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

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Systems with Infinite-Dimensional State Space: the Hilbert Space Approach 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-space systems 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 infmite dimensional 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- dimensionnl rime 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 though there 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 equally foolish 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 intrinsically finite-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 situations while [38] gives a lossy 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 which eliminates 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
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