Singular estimates and Riccati theory for thermoelastic plate models with boundary thermal control

Abstract: We consider an established thermoelastic plate model subject to boundary control in the thermal component. The model under investigation accounts for rotational forces, hence it is of predominantly hyperbolic type. Different sets of coupled (mechanical/thermal) boundary conditions are examined. It is shown that the quadratic optimal control problems associated with this system -- over both finite and infinite time horizon -- give rise to well-posed Riccati equations. In particular, it is proved that the gain operators are bounded. This type of result is rather unusual in the case of hyperbolic-like dynamics, where typically one obtains `non-standard’ Riccati equations with highly unbounded gain operators. What plays crucial role in the present analysis are specific regularity estimates obtained for the solutions to the thermoelastic problem, for each set of boundary conditions, which take advantage of the interactions between mechanical (hyperbolic) and thermal (parabolic) components of the dynamics. These regularity results allow to apply the Riccati theory recently developed for abstract control systems whose free dynamics operator A and control operator B yield singular estimates for the operator exp(At)B.