A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control (Q2843470)

The authors study the infinite horizon linear-quadratic (LQ) problem and the associated algebraic Riccati equations for systems with unbounded control actions. The operator-theoretic context is motivated by composite systems of partial differential equations with boundary or point control. Specific focus is placed on systems of coupled hyperbolic/parabolic PDEs with an overall ``predominant'' hyperbolic character, such as some models for thermoelastic or fluid-structure interactions. While unbounded control actions lead to Riccati equations with unbounded (operator) coefficients, unlike in the parabolic case solvability of these equations becomes a major issue, owing to the lack of sufficient regularity of the solutions to the composite dynamics.NEWLINENEWLINE The purpose of this paper is to complement the LQ theory by developing a complete infinite time horizon analysis. The task is not straightforward owing to a natural mixing of singularities occurring in short and long time. The interplay between the long time stability for the forward problem and the short time development of singularities for the adjoint problem lie at the heart of the problem. The authors emphasize at the outset that they establish solvability of the optimal control problem, as well as well-posedness of the corresponding algebraic Riccati equations, under minimal assumptions on the operators involved. 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 in thermoelasticity.