Quadratic optimal control of stable abstract linear systems

Abstract

We consider the nonstandard infinite horizon quadratic cost minimization problem for a stable abstract linear control system, and show that it can be reduced to a J-inner coprime factorization problem (or equivalently, to a canonical spectral factorization problem) in the (often finite-dimensional) control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular together with their adjoints, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the classical algebraic Riccati equation, but one of its coefficients differs from the expected one. We apply our main theorems to prove the first available versions of the positive real and bounded real lemmas for abstract linear systems. Similar results are true for unstable systems.KeywordsSpectral factorizationJ-inner coprime factorizationpositive real lemmabounded real lemma