Minimizers That Are Not Also Relaxed Minimizers

Relaxation is a widely used regularization procedure in optimal control, involving the replacement of velocity sets by their convex hulls, to ensure the existence of a minimizer. It can be an important step in the construction of suboptimal controls for the original, unrelaxed, optimal control problem (which may not have a minimizer), based on obtaining a minimizer for the relaxed problem and approximating it. In some cases the infimum cost of the unrelaxed problem is strictly greater than the infimum cost over relaxed state trajectories; we need to identify such situations because then the above procedure fails. The noncoincidence of these two infima leads also to a breakdown of the dynamic programming method because, typically, solving the Hamilton--Jacobi equation yields the minimum cost of the relaxed, not the original, optimal control problem. Following on from earlier work by Warga, we explore the relation between, on the one hand, noncoincidence of the minimum cost of the optimal control and its relaxation and, on the other, abnormality of necessary conditions (in the sense that they take a degenerate form in which the cost multiplier is set to zero). Two kinds of theorems are proved, depending on whether we focus attention on minimizers of the unrelaxed or the relaxed formulation of the optimal control problem. One kind asserts that a local minimizer which is not also a relaxed local minimizer satisfies an abnormal form of the Hamiltonian inclusion. The other asserts that a relaxed local minimizer that is not also a local minimizer also satisfies an abnormal form of Hamiltonian inclusion.