We begin by reviewing different stability types for abstract differential equations and strongly continuous semigroups on Hilbert spaces. We concentrate on exponential stability, polynomial stability, and strong stability with a finite number of singularities on the imaginary axis. We illustrate each stability type with examples from partial differential equations and control theory. In the second part of the talk we study the preservation of strong and polynomial stabilities of a semigroup under bounded perturbations of its generator. As the main results we present conditions for preservation of these two stability types under finite rank and trace class perturbations. In particular, the conditions require that certain graph norms of the perturbing operators are sufficiently small. In the final part of the talk we consider robust output tracking for linear systems, and explain how this control problem motivates the study of preservation of polynomial stability of semigroups. In particular, the solution of this problem requires determining which uncertainties in the parameters of the controlled system preserve the stability of the closed-loop system consisting of the system and the dynamic controller. We show that if the reference signal to be tracked is a nonsmooth periodic function, it is impossible to stabilize the closed-loop system exponentially, but polynomial stability is achievable under suitable assumptions. Subsequently, the uncertainties in the parameters of the system can be represented as a bounded perturbation to the system operator of the polynomially stable closed-loop system.
- Functional Analysis Seminar