Admissible Perturbation, Controllability And Robust Stability Of Distributed Parameter Systems

In this thesis, we consider the following problems:(i) The admissible perturbation for regular linear systems and absolutely regular non-autonomous systems.(ii) The controllability and its unbounded perturbation for infinite dimensional linear systems with unbounded control operator in Banach spaces.(iii) The robust stability with respect to small delays for autonomous and non-autonomous differential equations with unbounded delay operator.This thesis is divided into five chapters.In chapter 1, we consider the regularity of the closed-loop systems obtained from a regular linear system with an admissible state-feedback operator, and prove that the closed-loop system is again regular. Also, we transform a system with delay in the output, and apply our perturbation results to the delay system.In chapter 2, we consider the closed-loop system obtained from a absolutely regular non-autonomous linear system with an admissible state-feedback, and obtain its absolute regularity. Then, we apply this perturbation results to non-autonomous delay systems.In chapter 3, we, first, obtain a characterization (i.e., observability inequality) for exact controllability of the control system with unbounded control operator in non-reflexive Banach space. Secondly, we consider a class of unbounded perturbation of exact controllability in reflexive Banach space, and proved that the controllability is robust with respect to this perturbation. Also, we consider another class of unbounded perturbation of observability, and prove that the observability is robust with respect to this perturbation when the distancebetween the perturbation operator and the zero operator is small enough.In chapter 4, we obtain frequency-domain characterization for robustness with respect to small delays for exponential stability of partial differential equations with unbounded delay operator in L^p-phase space. For this purpose, the fundamental operator family and the operator-valued Fourier multipliers theory are applied. It is worthy to be noted that it is also the frequency-domain characterization for exponential stability of C₀-semigroup when the delay is zero. Furthermore, we obtain a sufficient condition for the robustness. The condition is independent of the delay, so it is easy to be verified. At last, we apply the the sufficient condition to the problem of robustly exponential stability with respect to small delays of a damping elastic systems with delay.In chapter 5, we obtain the robustness with respect to small delays for exponential stability of non-autonomous parabolic equation with unbounded delay operator.