Controllability and stabilizability of linear systems in hilbert spaces

Project Abstract

The abstract section is as follows Controllability and stabilizability are fundamental concepts in the control theory of linear systems. These concepts are traditionally studied in finite-dimensional spaces but have recently gained attention in the context of infinite-dimensional Hilbert spaces. In this research, we investigate the controllability and stabilizability of linear systems in Hilbert spaces. The study begins by defining the notions of controllability and stabilizability for linear systems defined on Hilbert spaces. We establish conditions under which a system is controllable, meaning that it can be steered from any initial state to any desired state in finite time using appropriate control inputs. Stabilizability, on the other hand, refers to the ability to stabilize the system to a desired equilibrium point using control inputs. We explore the controllability and stabilizability of linear systems by analyzing the properties of the system's dynamics and input-output behavior. By investigating the spectral properties of the system operator and the controllability Gramian, we derive criteria for determining controllability and stabilizability in Hilbert spaces. Furthermore, we examine the role of observability in the controllability and stabilizability of linear systems. Observability is crucial for determining the system's internal state based on the output measurements. We investigate the interplay between observability and controllability/stabilizability and establish conditions under which observability guarantees controllability and stabilizability. Moreover, we study the controllability and stabilizability of linear systems subject to constraints, such as input saturation or output disturbance. We develop control strategies that ensure the system remains controllable and stabilizable despite these constraints, thereby enhancing the robustness of the control system. In summary, this research contributes to the theoretical understanding of controllability and stabilizability of linear systems in Hilbert spaces. By establishing conditions and criteria for analyzing these properties, we provide valuable insights into the design and analysis of control systems operating in infinite-dimensional spaces. The results obtained in this study have implications for various applications, including robotics, aerospace systems, and distributed parameter systems, where control systems operate in continuous-time and infinite-dimensional spaces.

Project Overview

<p> </p><p>Introduction<br>Questions about controllability and stability arise in almost every dynamical system problem.<br>As a result, controllability and stability are one of the most extensively studied subjects in<br>system theory. A departure point of control theory is the dierential equation<br>x_ = f(x; u); x(0) = x0 2 Rn; (1.0.1)<br>with the right hand side depending on a parameter u from a set U Rm. The set U is<br>called the set of control parameters. Controls are of two types: open and closed loops. An<br>open loop control can be basically an arbitrary function u() : [0;+1) ô€€€! U; for which the<br>equation<br>x_ (t) = f(x(t); u(t)); t 0; x(0) = x0 2 Rn; (1.0.2)<br>has a well dened solution.<br>A closed loop control can be identied with a mapping k : Rn ô€€€! U, which may depend<br>on t 0, such that the equation<br>x_ (t) = f(x(t); k(x(t))); t 0; x(0) = x0 2 Rn; (1.0.3)<br>has a well dened solution. The mapping k() is called feedback. Controls are called also<br>strategies or inputs, and the corresponding solutions of (1.0.2) or (1.0.3) are outputs of the<br>system.<br>We do not consider all the system theory concepts here, we will concentrate mainly here<br>on controllability and stability of linear system. To motivate our approach we present a brief<br>survey of nite dimensional theory concepts and results which we will generalize later.<br>Linear System Theory A linear system is described by a (linear) dierential equation<br>x_ (t) = Ax(t) + Bu(t); t 0; x(0) = x0; (1.0.4)<br>where u is the control, x is the state. These functions take their values in linear spaces U<br>and X, respectively. Furthermore, A and B are linear mappings between appropriate spaces.