Uniqueness for Riccati equations with application to the optimal boundary control of composite systems of evolutionary partial differential equations

Abstract: In this article we address the issue of uniqueness for differential and algebraic operator Riccati equations, under a distinctive set of assumptions on their unbounded coefficients. The class of boundary control systems characterized by these assumptions encompasses diverse significant physical interactions, all modeled by systems of coupled hyperbolic–parabolic partial differential equations. The proofs of uniqueness provided tackle and overcome the obstacles raised by the peculiar regularity properties of the composite dynamics. These results supplement the theories of the finite and infinite time horizon linear–quadratic problem devised by the authors jointly with I. Lasiecka, as the unique solution to the Riccati equation enters the closed-loop form of the optimal control.