# A High-Order Semi-Lagrangian/Finite Volume Scheme for Dynamic Programming Equations

Maurizio Falcone and Dante Kalise*

The numerical approximation of solutions of dynamic programming equations is a fundamental task in the quest for feedback solutions in optimal control and differential games. In this talk, we present a numerical scheme for the approximation of Hamilton-Jacobi-Bellman and Isaacs equations of the form $\lambda v(x) + H(x,D v)=0\,,$ where $H$ can be either a convex Hamiltonian or a minmax operator. We introduce a scheme based on the combination of semi-Lagrangian in time discretization with a high-order finite volume approximation in space. The high-order character of the scheme provides an efficient way towards accurate approximations. We assess the performance of the scheme with a set of problems arising in optimal control and pursuit-evasion games.

Mathematics Subject Classification: 49L20 35Q93

Keywords: High-order methods; semi-Lagrangian schemes; finite volume methods; WENO interpolation; Hamilton-Jacobi-Bellman equations.

Minisymposion: On Optimal Feedback Control for Partial Differential Equations: Theory and Numerical Methods