Controllability And Stabilizability Of Linear Systems In Hilbert Spaces

Abstract

Questions about controllability and stability arise in almost every dynamical system problem. As a result, controllability and stability are one of the most extensively studied subjects in system theory. A departure point of control theory is the di erential equation x(0) = x0 ∈ Rn , x = f (x, u), ̇ (1.0.1) with the right hand side depending on a parameter u from a set U ⊂ Rm . The set U is called the set of control parameters. Controls are of two types: open and closed loops. An open loop control can be basically an arbitrary function u(·) : [0, +∞) −→ U, for which the equation x(t) = f (x(t), u(t)), ̇ t ≥ 0, x(0) = x0 ∈ Rn , (1.0.2) has a well de ned solution. A closed loop control can be identi ed with a mapping k : Rn −→ U , which may depend on t ≥ 0, such that the equation t ≥ 0, x(0) = x0 ∈ Rn , x(t) = f (x(t), k(x(t))), ̇ (1.0.3) has a well de ned solution. The mapping k(·) is called feedback. Controls are called also strategies or inputs, and the corresponding solutions of (1.0.2) or (1.0.3) are outputs of the system. We do not consider all the system theory concepts here, we will concentrate mainly here on controllability and stability of linear system. To motivate our approach we present a brief survey of nite dimensional theory concepts and results which we will generalize later. Linear System Theory A linear system is described by a (linear) di erential equation x(t) = Ax(t) + Bu(t), ̇ t ≥ 0, x(0) = x0 , (1.0.4) where u is the control, x is the state. These functions take their values in linear spaces U and X , respectively. Furthermore, A and B are linear mappings between appropriate spaces. If the spaces U and X are nite dimensional, then the system is called nite dimensional. Otherwise, we have an in nite dimensional linear system.