Max-plus summation of Fenchel-transformed semigroups for solution of nonlinear Bellman equations

Max-plus methods have been explored for solution of first-order, nonlinear Hamilton–Jacobi–Bellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. These methods exploit the max-plus linearity of the associated semigroups. In particular, although the problems are nonlinear, the semigroups are linear in the max-plus sense. These methods have been used successfully to compute solutions. Although they provide certain advantages, they still generally suffer from the curse-of-dimensionality. The natural analog to the Laplace transform in ordinary spaces is the Fenchel transform over max-plus spaces, the range space being referred to as the dual space. One can transform the semigroup operators into operators on the dual space. There are natural operations on the transformed operators which may be used to construct solutions of nonlinear control problems. Natural building blocks correspond to transforms of operators for linear/quadratic problems. In this paper, a method for exploiting operations in the Fenchel transform space is used to develop a method for certain problems such that the computational growth in the most time-consuming portion of the computations can be hugely reduced. Although the curse-of-dimensionality is not entirely avoided, the computational cost reductions are very high for some classes of problems.