Controllability

Abstract

This chapter is devoted to the controllability of linear systems. The analysis is based on characterizations of images of linear operators in terms of their adjoint operators. The abstract results lead to specific descriptions of approximately controllable and exactly controllable systems which are applicable to parabolic and hyperbolic equations. Formulae for controls which transfer one state to another are given as well.

Bibliographical notes

Theorems from §15.1 are taken from the paper by S. Dolecki andD.L. Russell [28] as well as from the author’s paper J. Zabczyk [109]. They are generalizations of a classical result due to R. Douglas [29].

An explicit description of reachable spaces in the case of finite-dimensional control operators B has not yet been obtained; see an extensive discussion of the problem in the survey paper by D.L. Russell [88] and in H.O. Fattorini and D.L. Russell [36].

Asymptotic properties of the sequences $(f_m )$ and $(\delta _m )$ were an object of intensive studies, see D.L. Russell [88] and S.A. Avdonin and S.A. Ivanov [3]. Theorem 15.9 is from author’s paper J. Zabczyk [109]. Part (i) is in the spirit of D.L. Russell [88]. For more recent results we refer to J.-M. Coron [20] and I. Lasiecka and R. Triggiani [62].