An approximation theory for strongly stabilizing solutions to the operator LQ Riccati equation

The linear-quadratic (LQ) control problem is considered for a class of infinite-dimensional systems with bounded input and output operators, that are not exponentially stabilizable, but only strongly stabilizable. A sufficient condition for the existence of a minimizing control and of a stabilizing solution to the associated LQ Riccati equation is given. The main contribution of this paper is the convergence of the stabilizing solutions of a sequence of finite-dimensional Riccati equations to the strongly stabilizing solution of the infinite-dimensional Riccati equation. We will illustrate the approximation result with an example of LQ control for a model of propagation of sound waves in a one-dimensional wave-guide.