Can Local Single-Pass Methods Solve Any Stationary Hamilton--Jacobi--Bellman Equation?

Abstract

The use of local single-pass methods (like, e.g., the fast marching method) has become popular in the solution of some Hamilton-Jacobi equations. The prototype of these equations is the eikonal equation, for which the methods can be applied saving CPU time and possibly memory allocation. Then some questions naturally arise: Can local single-pass methods solve any HamiltonJacobi equation? If not, where should the limit be set? This paper tries to answer these questions. In order to give a complete picture, we present an overview of some fast methods available in the literature and briefly analyze their main features. We also introduce some numerical tools and provide several numerical tests which are intended to exhibit the limitations of the methods. We show that the construction of a local single-pass method for general Hamilton-Jacobi equations is very hard, if not impossible. Nevertheless, some special classes of problems can actually be solved, making local single-pass methods very useful from a practical point of view.1. Introduction. The study of Hamilton-Jacobi (HJ) equations arises in several applied contexts, including classical mechanics, front propagation, control problems, and differential games, and has had a great impact in many areas, such as robotics, aeronautics, and electrical and aerospace engineering. In particular, for control/game problems, an approximation of the value function allows for the synthesis of optimal control laws in feedback form, and then for the computation of optimal trajectories. The value function for a control problem (resp., differential game) can be characterized as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation (resp., Hamilton-Jacobi-Isaacs (HJI) equation), and it is obtained by passing to the limit in Bellman's well-known dynamic programming (DP) principle. The DP approach can be rather expensive from a computational point of view, but in some situations it gives a real advantage over methods based on Pontryagin's maximum principle, because the latter approach allows one to compute only open-loop controls and locally optimal trajectories. Moreover, weak solutions to HJ equations are nowadays well understood in the framework of viscosity solutions, which offers the correct notion of solution for many applied problems.The above remarks have motivated the research of efficient and accurate numerical methods. Indeed, an increasing number of techniques have been proposed for

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