A Curse-of-Dimensionality-Free Numerical Method for Solution of Certain HJB PDEs

In previous works of the author and others, max-plus methods have been explored for the 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 computational-speed advantages, they still generally suffer from the curse of dimensionality. Here we consider HJB PDEs in which the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms. The approach to the solution will be rather general, but in order to ground the work, we consider only constituent Hamiltonians corresponding to long-run average-cost-per-unit-time optimal control problems for the development. We obtain a numerical method not subject to the curse of dimensionality. The method is based on construction of the dual-space semigroup corresponding to the HJB PDE. This dual-space semigroup is constructed from the dual-space semigroups corresponding to the constituent linear/quadratic Hamiltonians. The dual-space semigroup is particularly useful due to its form as a max-plus integral operator with a kernel obtained from the originating semigroup. One considers repeated application of the dual-space semigroup to obtain the solution.