An optimal uniqueness result for Riccati equations arising in abstract parabolic control problems

TL;DR

This paper establishes an optimal uniqueness result for integral Riccati equations in abstract parabolic control problems, extending previous work and filling a gap in the autonomous case, while also providing regularity results for the optimal state.

Contribution

It proves a broad uniqueness theorem for Riccati equations in nonautonomous parabolic control problems, enhancing understanding of solution properties and regularity.

Findings

Proves an optimal uniqueness result for the integral Riccati equation.

Provides regularity results for the optimal state.

Fills a gap in the autonomous case analysis.

Abstract

An abstract nonautonomous parabolic linear-quadratic regulator problem with very general final cost operator P_T is considered, subject to the same assumptions under which a classical solution of the associated differential Riccati equation was shown to exist, in two papers appeared in 1999 and 2000, by Terreni and the first named author. We prove an optimal uniqueness result for the integral Riccati equation in a wide and natural class, filling a gap existing in the autonomous case, too. In addition, we give a regularity result for the optimal state.