Uniqueness for Riccati equations with unbounded operator coefficients

TL;DR

This paper establishes the uniqueness of solutions for Riccati equations with unbounded operator coefficients, relevant to boundary control systems modeled by coupled PDEs, under specific assumptions on their coefficients.

Contribution

It introduces new uniqueness results for Riccati equations with unbounded coefficients in boundary control systems, extending existing theories to more complex PDE models.

Findings

Proved uniqueness of Riccati solutions under new assumptions.

Addressed challenges from regularity properties of coupled PDE dynamics.

Extended linear-quadratic control theory to systems with unbounded operators.

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 Lasiecka, as the unique solution to the Riccati equation enters the closed loop form of the optimal control.