recently 1471, [581. I. T. Beale and S. J. Rosencrans, “Acoustic boundary conditions,” If the operator eAt is a compact operator for t sufficiently Bull. AMS,vol. 80,no. 6, pp. 1276-1279, Nov. 1974. large, then eAr is exponentially stable ([37, Cor. 5.11). This M. S. Brodski, “Triangular and Jordan representations of linear operators,” Trans. Math. Monogr. AMs, vol. 32. stronger compactness condition is frequently satisfied, for ex- C. T. Chen, Introduction to Linear Systems Theory. New York: ample, if the obstacle is convex 1481 or if the bicharacteristic Holt, 1970. condition holds [ 53 ] P. Dewilde, “Roomyscattering matrix synthesis,” Univ. Cali- . fornia, Berkeley, Tech. Rep. 1971. Another class of results describes how close poles get to the -, “Input-output description of roomy systems.” ?rigin. Given p > 0 there exists a region N in the complex R. G. Douglas and J. W. Helton, “Innerdilations of analytic [ matrix functions and Darlington synthesis,” Acta Sci. Math., vol. plane containing the interval -n/p, n/p] so that the scattering 34, pp. 61-67,1973. matrix for any obstacle which is contained in the sphere about R. G. Dough, H. S. Shapiro, and A. L. Shields, “Cyclic vectors the origin of radius p has no poles in N;see [ 421 . and invariant subspaces for the backward shift operator,” Ann. Inst. Fourier,vol. 20, pp. 37-76, 1970. A third class of results concerns how the scattering matrix N. Dunford and J. Schwartz, Linear Operators, vol. 1. New changes when the coefficients are changed. This type of work York: Interscience, 1967. goes under the heading of ‘sensitivity analysis’ in engineering A. V. Efiiov and V. P. Potapov, “Jexpandingfunctions and their role in the analytic theory of electrical circuits,” Rum literature. Math. Surveys, Uspeki,vol. 28, no. 1, 1973. 1) Suppose the obstacle is starlike or that a andA decrease P. A. Fuhrmann, “On realization of linear systems and applica- a tions to some questions of stability,” to appear in Math. Syst. monotonically on rays through the origin. Then as decreases Theory. and A increases the poles of s(z) which are located on the real -, “Realization theory in Hilbert space for a class of transfer slide out the real Obstacle case [37]-[42] ;variable functions,” J. Math. AMI. Appl. axis axis. -, “Exactcontrollability and observability andrealization coefficient case [ 501. theory in Hilbert space,” J. Math. AM^. Appl. 2) An eigenvalue of s(jw) equals e ’6k(w) where 6k takes P. A. Fuhrmann, “On sums of Hankel operators,” to appear in l3al values and is called the phase shift. If the obstaclein- Proc. AMS. -, “On generalized Hankel operators induced by sumsand creases in size, or if A, a, and b/a increase, then 6k(w) in- products.” creases. Also under starlikenessand radial monotonicity -, “On Hankel operator ranges, meromorphic pseudocontinua- tions and factorization of operator valued analytic functions,” to assumptions as above, the derivative (d/dw) 6k(w) is positive. appear. For the singly infinite transmissionline (open- or short-cir- -, “Factorization theorems for a class of bounded measurable cuited finite end) this says that for fiied frequency input the functions,” toappear. -, “On series and parallel coupling of a class of infinite dimen- phase of the output is shifted by an amount which increases sional systems.” when the capacitance or inductance of the line decreases. The I. M. Gelfand, Generalised Functions. New York: Academic last fact is classical physics, the first two facts are in 321 Press, pp. 1964, 1968. I . H. Helson, InvariantSubspaces. New York: Academic Press, Quite a different topic is that of controllability and observ- 1964. ability. It has recently come of interest to mathematicians in J. W. Helton, “The characteristic functions of operator theory and electrical networkrealization,” IndioM J., vol. 22, no. 5, this group. Majda [46]studied observability of the system 1972. which arises (in the same way as the transmission line Section -, “Discrete timesystems, operator models,and scattering theory,” I. Funct. Anal., vol. 16, no. 1, pp. 15-38,May 1974. 