Local minimization algorithms for dynamic programming equations

TL;DR

This paper emphasizes the importance of accurate minimization in solving Hamilton-Jacobi-Bellman equations for optimal control and proposes algorithms to improve this process, especially for nonsmooth and sparse control problems.

Contribution

It introduces algorithms for precise minimization in dynamic programming equations, enhancing solutions for nonsmooth and sparse control problems.

Findings

Improved algorithms for minimization in Hamilton-Jacobi-Bellman equations.

Effective handling of nonsmooth control problems with sparsity.

Enhanced numerical accuracy in dynamic programming solutions.

Abstract

The numerical realization of the dynamic programming principle for continuous-time optimal control leads to nonlinear Hamilton-Jacobi-Bellman equations which require the minimization of a nonlinear mapping over the set of admissible controls. This minimization is often performed by comparison over a finite number of elements of the control set. In this paper we demonstrate the importance of an accurate realization of these minimization problems and propose algorithms by which this can be achieved effectively. The considered class of equations includes nonsmooth control problems with $\ell_1$-penalization which lead to sparse controls.