Necessary conditions for exponential stabilizability of distributed parameter systems with infinite dimensional unbounded feedback

Abstract

In this paper we consider distributed parameter control systems of the form x′(t) = Ax(t) + Bu(t), where x(t) is in the state space X, u(t) is in an infinite dimensional control space U, and B is an unbounded input operator which is admissible in a standard sense. We find necessary conditions for there to exist a compact operator K : X → U such that A + BK generates a C 0- semigroup which is of exponential growth type μ. In particular, for every γ > μ, A must have at most finitely many eigenvalues of finite multiplicity in { Re(z) ≥ γ} . Furthermore, the part of A corresponding to the remaining spectrum must generate a semigroup with type at most γ. We apply this to a model for a clamped Euler-Bernoulli plate with Dirichlet control applied to the entire boundary. We show that a compact state feedback cannot exponentially stabilize the plate in the state space of maximum regularity.