11) from scattering for (5.1) with A = l/a in the extension of -, “operator theory techniques for distributedsystems,” in an obstacle 52 withboundary conditions &/an +?(x) ut + Conf. Rec. 11th AUertonConf., pp. 131-140, 1973. (N = 0 on the boundary 252 of the obstacle. He proves when -, “A spectral factorization approach to the distributed stable 160 PROCEEDINGS OF THE IEEE, VOL. 64, NO. 1, JANUARY 1976

regulator problem; the algebraic Riccati equation,” to appear in [45] -, “Theoryof operator nodes on Hilbert spaces,” 1971, in SIAM J. Contr. Russian. [31] F. J. Helton and J. W. Helton, “Scattering theory for computers 1461 A. Majda, “Disappearing solutions for the dissipative wave equa- and other nonlinear systems,” to appear. tion, toappear. I321 J. W. Helton and I. V. Ralston, “The fmt variation of the scat- 1471 C. Morawetz, “Decay for solutions of Dirchlet exterior problem tering matrix,” to appear. for thewave equation.” [ 331 E. Hille and R. S. Phillips, “Functional analysis and semigroups,” [48] C. Morawetz and D. Ludwig. AMS Colloq.,vol. 3 1. [49] B. Sz-Nagy and C. Foias, Harmonic A~lysisof Operatorson [ 341 E. Kamen, “On an algebraic theory of systems defined by con- HilbertSpace. Amsterdam, The Netherlands: North Holland, volution operators,” to appear in Math Syst. Theory. 1970. [ 351 -, “Topological module structure of linear continuoustime [SO] J. V. Ralston,“Solutions of the wave equationwith localized systems.” energy,” Commun. Atre Appl. Math., vol. XXII, pp. 807-825, [36] -, “Applicationsof modem algebra to infinitedimensional 1969. systems.” [ 5 1 ] -, “Trapped rays in spherically symmetric media and poles of [37] P.D. Lax and R. S. Phillips, Scattering Theory. New York: the scattering matrix,” Commun. Pure Appl. Math., vol. XXIV, Academic Press, 1967. pp. 571-582,1971. [ 381 -, “Scattering theoryfor dissipative hyperbolicsystems,” J. [S2] -, “Variationof the transmissioncoefficient and comparison Funct. A~l.,vol.14, pp. 172-235, 1973. theorems for thepurely imaginary polar of the scattering matrix,” (391 -, “Decaying modes for the wave equation in the exterior of Commun. Pure Appl. Math.,vol. XXV, pp. 45-61, 1972. an obstacle,” Commun. Pure Appl. Math., vol. XXVII, pp. 737- (531 -, “Local decay of solutions of conservative fmt order hyper- 787,1969. bolic systems in odd dimensional space,” Trans. AMS, vol. 194, [40] -, “A logrithmic bound on the location of the poles of the scat- 1974. tering matrix,” Arch. Znst. Mech. AMI., vol. 40, no. 4, pp. 268- [SS] -, “On the propagationof singularities in solutions ofsyrq 280,1971. metric hyperbolic partial differential equations.” [41] -, “Scattering theory for acoustic equation in an even number [55] M. Reed and B. Simon, Methods of ModernMathematics and of space dimensions,”Indiana J., vol. 22, pp. 101-134, 1972. Physics: Fumti0~1Analysis. New York: Academic Press, 1972. [42] -, “On the scatteringfrequencies of the Laplace operator for [56] F. Riesz and B. Sz. Nagy, Functional AMJYS~S.New York: exterior domains.” Ungar, 1955. (431 J. L. Lions, OptimalControl ofSystems Governed by Partial [57] W. Rudin, FunctionalAnalysis. New York:McGrawNill, 1973. Differential Equations. New York: Springer, 1971. [ 581 W. Strauss, “Dispersion of waves vanishing on the boundaryof an [44] M. S. Livsic, “Operators oscillations, waves, open systems,” AMS exterior domain,” to appear. Trans. Monogr., vol. 34, 1973. [ 591 Yosida, Functional Analysis. New York: Springer, 1966.

