Abstract
The Linear-Quadratic optimal control problem with a partial stabilization constraint (LQPS) is considered for exponentially stabilizable infinite dimensional semigroup state-space systems with bounded sensing and control (having their transfer function with entries in the algebra\(\mathop \mathcal{B}\limits^ \wedge \). It is reported that the LQPS-optimal state-feedback operator is related to a 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\(\mathop \mathcal{A}\limits^ \wedge \)_, and (2) by obtaining any constant solution of a diophantine equation over\(\mathop \mathcal{A}\limits^ \wedge \)_. These theoretical results are applied to a simple model of heat diffusion, leading to an approximation procedure converging exponentially fast to the LQPS-optimal state feedback operator.
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