On the Perturbations of Regular Linear Systems and Linear Systems with State and Output Delays

Abstract

This paper is concerned with perturbation problems of regularity linear systems. Two types of perturbation results are proved: (i) the perturbed system (A + P, B, C) generates a regular linear system provided both (A, B, C) and (A, B, P ) generate regular linear systems; and (ii) the perturbed system ((A−1 + ΔA)|X , B, C A Λ ) generates a regular linear system if both (A, B, C) and (A, ΔA, C) generate regular linear systems. These allow us to establish a new variation of constants formula of the control system (A + P, B). Moreover, these results are applied to the linear systems with state and output delays. The regularity and the mild expressibility is deduced, and a necessary and sufficient condition for stabilizability of the delayed systems is proved.Mathematics Subject Classification (2010). Primary 47A55; Secondary 93C23.

Perturbations of Admissibility, Exact Controllability, Exact Observability and Regularity

“…generates a regular linear system, particularly, J A,A+C ∆B is admissible for A + C, such result has been proved by Mei and Peng [27]. This means that our result is a generalization of [16] and [27].…”

“…) is exactly observable provided (A, C) is exactly observable and ∆A is "small" enough. In their paper [27],…”

Well-posedness of boundary feedback systems with unbounded feedback operator

“…Since both (A, B, C) and (A, B, λI) generate regular linear systems, by [12,Theorem 3.4], it follows that (A λ , B, C) generates a regular linear system, too.…”

A Class of Linear Boundary Systems with Delays in State, Input and Boundary Output

“…Lemma 2.4 [25] Assume that (A, B, C) and (A, B, P ) generate regular linear systems on (X, U, Y ) and (X, U, X), respectively. Then (A + P, J…”

“…However, we observe that the unboundedness of L will bring us essential difficulties. In order to settle such problem, we plan to use the perturbation theory developed by our recent paper [25]. Moreover, some other theorems will be proved, which are useful to deal with our problem.…”