Invariant Subspace Methods in Linear Multivariable=DistributedSystems and Lumped- Distributed Network Synthesis

Abstmct-Linea multivariable-dhiuted systems and synthesis These factorizations play an essential role in the development and show problems for lumpeddistributed networks are analyzed. The methods that properties of Hardy functions are of fundamental importance for used center around the invariant subspace theory of Helson-Lax and this class of distributed systems as properties of rational functions se the theory of vectorial Hardy functions. State-space and transfer func- for lumped systems. tion models are studied and their relations analyzed. We single out a class of systems andnetworks with nonrational transfer functions INTRODUCTION (scattering matrices), for which several of the well-known results for lumped systems and networks aregeneralized. In particular we develop REQUENCY DOMAIN methods in hmped-multivariable the relations between singuhrities of transfer functions and ‘‘natural systems have been developed in the last few years for the modes” of the systems, a degree theory for inf~tedimensionallinear analysis and design of control systems [ 1 1. These systems and a synthesis via lossless embedding of the scattering matrix. IF methods provided a clear understanding of the interrelations FinaUy coprime factorizations for this class of systems are developed between statespace and transfer function models for systems and networks and proved to be extremely useful in practical Manuscript received April 1, 1975; revised August 1, 1975. The work design applications [ 821. of J. S. Baras was supported in part by a General Research Board Award Recently, several researchers, Baras [2] -[8], Brockett [31- from the University of Maryland and the work of P. Dewilde was sup- ported in part by the Belgian National Fundfor Scientific Research [4], Dewilde [9]-[16],Fuhrmann f171-[23], Helton [24] (NFWO). and [25], have been investigating a similar approach to the I. S. Baras is with the Electrical Engineering Department, University analysis and synthesis of distributed systemsand networks. of Maryland, College Park, MD 20742. P. Dewilde is withthe Departement Elektrotechniek, Katholieke This theory applies to situations where energy considerations Universiteit Leuven, Heverlee, Belgium. provide the setting of a Hilbert space for the state space of the 1436 SEPTEMBER IEEE,PROCEEDINGS OF THE 1976

Comments on “Systems with Infinite-Dimensionai REFERENCES State Space: The HilbertSpace Approach” [ 11 R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, vol. 111. Reading, PA: Addison-Wesley, 1965. [2] J. B. Keller and D. W. McLaughlin, “The Feynman integral,” JAMES A. DYER Amer. Math. Monthly, vol. 82, pp. 451-465, May 1975. 131 H. J. White and S. Tauber, Systems Analy&, Philadelphia: W. B. Professor Helton has written a very interesting expository paper’ on Saunders, 1969. the applications of Hilbert space techniques to systems having infiite- dimensional state spaces. I find it, however, to be rather surprising that auantum mechanics. which is the best example in all mathematical Reply2 by J. W. Helfon physik of the applications of Hilbext space t&hniques to the study of infiitedimensional systems, is not mentioned. It is pleasing to hear that my article seemed natural enough to be Not only does quantum mechanics provide excellent examples of termed expository, since sections II, III, and IV were new. Professor infiitedimensional systems, the language currently used by many Dyer gives a nice discussion of how basic quantum mechanics itself can physicists to describe a quantum system is the same as that used by a be described as an infinitedimensional system. Quantum mechanics systems engineer to descriie an engineeringsystem from thestate was indeed not mentioned in the paper and so here are some comments viewpoint. To illustrate this, consider a singleparticle quantum me on the scattering theory aspects of it. chanical system. Such a system is completely descri-bed by a probabil- The paper introduces a particular definition of infinitedimensional A, B, C, D-type linear systems and in Section 111 proves thatit is ity amplitude function 6 (x, y,z, f). For fixed r, (f ) = (x, y, z, I)is required to be an element of the unit sphere of the Hilbert space equivalent to Lax-PWips scattering. Now their theory contains L2(R3). For fmed r, this function is usually called the state of the quantum mechanical scattering, although the examples they empha- system even by physicists. The propagation in