Stability Radius Of Nonsmooth Pritchard-Salamon Systems And The Algebraic Riccati Equation

A system always has structured perturbations because of reasons of its own. So, the robustness under structured perturbations has been widely studied ([5],[6],[9]). In order to analyze the robustness of a system under structured perturbations, Hinrichsen Pritchard et al have introduced the notion of stability radius. In our paper, by using the latest results on nonsmooth Pritchard-Salamon Systems([3],[8]), we have extended the results of Hinrichsen Pritchard et al([5],[6],[9]) to the situation of nonsmooth Pritchard-Salamon Systems, so we have got the relationship between stability radius and input and output operators. Also, we have got the relationship between stability radius and some type of algebraic Riccati equations. Besides, our paper has discussed another problem, which is the non-exponential stability of infinite-dimensional linear systems under generating compact perturbation. As we know, the problem of aninfinite-dimensional linear system is to decide a feedback control, such that the C₀ - semigroupcorresponding to the closed-loop system is stable. Russell([17]) has showed that, a compact perturbation, generated from unitary operator groups in infinite-dimensional linear Hilbert spaces, is unable to make the perturbed semigroup exponentially stable. In this paper, we have proved thatan anti-bounded C₀ - group in infinite-dimensional reflexive Banach spaces is not exponentiallystable under generating compact perturbation, and that a C₀ - isometric semigroup ininfinite-dimensional Hilbert spaces has the same property. So, we have extended and improved Russell Theorem.