<br>If the spaces U and X are nite dimensional, then the system is called nite dimensional.<br>Otherwise, we have an innite dimensional linear system.<br>1<br>1.1 Finite dimensional linear systems theory<br>Consider the linear nite dimensional control system:<br>x_ (t) = Ax(t) + Bu(t); t 0; x(0) = x0; (1.1.1)<br>where x0; x(t) 2 Rn, u(t) 2 Rm, the linear transformations A : Rn ô€€€! Rn and B : Rm ô€€€!<br>Rn will be identied with the representing matrices. System (1.1.1) is a dierential equation<br>on the state space Rn and the unique solution<br>x(t; x0; u) = eAtx0 +<br>Z t<br>0<br>eA(tô€€€s)Bu(s) ds: (1.1.2)<br>The following concepts of controllability, stability and stablizability are now standard.<br>Controllability One says that a state x0 2 Rn is reachable (or attainable) from x1 in<br>time T; if there exists an open loop control u() such that, for the output x(), one has<br>x(0) = x0; x(T) = x1: If an arbitrary state x0 is reachable from an arbitrary state x1 in a<br>time T; then the system (1.1.1) is said to be controllable.<br>The proposition below gives a formula for a control transferring x0 to x1. In this formula<br>the matrix QT , called the controllability matrix or controllability Gramian, appears:<br>QT =<br>Z t<br>0<br>eA(Tô€€€s)B(s)eA(Tô€€€s)B(s) ds; T &gt; 0;<br>where A is the transpose matrix of A. Then QT is symmetric and nonnegative denite.<br>Proposition 1.1.1. Assume that for some T &gt; 0 the matrix QT is nonsingular. Then for<br>arbitrary x0; x1 2 Rn the control<br>^u(s) = ô€€€BeA(Tô€€€s)Qô€€€1<br>T (eA(Tô€€€s)x1 ô€€€ x0); s 2 [0; T];<br>transfers x0 to x1 at time T:<br>We now formulate algebraic conditions equivalent to controllability of (1.1.1).<br>Theorem 1.1.2. The folowing conditions are equivalent:<br>(i) An arbitrary state x1 2 Rn is attainable from 0.<br>(ii) System (1.1.1) is controllable.<br>(iii) System (1.1.1) is controllable at a given time T &gt; 0:<br>(iv) Matrix QT is nonsingular for some T &gt; 0.<br>(v) Matrix QT s nonsingular for an arbitrary T &gt; 0.<br>(vi) rank</p><p>B : BA : BA2 : : BAnô€€€1</p><p>= n:<br>Condition (vi) is called the Kalman rank condition or the rank condition for short.<br>2<br>Stability An important problem in the control of linear systems is the study of its sability.<br>Consider the stability problem for the linear system<br>x_ (t) = Ax(t); t 0; x(0) = x0 2 Rn and A : Rn ô€€€! Rn: (1.1.3)<br>The system (1.1.3) is said to be assymptotically stable or strongly stable if for arbitrary<br>x0 2 Rn<br>lim<br>t!+1<br>kx(t; x0)k = 0;<br>and exponentially stable or uniformly stable if there exist positive constants M and such<br>that<br>kx(t; x0)k Meô€€€tkx0k; t 0:<br>Instead of saying that (1.1.3) we will often say that the matrix A is stable. We have the<br>following well known result for nite dimensional linear system such as (1.1.3).<br>Theorem 1.1.3. The following condition are equivalent:<br>(i) System (1.1.3) is assymptotically stable.<br>(ii) System (1.1.3) is exponentially stable.<br>(iii) supfRe; 2 (A)g &lt; 0:<br>(vi)<br>R +1<br>0 kx(t; x0)k2 dt &lt; +1:<br>(v) The matrix equation<br>AQ + AQ = ô€€€I (1.1.4)<br>has a nonnegative symmetric solution Q.<br>Equation (1.1.4) is called Lyapunov equation.<br>Stabilizability The central theme within the system theory is to design a control u such<br>that the corresponding state has desired behaviour. A typical example is stabilization:<br>Design an control u such that x(t) ! 0 as t ! +1: A control designed by feedback achieves<br>this goal. Feedback means that we do not calculate u only based on x0; A; and B but also<br>on the current state. This results in a control of the form u(t) = Dx(t); and the question of<br>stabilization turns into a question of nding D.<br>One says that the system (1.1.1) is stablizable if there exists an mn marix D such that<br>A + BD is stable.