On robustness of exact controllability and exact observability under cross perturbations of the generator in Banach spaces

Abstract

Abstract. This paper is concerned with the exact controllability and exact observability of linear systems in the Banach space setting. It is proved that both the admissibility of control operators and the admissibility of observation operators are invariant to cross perturbations of the generator of a C 0 -semigroup. Moreover, under the admissibility invariance premise, the robustness of the exact controllability as well as the exact observability to such cross perturbations is verified. An illustrative example is presented.

Well-posedness of boundary feedback systems with unbounded feedback operator

“…In order to obtain the main results, we have to introduce an admissibility invariance theorem under admissible perturbations by Hadd [5] and under cross perturbations by Mei and Peng [11], respectively. Theorem 3.3.…”

“…Theorem 3.4. [11] Assume that (A, ΔA, C) generates a regular linear system on the triple of Banach spaces (X, X, Y). Then, ((A −1 +ΔA)| X , C A Λ ) generates an abstract linear observation system.…”

“…By [11,Theorem 3.9], it follows that (A λ , B) generates an abstract linear control system. Therefore,…”

“…We only have to prove that A K generates a C 0 -semigroup. The admissibility invariance theorem under cross perturbation [11] will be the main tool. As an application, we consider Euler-Bernoulli plate with partial boundary Neumann feedback operator in the dynamic boundary.…”

Radius of approximate controllability of linear retarded systems under structured perturbations

“…While exact controllability is generally preserved under small perturbations (see, e.g. [9] and references therein) the approximate controllability, unfortunately, can be destroyed by arbitrarily small perturbations of the system parameters. To see this, let us consider the control system (A, b) described by linear differential equatioṅ x = Ax + bu, x ∈ X , u ∈ C where X = l 2 , the Banach space of all square-summable sequences of complex numbers with the standard basis {e i }, i = 1, 2, .…”