A max-plus dual space fundamental solution for a class of operator differential Riccati equations

TL;DR

This paper introduces a novel max-plus dual space fundamental solution semigroup for operator differential Riccati equations, enabling explicit solution propagation through max-plus integral operators and quadratic kernels.

Contribution

It develops a new fundamental solution semigroup using max-plus linearity and Legendre-Fenchel transform, advancing solution propagation methods for operator differential Riccati equations.

Findings

Constructed a max-plus integral operator semigroup for Riccati equations.

Demonstrated solution propagation from a class of initial conditions.

Provided an explicit recipe for solution propagation via quadratic kernels.

Abstract

A new fundamental solution semigroup for operator differential Riccati equations is developed. This fundamental solution semigroup is constructed via an auxiliary finite horizon optimal control problem whose value functional growth with respect to time horizon is determined by a particular solution of the operator differential Riccati equation of interest. By exploiting semiconvexity of this value functional, and the attendant max-plus linearity and semigroup properties of the associated dynamic programming evolution operator, a semigroup of max-plus integral operators is constructed in a dual space defined via the Legendre-Fenchel transform. It is demonstrated that this semigroup of max-plus integral operators can be used to propagate all solutions of the operator differential Riccati equation that are initialized from a specified class of initial conditions. As this semigroup of max-plus integral operators can be identified with a semigroup of quadratic kernels, an explicit recipe for the aforementioned solution propagation is also rendered possible.