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We use here the notation of Chapter 3 in Part IV. We assume that $$ (\mathcal{H}\mathcal{P})\infty \left\{ \begin{gathered} ({\text{i}}){\text{ }}A{\text{ generates an analytic semigroup e}}^{tA} {\text{ in }}H \hfill \\ {\text{ of type }}\omega _{\text{0}} {\text{ and }}\lambda _0 {\text{ is a real number in }}\rho {\text{(}}A{\text{)such}} \hfill \\ {\text{ that }}\omega _{\text{0}} < \lambda _0 , \hfill \\ {\text{(ii) }}E \in \mathcal{L}(U;H), \hfill \\ {\text{(iii) }}\forall T {\text{ > 0, }}\exists K_T {\text{ > 0 such that}} \hfill \\ {\text{ }}\int_{\text{0}}^t {{\text{}}E{\text{*}}A{\text{*}}e^{sA{\text{*}}} x^2 ds \leqslant K_T^2 x^2 } ,{\text{ }}\forall x \in D(A*),t \geqslant 0, \hfill \\ {\text{(iv) }}C \in \mathcal{L}(H;Y). \hfill \\ \end{gathered} \right. $$ Clearly, if $$ (\mathcal{H}\mathcal{H})\infty $$ hold, then the hypotheses $$ (\mathcal{H}\mathcal{H}) $$ of Chapter 3 in Part IV are fulfilled with P0 = 0. We want to minimize the cost function: (1.1)$$ J_\infty (u) = \int_0^\infty {\{ Cx(s)^2 + u(s)^2 \} ds,} $$ over all controls u ∈ L2(0,∞;U) subject to the equation constraint (1.2)$$ \begin{array}{*{20}c} {x(t) = e^{tA} x_0 + G(u)(s),} \\ {G(u)(s) = (\lambda _0 - A)\int_0^t {e^{(t - s)A} Eu(s)ds.} } \\ \end{array} $$ Moreover, x0 ∈ H and u ∈ L2(0,∞;U). We recall that by Proposition 3.1 in Chapter 1 in Part II, x ∈ C([0, T];H) for all ∈ L2(0, T;U); more precisely $$ G \in \mathcal{L}(L^2 (0,T;U);C([0,T];H)),{\text{ }}\forall T > 0. $$ [ABSTRACT FROM AUTHOR]

As in Chapter 2 of Part IV we consider a dynamical system governed by the following equation: $$ \left\{ \begin{gathered} x'(t) = Ax(t) + (\lambda _0 - A)Du(t),{\text{ }}t \geqslant 0, \hfill \\ x(0) = x_0 , \hfill \\ \end{gathered} \right. $$ or equivalently (1.1)$$ x(t) = e^{tA} x_0 + (\lambda _0 - A)\int_0^t {Du(s)ds,} $$ where x0 ∈ H and u ∈ L2(0,∞;U). We assume that $$ (\mathcal{H}\mathcal{P})\infty \left\{ \begin{gathered} ({\text{i}}){\text{ }}A{\text{ generates an analytic semigroup e}}^{tA} {\text{of type }}\omega _{\text{0}} \hfill \\ {\text{ and }}\lambda _0 {\text{ is a real number in }}\rho {\text{(}}A{\text{)such that }}\omega _{\text{0}} < \lambda _0 , \hfill \\ {\text{(ii) }}\exists \alpha \in {\text{]0,1[ such that }}D \in \mathcal{L}(U;D([\lambda _0 - A]^\alpha )), \hfill \\ {\text{(iii) }}C \in \mathcal{L}(H;Y), \hfill \\ \end{gathered} \right. $$ Clearly, if hypotheses $$ (\mathcal{H}\mathcal{P})_\infty $$ hold, then the hypotheses $$ (\mathcal{H}\mathcal{P}) $$ of Chapter 2 of Part IV are fulfilled with P0 = 0. If α ≤ 1/2, we will choose once and for all a number β belonging to ]1 − α/2, 1 − α/2[. We want to minimize the cost function: (1.2)$$ J_\infty (u) = \int_0^\infty {\{ Cx(s)^2 + u(s)^2 \} ds} $$ over all controls u ∈ L2(0,∞;U) subject to the differential equation constraint (1.1). We say that the control u ∈ L2(0,∞;U) is admissible if J∞(u) < ∞. The definitions of optimal control, optimal state, and optimal pair are the same as in Chapter 1. When, for any x0 ∈ H, an admissible control exists, we say that (A,AD) is C-stabilizable. [ABSTRACT FROM AUTHOR]

As in Chapter 1 (Part IV) we consider a dynamical system governed by the following state equation: 1.1$$ \left\{ \begin{gathered} x'(t) = Ax(t) + Bu(t),{\text{ }}t \geqslant 0, \hfill \\ x(0) = x_0 \in H, \hfill \\ \end{gathered} \right. $$ and we use the notation introduced in §1 of that chapter. We assume that $$ (\mathcal{H})\infty \left\{ \begin{gathered} (i){\text{ }}A{\text{ generates a }}C_0 {\text{ semigroup }}e^{tA} {\text{ in }}H{\text{,}} \hfill \\ (ii){\text{ }}B \in \mathcal{L}(U;H), \hfill \\ (iii){\text{ }}C \in \mathcal{L}(U;H). \hfill \\ \end{gathered} \right. $$ Clearly, if the assumptions $$ (\mathcal{H})_\infty $$ hold, then the assumptions $$ (\mathcal{H}) $$ of §1 in Chapter 1 (Part IV) are verified with P0 = 0. [ABSTRACT FROM AUTHOR]

