Optimal boundary control and Riccati theory for abstract dynamics motivated by hybrid systems of PDEs

Abstract

We study the quadratic optimal control problem over a finite time horizon for a class of abstract systems with analytic underlying semigroup $e^{tA}$ and unbounded control operator $B$. It is assumed that a suitable decomposition of the operator $B^*e^{tA^*}$ is valid, where only one component satisfies a `singular estimate', whereas for the other component specific regularity properties hold. Under these conditions, we prove well posedness of the associated differential Riccati equation, and in particular that the gain operator is bounded on a dense set. In spite of the unifying abstract framework used, the prime motivation (and application) of the resulting theory of linear-quadratic problems comes from optimal boundary control of a thermoelastic system with clamped boundary conditions. The non-trivial trace regularity estimate showing that this PDE mixed problem fits into the distinct class of models under examination---for which we have developed the present, novel optimal control theory---is established, as well.