<br>1.2 Motivation<br>From nite to innite dimensional system To get an appreciation of the technical<br>problems associated with the control of innite dimensional systems, consider the distributed<br>control of wave motion. Let y(x; t) denote the transverse displacement at timt t 0 of<br>3<br>a vibrating medium in an nô€€€dimensional bounded open region<br>with smooth boundary<br>@<br>Rn: Let us assume that<br>y(x; t) = 0; for x 2 @<br>and t 0<br>and let the initial data at time t = 0 be<br>y(x; 0) = y0 and<br>@<br>@t<br>y(x; 0) = y1;<br>for some suciently smooth functions y0 and y1 dened on<br>. Consider the partial dierential<br>equation<br>@2y<br>@t2 (t) + Ay(t) = Bu(t) (1.2.1)<br>or the equivalent rst order system<br>8&gt;&gt;&gt;&lt;<br>&gt;&gt;&gt;:<br>@y<br>@t<br>(t) ô€€€ z(t) = 0<br>@z<br>@t<br>(t) + Ay(t) = Bu(t);<br>(1.2.2)<br>where the operator<br>A := ô€€€<br>Xn<br>i;j=1<br>@<br>@xi</p><p>aij(x)<br>@<br>@xj</p><p>+ a0(x); a0(x) &gt; 0<br>is uniformly elliptic operator with C1ô€€€coecients in<br>, that is aij 2 C1(<br>). Let the<br>controller u() 2 L2(0;1;Rn) and let<br>B(x) :=</p><p>B1(x) Bm(x)</p><p>be an n m matrix with each column in C1(<br>). We consiedr the state space<br>w :=</p><p>y<br>z</p><p>2 W := H1<br>0(<br>)<br>L2(<br>);<br>where H1<br>0(<br>) denotes the usual Sobolev space of L2ô€€€functions, with derivatives, in the<br>distribution sense, belong to L2(<br>) and vanishing at the boundary. Then (1.2.2) can be<br>formally written as<br>dw<br>dt<br>(t) + ~ Aw(t) = ~B u(t); (1.2.3)<br>where<br>A~ =</p><p>0 ô€€€I<br>A 0</p><p>and ~B =</p><p>0<br>B</p><p>:<br>Here ~B2 L(Rm;W) and the operator ~ A is an unbounded opertor with a dense domain<br>D(A~) = [H1<br>0(<br>) H2(<br>)]<br>H1<br>0(<br>) W:<br>4<br>Because of the presence of the unbounded operator A~; it is clear that the concept of a<br>solution for (1.2.3) is not immediate. Intuitively, if we want (1.2.3) to have classical solution,<br>then we would need<br>w0 =</p><p>y0<br>z0</p><p>2 D(A~)<br>and u() 2 C1(<br>): On the other hand, ô€€€A~ generates a strongly continuous semigroup<br>(S(t))t0 on W, i.e., S(t) is a bounded linear opearator on W satisfying<br>(i) S(0) = I.<br>(ii) S(t1 + t2) = S(t1)S(t2); 0 &lt; t1; t2 &lt; +1:<br>(iii) t 7ô€€€! S(t)w : [0;+1) ô€€€! W is continuous for each w 2 W:<br>Then we say that w(t) is a solution of (1.2.3) if it satises<br>w(t) = S(t)w0 +<br>Z t<br>0<br>S(t ô€€€ s)~B u(s) ds; (1.2.4)<br>where the integral is interpreted in the Bochner sense.<br>Now, given a strongly continuous semigroup S(t) on X (say a Banach space), it has a<br>unique innitesimal generator A, a closed unbounded operator that is dened on some dense<br>domain D(A). The converse question, namely, when does A generate a strongly continuous<br>semigroup, is a far more dicult issue and is the content of the Hille-Yosida theorem. Furthermore,<br>there are dierent kinds of semigroups, and each plays an important role in the<br>study of controllability and stability of innite dimensional systems.<br>1.3 Organization of the work<br>The project is divided into two parts. Firstly, a general framework for the concepts of<br>controllability and stability of linear systems in a Hilbert spaces is developed. Secondly, the<br>concepts of controllability and stability of linear system in a Banach space is discussed.<br>Chapter 2<br>In chapter 2 we review some basic concepts of spectral theory and semigroup of linear<br>operators, all the results discussed in this chapter are preliminary and can be found in any<br>materials on the subject. The materials follows from Khalil Ezzinbi [3]; Klaus-Jochen Engel<br>and Rainer Nagel [5]; R. F. Curtain and H. J. Zwart [6]; and Zheng-Hua Luo, Zhu Guo and<br>Ömer Mörgul [8].<br>Chapter 3<br>Chapter 3 is dealed with the controllability and stabilizability of linear systems in Hilbert<br>spaces. A general framework of concept of controllability of linear systems with bounded<br>5<br>control is discussed and then the various concepts of controllability and sabilizability of<br>linear systems and the the relationships between them is stated. almost all the results in<br>this chapter are from A. J. Pritchard and T. Zabczyk [1]; Alain Bensoussen, Guiseppe Da<br>Prato, Michel C. Delphour, Sanjoy K. Mitter [2]; Jerzy Zabczyk [4] and Ruth F. Curtain<br>and Anthony J. Pritchard [7].<br>6</p> <br><p></p>