A refinement on existence results in nonconvex optimal control

An alternative approach for non-linear optimal control problems based on the method of moments

“…which is the maximum principle's necessary condition of the convex formulation (11). Since the relaxation (11) is convex, the maximum principle's necessary conditions are also sufficient to guarantee optimality.…”

“…We face the question of the existence of minimizers from the point of view of absolutely continuous functions [3,5,6,10,14,42,45]. Since relaxation (11) is an optimal control problem with convex structure in the control variable m, we can suppose under mild assumptions that (11) has a minimizer m * [4,5,7,10,11,42]. We will show that the minimizers m * of the formulation (11) may determine the existence of minimizers of the non-linear optimal control problem (1)- (2).…”

“…Let us assume that m * is a minimizer of (11). According to the maximum principle, m * satisfies (17) where x * (t) and p * (t) satisfy the boundary value problem (16).…”

“…See [1,2,16,29,30,31,53] for classical and modern results on the problem of moments and [17,18,19,20,21,22,23,24,25,26,27,53,54,55,56,57,58,59,60,61] for recent results on global optimization of polynomial expressions by the characterizations of moments. Convexification of non-convex, optimal control problems by using probability measures is a well known tool exploited by other authors [3,4,5,8,9,10,11,12,13,14]. What is new in this approach is the convexification of the control variable by using moment variables, which allow us to obtain an equivalent, convex formulation more appropriated to be solved by high performance numerical computing.…”

Optimization, relaxation and Young measures

“…A formulation in terms of Young measures has been analyzed in [8] and [9] where particular forms of the relaxation have been given. Our reference for this problem is [48] where the optimal relaxation with as few design variables as possible is given and proved.…”