In previous work of the author and others, 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 computational-speed advantages, they still generally suffer from the curse-of-dimensionality. Here we consider HJB PDEs where the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms. The approach to 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 kernel obtained from the originating semigroup. One considers repeated application of the dual-space semigroup to obtain the solution.
Abstract-Recently, a curse-of-dimensionality-free method was developed for solution of Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs) for nonlinear control problems, using semiconvex duality and max-plus analysis. The curse-of-dimensionality-free method may be applied to HJB PDEs where the Hamiltonian is given as (or well-approximated by) a pointwise maximum of quadratic forms. Such HJB PDEs also arise in certain switched linear systems. The method constructs the correct solution of an HJB PDE from a maxplus linear combination of quadratics. The method completely avoids the curse-of-dimensionality, and is subject to cubic computational growth as a function of space dimension. However, it is subject to a curse-of-complexity. In particular, the number of quadratics in the approximation grows exponentially with the number of iterations. Efficacy of such a method depends on the pruning of quadratics to keep the complexity growth at a reasonable level. Here we apply a pruning algorithm based on semidefinite programming. Computational speeds are exceptional, with an example HJB PDE in six-dimensional Euclidean space solved to the indicated quality in approximately 30 minutes on a typical desktop machine.
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