Convergence Rate for a Curse-of-Dimensionality-Free Method for a Class of HJB PDEs
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McEneaney-2009-CONVERGENCE RATE FOR.pdf
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Author(s)
Kluberg, Lionel J.
McEneaney, William M.
Alternative Title
CONVERGENCE RATE FOR A CURSE-OF-DIMENSIONALITY-FREE METHOD FOR A CLASS OF HJB PDES
Date Issued
December 2009
Journal
SIAM Journal on Control and Optimization
Publisher
Society for Industrial and Applied Mathematics
Citation
McEneaney, William M. and L. Jonathan Kluberg. "Convergence Rate for a Curse-of-Dimensionality-Free Method for a Class of HJB PDEs." SIAM J. Control Optim. Volume 48, Issue 5, pp. 3052-3079 (2009) ©2009 Society for Industrial and Applied Mathematics.
Version
Final published version
Abstract
In previous work of the first 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. Although max-plus basis expansion and max-plus finite-element methods can provide substantial 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 consider a previously obtained 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. Although previous work indicated that the method was not subject to the curse-of-dimensionality, it did not indicate any error bounds or convergence rate. Here we obtain specific error bounds.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
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DOI of Published Version
https://doi.org/10.1137/070681934