In §2.1 and §2.2 of Chapter 1 of Part I, we have discussed criteria for controllability, and observability for finite dimensional systems and have also shown that when the system is controllable we can transfer the state z0 ∈ H at time t0 to the state z1 ∈ H at time t1 using minimum energy controls. These results were obtained by considering the controllability operator $$ \begin{gathered} L_T :L^2 (0,T;U) \to H \hfill \\ {\text{ }}:u \mapsto \int_0^T {e^{(T - s)A} Bu(s)ds,} \hfill \\ \end{gathered} $$ and its adjoint $$ \begin{gathered} L_T^* :H \to L^2 (0,T;U) \hfill \\ {\text{ }}:y \mapsto B*e^{(T - \cdot )A*} y, \hfill \\ \end{gathered} $$ and studying the relation between the ranges and null spaces of these two operators and by showing that controllability is equivalent to invertibility of LTLT*. As we have remarked (see Remark 2.1, Chapter 1 of Part I) in some sense the same ideas can be used to obtain characterizations of controllability when the spaces U and X are infinite dimensional Hilbert spaces, but at the expense of using much elaborate technical machinery. In this chapter we discuss questions of controllability for parabolic and second-order hyperbolic equations, the plate equation, and Maxwell's equations. [ABSTRACT FROM AUTHOR]

In this chapter our objective is to present a modern approach that provides a unifying framework for a large family of differential and integro-differential systems with delays. This point of view is of paramount importance for the Control Theory, Filtering Theory, and Realization Theory of such systems. The material presented in this chapter is an outgrowth of the lecture notes in French presented at the INRIA School on Representation and Control of Delay Systems in June 1984 by M. C. DPELFOUR [14]. The original material has been restructured, the proofs have been streamlined, and new results have been introduced in §6. [ABSTRACT FROM AUTHOR]

Let S(t) be a strongly continuous semigroup on the Hilbert space H. Let · and (·, ·) be the norm and inner product in H. Denote by A the infinitesimal generator of S(t) and by D(A) its domain. When D(A) is endowed with the graph norm of A(1.1)$$ h_A^2 = h^2 + Ah^2 ,{\text{ }}h \in D(A), $$ it becomes a Hilbert space and (1.2)$$ A:D(A \to H $$ is a continuous linear operator. [ABSTRACT FROM AUTHOR]

A complete treatment of variational differential equations is beyond the scope of this book. We shall mainly quote some results from the books of J. L. LIONS and E. MAGENES [1]. In order to motivate the chosen constructions and models, we give a series of classical examples. We assume that the reader is familiar with Sobolev spaces and their properties. [ABSTRACT FROM AUTHOR]

We shall denote by X a complex Banach space of norm · , and by $$ \mathcal{L}{\text{(}}X{\text{)}} $$ the Banach algebra of all linear continuous mappings T: X → X. The linear space $$ \mathcal{L}{\text{(}}X{\text{)}} $$ is endowed with the usual norm: For any $$ T \in \mathcal{L}{\text{(}}X{\text{)}} $$(1.1)$$ T{\text{ = sup \{ }}Tx{\text{:}}x \in X,x \leqslant 1\} . $$ [ABSTRACT FROM AUTHOR]

In this chapter we broaden the general perspective of the book and consider two-player zero-sum games with linear dynamics and a quadratic utility function over a finite time horizon. They can be viewed as a natural extension of the single player linear quadratic optimal control problem. In particular they illustrate the occurrence of symmetrical solutions to the matrix Riccati differential equation that are not necessarily positive semi-definite. It also connects with the glimpse of H∞-theory given in the previous chapter. [ABSTRACT FROM AUTHOR]

This Part I serves the purpose of an introduction to Parts III to V of the book, which are mainly concerned with the quadratic cost optimal control problem for distributed parameter systems and systems with time delay, both over a finite and an infinite time interval. For problems over a finite time interval, the main tool used is Dynamic Programming, which leads to a Hamilton-Jacobi equation for the value function. For the class of control problems considered, the Hamilton-Jacobi equation can be explicitly solved via the study of an operator Riccati equation. The study of the operator Riccati equation when control is exercised through the boundary in the case of distributed parameter systems or when delays are present in the control in the case of systems with time delay poses additional technical difficulties. The results of Part II are needed to overcome these difficulties. For problems over an infinite time interval, the concepts of controllability and observability (and the weaker concepts of stabilizability and detectability) play an essential role in the development of the theory. [ABSTRACT FROM AUTHOR]

The primary concern of this book1 is the control of linear infinite dimensional systems, that is, systems whose state space is infinite dimensional and its evolution is typically described by a linear partial differential equation, linear functional differential equation or linear integral equation. [ABSTRACT FROM AUTHOR]