Linear conic optimization for nonlinear optimal control

AbstractInfinite-dimensional linear conic formulations are described for nonlinear optimalcontrol problems. The primal linear problem consists of finding occupation mea-sures supported on optimal relaxed controlled trajectories, whereas the dual linearproblem consists of finding the largest lower bound on the value function of theoptimal control problem. Various approximation results relating the original opti-mal control problem and its linear conic formulations are developed. As illustratedby a couple of simple examples, these results are relevant in the context of finite-dimensional semidefinite programming relaxations used to approximate numericallythe solutions of the infinite-dimensional linear conic problems. 1 Motivation In [8, 9], J.-B. Lasserre described a hierarchy of convex semidefinite programming (SDP)problems allowing to compute bounds and find global solutions for finite-dimensionalnonconvex polynomial optimization problems. Each step in the hierarchy consists ofsolving a primal moment SDP problem and a dual polynomial sum-of-squares (SOS) SDPproblem corresponding to discretizations of infinite-dimensional linear conic problems,namely a primal linear programming (LP) problem on the cone of nonnegative measures,and a dual LP problem on the cone of nonnegative continuous functions. The number ofvariables (number of moments in the primal SDP, degree of the SOS certificates in the dualSDP) increases when progressing in the hierarchy, global optimality can be ensured bychecking rank conditions on the moment matrices, and global optimizers can be extractedby numerical linear algebra. For more information on the moment-SOS hierarchy and itsapplications, see [11].This approach was then extended to polynomial optimal control in [10]. Whereas the keyidea in [8, 9] was to reformulate a (finite-dimensional) nonconvex polynomial optimiza-tion on a compact semi-algebraic set into an LP in the (infinite-dimensional) space of