On the Convergence of Min/sup Points in Optimal Control Problems

In case f0 and C are convex, a variety of primal/dual numerical methods can be used to find the saddle points of a Lagrangian L associated with this problem [4, 7]. These methods take advantage of the fact that the search for the saddle points of L is unconstrained, or conducted over sets much simpler than C. Furthermore, we can introduce penalties on f0 in a way that regularizes L and makes it smoother. This in turn leads to a more convenient optimality condition of the form (0,0) ∈ ∂L. In the case of a non-convex problem, similar methods can be used to find the saddle points of an augmented Lagrangian, and these points can be used to obtain a solution to the original problem (see [7, 8], and [11, Chapter 10, Sections I and K∗]). The primal/dual methods are often combined with approximating L. The notion of epi/hypo convergence introduced by Attouch and Wets in [2] provides a setting for constructing a sequence Ln in such way that the saddle points (x̄n, ȳn) of Ln converge to the saddle points of L. In this paper, however, we are interested in problems where L has a min/sup point rather than a saddle point,