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

The linear-quadratic (LQ) control problem is considered for a class of infi nite-dimensional systems with bounded input and output operators, that are not exponentially stabilizable, but only strongly stabilizable. A sufficien t condition for the existence of a minimizing control and of a stabilizing solution to the associated LQ Riccati equation is given. The main contribut ion of this paper is the convergence of the stabilizing solutions of a sequ ence of finite-dimensional Riccati equations to the strongly stabilizing so lution of the infinite-dimensional Riccati equation. The result is applied to a model of propagation of sound waves in a one-dimensional wave-guide.