Max-plus fundamental solution semigroups for a class of difference Riccati equations

TL;DR

This paper introduces a new max-plus primal space fundamental solution semigroup for difference Riccati equations, clarifying its relationship with the existing dual space semigroup and enhancing understanding of the value function evolution in optimal control.

Contribution

A novel max-plus primal space fundamental solution semigroup is developed, linking it with the existing dual space semigroup for difference Riccati equations.

Findings

Established connections between primal and dual space semigroups.

Provided a deeper understanding of the value function evolution.

Enhanced the theoretical framework for difference Riccati equations.

Abstract

Recently, a max-plus dual space fundamental solution semigroup for a class of difference Riccati equation (DRE) has been developed. This fundamental solution semigroup is represented in terms of the kernel of a specific max-plus linear operator that plays the role of the dynamic programming evolution operator in a max-plus dual space. In order to fully understand connections between this dual space fundamental solution semigroup and evolution of the value function of the underlying optimal control problem, a new max-plus primal space fundamental solution semigroup for the same class of difference Riccati equations is presented. Connections and commutation results between this new primal space fundamental solution semigroup and the recently developed dual space fundamental solution semigroup are established.