Nonlinear optimal control: Numerical approximations via moments and LMI-relaxations

We consider the class of nonlinear optimal con- trol problems with all data (differential equation, state and control constraints, cost) being polynomials. We provide a simple hierarchy of LMI-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Preliminary results show that good approximations are obtained with few moments. I. INTRODUCTION In general, solving a general nonlinear optimal control problem (OCP) is a difficult challenge, despite powerfull theoretical tools are available, e.g. the maximum principle and Hamilton-Jacobi-Bellman optimality equation. How- ever there exist many numerical methods to approximate the solution of a given optimal control problem. For in- stance, Multiple shooting techniques which solve two-point boundary value problems as decribed in e.g. (17), (7), or direct methods, as in e.g. (18), (5), (6), which for instance, use descent algorithms, among others. Contribution. In this paper, we consider the particular class of nonlinear OCPs for which all data describing the problem (dynamics, state and control constraints) are polynomials. We propose a completely different approach to provide a good approximation of (only) the optimal value of the OCP, via a sequence of increasing lower bounds. As such, it could be seen as a complement to the above shooting or direct methods which provide an upper bound, and when the sequence of lower bounds converges to the optimal value, a test of their efficiency. We first adopt an infinite-dimensional linear program- ming (LP) formulation based on the Hamilton-Jacobi- Bellman equation, as developed in e.g. Hernandez- Hernandez et al. (11). We then follow a numerical approx- imation scheme (a relaxation of the original LP) in the vein but different from the general framework developed in Hernandez and Lasserre (10) for infinite-dimensional linear programs. Our contribution is to here exploit the fact that all data are polynomials to provide a hierarchy of semidefinite programming (SDP) (or, LMI) relaxations, whose optimal values form a monotone nondecreasing sequence of lower bounds on the optimal value of the OCP. The first numerical experiments show that good approximations are obtained early in the hierarchy, i.e., with few moments, confirming