  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.
