WP2 - Hamilton-Jacobi approach for optimal control problems

• State of the art

The Hamilton-Jacobi approach to optimal control identifies the value function, describing how the minimum cost depends on initial conditions, as the solution to a Hamilton-Jacobi type partial differential equation. Solving this partial differential equation gives both the minimum cost and also yields the optimal control in a feedback form suitable for many engineering applications. The approach has its roots in classical verification techniques in the Calculus of Variations. Richard Bellman in the 1950’s showed that the underlying ideas covered a very wide range of optimization problems under the banner of ‘Dynamic Programming’. Two distinct methodologies have been separately developed to establish existence, uniqueness and regularity properties of the solution to the Hamilton-Jacobi equation and to link it to the value functions. One is based on nonsmooth analysis and viability theory (team of UPMC) and another on the notion of viscosity solutions introduced by Crandall-Evans-Lions. Viscosity solutions for the Hamilton-Jacobi equation of Optimal Control have been extensively investigated by the Rome team and, in the broader context of differential games, by USP (Padova) team.

• Objectives

The Hamilton-Jacobi approach, when it can be applied, yields a global solution to the optimal control problem. It also provides a feedback representation of the optimal control. Despite of these advantages, this approach is seldom used in the industrial applications, because of the difficulties of computing the solution to the Hamilton-Jacobi equation in higher dimensions. The aim of this WP is to broaden the applicability of the Hamilton-Jacobi approach, by developing analytic techniques and computational methods for solving high dimensional problems.

• Planned person-months: 198