Analysis and Control of Partial Differential Equations using Occupation Measures

Context This work is in the line of research with the following issue: how to develop new convex optimization techniques based on semidefinite programming (SDP) and real algebraic geometry to solve optimal control problems (OCP) in a nonlinear setting. Recently, several research efforts allowed to solve numerically certain optimal control problems with polynomial data. The general idea is to reformulate such a nonlinear problem into an infinite dimensional linear optimization problem, called primal over the moments of so-called occupation measures. This is inspired from a line of research initiated by Rubio [6]. This problem being convex but infinite dimensional, one way to handle it in practice is to solve a hierarchy of convergent semidefinite relaxations, each relaxation being solved with SDP numerical solvers. This approach allows to obtain suboptimal polynomial solutions of the value function of the OCP, this function being a viscosity solution of a nonlinear Hamilton-Jacobi-Bellman partial differential equation (PDE). Each sub-optimal solution is provided through solving a strengthening of the dual sums-of-squares problem, associated to the primal reformulation over moments of occupation measures. Further works allowed to apply the same framework to numerous problems related to optimal control: impulsive linear or nonlinear systems [1] or switch systems [2], certified over approximations of region of attraction [3] for polynomial systems or image set of polynomial dynamics [4], synthesis of control laws for infinite horizon problems.