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We study the infinite horizon Linear-Quadratic problem and the associatedalgebraic Riccati equations for systems with unbounded control actions. Theoperator-theoretic context is motivated by composite systems of PartialDifferential Equations (PDE) with boundary or point control. Specific focus isplaced on systems of coupled hyperbolic/parabolic PDE with an overall`predominant' hyperbolic character, such as, e.g., some models forthermoelastic or fluid-structure interactions. While unbounded control actionslead to Riccati equations with unbounded (operator) coefficients, unlike theparabolic case solvability of these equations becomes a major issue, owing tothe lack of sufficient regularity of the solutions to the composite dynamics.In the present case, even the more general theory appealing to estimates of thesingularity displayed by the kernel which occurs in the integral representationof the solution to the control system fails. A novel framework which embodiespossible hyperbolic components of the dynamics has been introduced by theauthors in 2005, and a full theory of the LQ-problem on a finite time horizonhas been developed. The present paper provides the infinite time horizontheory, culminating in well-posedness of the corresponding (algebraic) Riccatiequations. New technical challenges are encountered and new tools are needed,especially in order to pinpoint the differentiability of the optimal solution.The theory is illustrated by means of a boundary control problem arising inthermoelasticity.

A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control