time of the state is given size are classical. This correspondence is explained in Chapter VI, by Schriidinger’s equation Section 4, of their book. Roughly, to a Schrodinger equation scatter- ing problem with s-matrix S there is associated a scattering problem for a wave equation with scattering matrix r. men S(Z) = r (&I. Functions with branch cuts are not pseudomeromorphic in the sense of Section IV-D of the paper and so S(z) is not pseudomeromorphic. @ (0) = eo Also note that most results demiin the exposition, Section V, hold not only for classical but for quantum mechanical scattering. We where i = fi, h is (2n)‘l times Planck’s constant, and X known as should emphasize that when wesay quantum mechanical scattering the Hamiltonian operator, is a self-ad‘oint unbounded linear operator we automatically mean very special input and output operators B, C defined on a dense subspace of Lz(R1 ). It will be observed that this and so these comments do not address many systems based on equation has the form of the usual state propagation equation of linear Schrodinger’s equation. systems theory. It will be observed from this equation also that quantum mechanics concerns itself only with zero-input solutions. The quantities which a systems engineer would call outputs of this system ’Manuscript received A ril7, 1976. ’J. W. Helton is with tge Department of Mathematics, University of would be called observables by a quantum physicist. An observable y California at San Diego, LaJolla, CA. of a singleparticle system is related to the state of the system by the equation Comments on ‘‘Geometric progression Ladder Networks” Y = (e, I: e) RC L. GRUNER where (, ) denotes the inner product on L2(R3) and L is a linear self- adjoint unbounded operator defmed on a dense subspace of L2(R3). Aktmt-It is ahown that a large class of nonrecunent RC Wer For example, if K is the Hamiltonian operator then the observable networks including, as a special case, the geometric ladder networks E = (e, K I$) is the energy of the system. Again, one finds here a discussed by Bha-uyya and Swmy an be represented as an standard type of output equation for a systemhaving an infinite- equivalent RCG ladder network. dimensional state space and a onedimensional output. An excenent elementary introduction to quantum mechanics from the state view- Bhattacharyya and Swamy’s [l]conjecture that other nonrecurrent pointisgivenin [l]. ladder networks may be represented in terms of recurrent ladder net- In the Feynman integral approach to quantum mechanics one also works is readily answered with reference to the author’s earlier publica- makes use of the concept of state transition operator. Here one writes tions [ 21 -[ 31. the state@(f) as The application of Kirchhoff‘s law to the nth node of the ladder net- work of Fig. 1 shows that the voltage V(n) satisfies the difference OD-- equation K(t, a,?, 2) ~$0(a,?, 2) dady”d2, @(t)=LLL where J( is an Lz valued function known as the propagator. In fact x is the impuk response of the quantum system and is the solution to (1) ~chriidinger’sequation for the initial state eo e,?,9) = 6 (X - 2) . 6 (y -9)6 (z - 2). SinceSchrddinger’s equation does not have a where for a nonrecurrent RC network the series impedance Z(n) = R(n) Green’s function in the usual mathematical sense, x is determined as and the shunt admittance Y(n)= sC(n). the value of an abstract integral known as a Feynman integral. An In the special case of a recurrent ladder network, Z(n) and Y(n) are elementary introduction to this approach to quantum theory is given invariant with respect to n; in particular, if Z(n) = R, and Y(n)= in [2]. Gu + sC,, then replacing V(n) by V,(n), the solution of the difference Thereason forthe parallelism between the usual formulations of equation (1) is given by quantum mechanics and the present formulations of systems theory is that both subjects have a common parent in classical Hamiltonian mechanics. These origins are discussed in detail in [3], part 11. Since both systems theory and quantum mechanicshave followed inde- pendent but approximately parallel paths since their origins it is pos- ablethat cross-fertilization between the two subjects could prove beneficial to both.

Manuscript received January 30, 1976. The author is with the Department of Mathematics, Iowa State Uni- Manuscript received November1 1, 197 5. versity Ames, IA 50010. The author is with the Department of Electrical Engineering,Monash ‘J.W. Helton, Roc. BEE, vol. 64, pp. 145-160, Jan. 1976. University, Clayton, Victoria, Australia.