Max-plus fundamental solution semigroups for optimal control problems

Abstract

Recent work concerning the development of fundamental solution semigroups for specific classes of optimal control and related problems is unified and generalized. By exploiting max-plus linearity, semiconvexity, and semigroup properties of the corresponding dynamic programming evolution operator, two types of max-plus fundamental solution semigroup are presented. These semigroups, referred to respectively as max-plus primal and max-plus dual space fundamental solution semigroups, consist of horizon indexed max-plus linear max-plus integral operators that facilitate the propagation of value functions, or (respectively) their semiconvex transform, to longer time horizons via simple max-plus convolutions. Properties of these semigroups, and interconnections between them, are established. Their application to solving specific problems, including a class of operator differential Riccati equations, is summarized.