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

@article{citeulike:1650040, abstract = {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.}, author = {Mceneaney, William M. }, citeulike-article-id = {1650040}, journal = {SIAM Journal on Control and Optimization}, keywords = {control-numeric, control-optimal, no-subscribe}, number = {4}, pages = {1239--1276}, posted-at = {2007-09-13 01:45:44}, priority = {2}, publisher = {SIAM}, title = {A Curse-of-Dimensionality-Free Numerical Method for Solution of Certain HJB PDEs}, url = {http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal\&id=SJCODC000046000004001239000001\&idtype=cvips\&gifs=yes}, volume = {46}, year = {2007} }

TY - JOUR ID - citeulike:1650040 L3 - citeulike-article-id:1650040 TI - A Curse-of-Dimensionality-Free Numerical Method for Solution of Certain HJB PDEs JF - SIAM Journal on Control and Optimization VL - 46 IS - 4 SP - 1239 EP - 1276 PB - SIAM N2 - 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. KW - control-numeric KW - control-optimal KW - no-subscribe AU - Mceneaney, William PY - 2007/// UR - http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJCODC000046000004001239000001&idtype=cvips&gifs=yes ER -