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Infinite dimensional systems is now an established area of research. Given the recent trend in systems theory and in applications towards a synthesis of time- and frequency-domain methods, there is a need for an introductory text which treats both state-space and frequency-domain aspects in an integrated fashion. The authors' primary aim is to write an introductory textbook for a course on infinite dimensional linear systems. An important consideration by the authors is that their book should be accessible to graduate engineers and mathematicians with a minimal background in functional analysis. Consequently, all the mathematical background is summarized in an extensive appendix. For the majority of students, this would be their only acquaintance with infinite dimensional systems.



1. Introduction

In order to motivate the usefulness of developing a theory for linear infinite-dimensional systems, we present some simple examples of control problems that arise for delay and distributed parameter (those described by partial differential equations) systems. These two special classes of infinite-dimensional systems occur most frequently in the applications.
Ruth F. Curtain, Hans Zwart

2. Semigroup Theory

The examples of infinite-dimensional systems we introduced in Chapter 1 were either partial or delay differential equations. The theme of this book is to describe them through an abstract formulation of the type
$$\begin{array}{*{20}{c}}{\dot z\left( t \right) = Az\left( t \right) + Bu\left( t \right),{\text{ t}} \geqslant {\text{0,}}}&{z\left( 0 \right) = {z_0},}\end{array}$$
on a separable complex Hilbert space Z to enable us to present a unified treatment of these and finite-dimensional systems. Let us first consider a simple example.
Ruth F. Curtain, Hans Zwart

3. The Cauchy Problem

In Theorem 2.1.10 we saw that if A is the infinitesimal generator of a C0-semigroup T(t), the solution of the abstract homogeneous Cauchy initial value problem
$$\begin{array}{*{20}{c}}{\dot z\left( t \right) = Az\left( t \right),}&{t \geqslant 0,}&{z\left( 0 \right) = {z_0} \in D\left( A \right)}\end{array}$$
is given by
$$z\left( t \right) = T\left( t \right){z_0}.$$
Ruth F. Curtain, Hans Zwart

4. Inputs and Outputs

In this chapter, we shall consider the following class of infinite-dimensional systems with input u and output y:
$$\dot z\left( t \right) = Az\left( t \right) + Bu\left( t \right),\,t \geqslant 0,\,z\left( 0 \right) = {z_0},$$
$$y\left( t \right) = Cz\left( t \right) + Du\left( t \right).$$
Ruth F. Curtain, Hans Zwart

5. Stability, Stabilizability, and Detectability

One of the most important aspects of systems theory is that of stability and the design of feedback controls to stabilize or to enhance stability. In this chapter, by stability we mean exponential stability.
Ruth F. Curtain, Hans Zwart

6. Linear Quadratic Optimal Control

In this section, we shall consider a control problem for the state linear system Σ (A, B, C), where Z, U, and Y are separable Hilbert spaces, A is the infinitesimal generator of a C0-semigroup T(t) on Z, BL(U, Z), and CL(Z, Y). In contrast with the previous chapters, we shall consider the time interval (t0, te] instead of the interval [0, τ]. We recall that the state and the output trajectories of the state linear system are given by
$$\begin{gathered}\begin{array}{*{20}{c}}{z\left( t \right)}& = &{T\left( {t - {t_0}} \right){z_0} + \int\limits_{{t_0}}^t {T\left( {t - s} \right)Bu\left( s \right)} ds,}\end{array} \hfill \\\begin{array}{*{20}{c}}{y\left( t \right)}& = &{Cz\left( t \right),}\end{array} \hfill \\\end{gathered}$$
where z0Z is the initial condition. We associate the following cost functional with the trajectories (6.1)
$$J\left( {{z_0};{t_0},{t_e},u} \right) = \left\langle {z\left( {{t_e}} \right),Mz\left( {{t_e}} \right)} \right\rangle + \int\limits_{{t_0}}^{{t_e}} {\left\langle {y\left( s \right),y\left( s \right)} \right\rangle } + \left\langle {u\left( s \right),Ru\left( s \right)} \right\rangle ds,$$
where z(t) is given by (6.1) and \(u{\text{ }} \in {\text{ }}{L_2}\left( {\left[ {{t_0},{\text{ }}{t_e}} \right];U} \right).\). Furthermore, ML(Z) is self-adjoint and nonnegative, RL(U) is coercive, that is, R is self-adjoint, and R ≥ ε I for some ε > 0 (see A.3.71).
Ruth F. Curtain, Hans Zwart

7. Frequency-Domain Descriptions

In Definition 4.3.5, we introduced the notion of a transfer function for a state linear system Σ(A, B, C. D) and showed that it was equal to D + C(sIA)-1 B. In this section, we study the input-output relationship directly in the frequency domain without reference to any state-space descriptions. More specifically, we suppose that we have a scalar input function of time u: \(0,\infty ) \mapsto\) ℂ and a scalar output function of time y: \(0,\infty ) \mapsto \) ℂ, which arc Laplace transformable and we suppose that their Laplace transforms û(.) and ŷ(.) are related by
$$\hat y\left( s \right) = g\left( s \right)\hat u\left( s \right),$$
where g(s) is an irrational function of the complex variable s. We call the g(s) the transfer function.
Ruth F. Curtain, Hans Zwart

8. Hankel Operators and the Nehari Problem

The Nehari problem is naturally formulated in frequency-domain terms: given a matrix-valued function \(G \in {L_\infty }(( - J\infty ,J\infty );{\mathbb{C}^{kxm}})\), find the distance of G from the antistable matrix-valued functions K such that \(K( - s) \in {H_\infty }({C^{kxm}});\); that is, find
$$\begin{array}{*{20}{c}}{\inf } \\{K\left( { - s} \right) \in {H_\infty }\left( {{C^{kxm}}} \right)}\end{array}\begin{array}{*{20}{c}}{{{\left\| {G + K} \right\|}_\infty }: = } \\{}\end{array}\begin{array}{*{20}{c}}{\inf } \\ {K\left( { - s} \right) \in {H_\infty }\left( {{C^{kxm}}} \right)}\end{array}\begin{array}{*{20}{c}}{ess{\text{ sup}}} \\{\omega \in R}\end{array}\begin{array}{*{20}{c}}{\left\| {G\left( {jw} \right) + K\left( {j\omega } \right)} \right\|.} \\{}\end{array}$$
Ruth F. Curtain, Hans Zwart

9. Robust Finite-Dimensional Controller Synthesis

In Chapter 7, we saw that the algebra \(M\hat B\left( \beta \right) \) of infinite-dimensional transfer matrices is large enough to contain all β-exponentially stabilizable and detectable state linear systems with finite-rank inputs and outputs and many systems with unbounded input and output operators. In this chapter, we shall develop a theory for robust controllers for transfer matrices in \(M\hat B\left( \beta \right) \), using the mathematical structure outlined in the last two chapters. First, we define the appropriate stability concepts for transfer matrices.
Ruth F. Curtain, Hans Zwart


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