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This paper deals with linear shift-invariant distributed systems. By this we mean systems described by constant coefficient linear partial differential equations. We define dissipativity with respect to a quadratic differential form, i.e., a quadratic functional in the system variables and their partial derivatives. The main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate. The proof of this result involves the construction of the dissipation rate. We show that this problem can be reduced to Hilbert's 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions.

This paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems. In many applications, we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables. The obvious cases where such functionals are important are in Lyapunov theory and in LQ and $H_{\infty}$ optimal control. With some exceptions, these theories have almost invariably concentrated on first order models and state representations. In this paper, we develop a theory for linear time-invariant differential systems and quadratic functionals. We argue that in the context of systems described by one-variable polynomial matrices, the appropriate tool to express quadratic functionals of the system variables are two-variable polynomial matrices. The main achievement of this paper is a description of the interaction of one- and two-variable polynomial matrices for the analysis of functionals and for the application of higher order Lyapunov functionals.

Modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential-algebraic equations. As such, state-space models are not a natural starting point for modeling, while they have utmost importance in the simulation and control phase. The paper addresses the problem of computing state variables for systems of linear differential-algebraic equations of various forms. The point of view from which the problem is considered is the behavioral one, as put forward in [J. C. Willems, Automatica J. IFAC, 22 (1986), pp. 561--580; Dynamics Reported, 2 (1989), pp. 171--269; IEEE Trans. Automat. Control, 36 (1991), pp. 259--294].

The symmetries of dynamical systems in a special class of dynamical systems referred to as L/sup q/ are studied. A symmetry is induced by a transformation group, the basic idea being that there is a group of transformations mapping one dynamical system Sigma in L/sup q/ into another. If this transformation leaves the behavior invariant, then Sigma is called symmetric. A necessary and sufficient condition for Sigma to exhibit a type of symmetry called v-symmetry is obtained, and an example showing a canonical form for symmetric systems is given. >

This paper is devoted to an investigation of controllability and almost controllability of $l^2 $-systems. These concepts are defined in terms of the possibility of steering one system trajectory to another. It is proved that a controllable $l^2 $-system always has finite memory. The main result on almost controllability states that this is equivalent to the existence of a scattering representation. The paper ends with an investigation of the relation of almost controllability and state representations.

In this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system. The problem is attacked using the geometric approach through the concepts of $A(\bmod \mathcal{B})$-invariant and controllability subspaces and their duals, $ A \mid K$-invariant and complementary observability subspaces. The solution of this problem has an interesting structure consisting of a feedback processor which decomposes into (i) a disturbance decoupling loop; (ii) a disturbance input stabilization or pole placement loop, and (iii) a controlled output stabilization or pole placement loop.

1. Introduction. A number of interesting stability criteria for feedback systems have recently appeared in the control theory literature. The procedures used in proving these criteria can roughly be divided into three classes; the first based on Popov-like methods, the second using Lyapunov theory with Lyapunov functions derived from spectral factorizations or Riccati-type algebraic matrix equations, and the third treating the stability problem from a functional analysis point of view. Each of these methods has relative merits, e.g., the Lyapunov methods seem to be the only ones which allow us to obtain an estimate of the domain of attraction in the case of nonglobal stability. However, the method based on functional analysis appears to be the more satisfactory one, in view of the essential simplicity, of the intuitive nature of the results (loop gain less than one, passivity conditions), and of the fact that it unifies the various criteria (as, e.g., the circle criterion and the Popov criterion). It therefore deserves more investigation and exposition than it has thus far been given. A peculiarity of this method, as presently employed, is that most of the analysis and estimates have to be made on extended spaces which, although derived from normed spaces, are themselves not normed. This entails in general rather cumbersome mathematical manipulations. One however suspects the