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Perspectives on characteristics based curse-of-dimensionality-free numerical approaches for solving Hamilton-Jacobi equations. (arXiv:1711.03314v1 [math.OC])
来源于:arXiv
This paper extends the considerations of the works [1, 2] regarding
curse-of-dimensionality-free numerical approaches to solve certain types of
Hamilton-Jacobi equations arising in optimal control problems, differential
games and elsewhere. A rigorous formulation and justification for the extended
Hopf-Lax formula of [2] is provided together with novel theoretical and
practical discussions including useful recommendations. By using the method of
characteristics, the solutions of some problem classes under
convexity/concavity conditions on Hamiltonians (in particular, the solutions of
Hamilton-Jacobi-Bellman equations in optimal control problems) are evaluated
separately at different initial positions. This allows for the avoidance of the
curse of dimensionality, as well as for choosing arbitrary computational
regions. The corresponding feedback control strategies are obtained at selected
positions without approximating the partial derivatives of the solutions. The
results of numerical 查看全文>>