LQ-optimal control of infinite-dimensional systems by spectral factorization

We consider the standard LQ-optimal control of an exponentially stabilizable and detectable infinite-dimensional semigroup state space system with bounded sensing and control. It is reported that the reachable restriction of the LQ-optimal state feedback operator 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 A, and (2) by obtaining any constant solution of an operator diophantine equation over A. The properties of the restricted solution feedback operator are discussed. It is shown in particular that the LQ-optimal state feedback operator and its reachable restriction coincide whenever the unreachable state components are unobservable. These theoretical results are applied to a simple model of heat diffusion with "collocated" sensor and actuator, leading to an approximation procedure converging exponentially fast to the LQ-optimal state feedback operator. This procedure is based on approximate spectral factorization by symmetric extraction and on simple residue calculus.