Optimal control and nonlinear PDEs
Here the goal is to show how useful can be the Moment-SOS hierarchy to help solve a class of optimal control problems and some non-linear PDE's as well, whose data are described by polynomials and semi-algebraic sets. A foundational approach for optimal control is described and summarized in the paper "Lasserre J.B., Henrion D., Prieur C., Trélat E. (2008) Nonlinear optimal control via occupation measures and LMI-relaxations SIAM J. Contr. Optim. 47, pp. 1649--1666."
The point of departure is to consider a weak-formulation associated with the intial optimal control problem; such formulations had been already considered in the past, but mainly as a tool to investigate existence of generalized solutions (e.g. a concept that extends the notion of "classical solution" to consider more general objects, e.g., relaxed controls). Then integrating on the domain (in the sense of distributions) with respect to polynomial test functions yields linear constraints on moments of the occupation measure on the domain as well as on the occupation measure on its boundary (where we also use the boundary information on the solution). Therefore we next only consider the larger familly of all "scalars" (pseudo-moments) that satisfy such linear constraints. Then to ensure that these pseudo-moments are indeed moments of some measures on the appropriate domains, we impose semidefinite constraints on the so-called moment and localizing matrices associated with the linear functional represented by these pseudo-moments. By restricting to finitely many moments up to order "d", we end up with solving a semidefinite program which is a relaxation of the intial optimal control problem. When "d" increases then under some conditions on the initial problem, the resulting sequence of optimal values (rd ) converges to the optimal value of the initial problem. A similar strategy was also used for the optimal control and analysis of a class of PDE's with polynomial data. In particular for the the Burgers equation (a class of hyperbolic PDE's) the approach was shown. (In the latter one has to also consider additional "entropy constraints" on the occupation measures to ensure that we consider appropriate solutions of the PDE.)
This work has been done in close collaboration with D. Henrion, C. Prieur, E. Trélat and more recently with Milan Korda, Swann Marx and Tillmann Weisser.