Spectral factorization and partially stabilizing LQ-optimal control for stabilizable semigroup systems

The main goal of this conference paper is to report some results concerning the solution of the Linear-Quadratic optimal control problem with partial stabilization constraint (LQPS). This problem is considered for the class of exponentially stabilizable infinite-dimensional semigroup state-space systems with bounded sensing and control, having their transfer functions with entries in the algebra B, [1]-[3]. It is reported that the LQPS-optimal state feedback operator is related to a particular nonnegative self-adjoint solution of an operator Riccati equation and it can be identified (1) by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper-stable transfer function algebra Â-; (2) by obtaining any constant solution of an operator diophantine equation over Â-; and (3) by solving an operator Lyapunov equation.