PARTIALLY STABILIZING LQ-OPTIMAL CONTROL FOR STABILIZABLE SEMIGROUP SYSTEMS

The Linear-Quadratic optimal control problem with a partial stabilizat ion constraint (LQPS) is considered for exponentially stabilizable inf inite dimensional semigroup state-space systems with bounded sensing a nd control (having their transfer function with entries in the algebra (B) over cap. It is reported that the LQPS-optimal state-feedback ope rator is related to a nonnegative self-adjoint solution of an operator Riccati equation and it can be identified (1) by solving a spectral f actorization problem delivering a bistable spectral factor with entrie s in the distributed proper-stable transfer function algebra (A) over cap_, and (2) by obtaining any constant solution of a diophantine equa tion over (A) over cap_. These theoretical results are applied to a si mple model of heat diffusion, leading to an approximation procedure co nverging exponentially fast to the LQPS-optimal state feedback operato r.