A max-plus method for optimal control of a diffusion equation

Recent work concerning the fundamental solution semigroup for a class of infinite dimensional Riccati equations is extended to include a diffusion term. By exploiting max-plus linearity and semiconvexity of the value function of the associated optimal control problem in this new case, the fundamental solution semigroup is constructed in a dual space via the Legendre-Fenchel transform. In particular, it is shown that the semigroup property in the dual space follows from a corresponding property arising from dynamic programming in the primal space. This fundamental solution semigroup is shown to take the form of a max-plus integral operator with an explicit quadratic functional kernel defined with respect to three time-indexed integral operators. Using this fundamental solution semigroup, a recipe for the evolution of general solutions of the infinite dimensional Riccati equation is proposed.