We consider the general optimization problem (P) of selecting a continuous function x over a σ-compact Hausdorff space T to a metric space A, from a feasible region X of such functions, so as to minimize a functional c on X. We require that X consists of a close equicontinuous family of functions lying in the product (over T) of compact subsets Yt of A. (An important special case is the optimal control problem of finding a continuous time control function x that minimizes its associated discounted cost c(x) over the infinite horizon.) Relative to the uniform-on-compact topology on the function space C(T, A) of continuous functions from T to A, the feasible region X is compact. Thus optimal solutions x* to (P) exist under the assumption that c is continuous. We wish to approximate such an x* by optimal solutions to a net {Pi}, i ∈ I, of approximating problems of the form minx∈Xi ci(x) for each i ∈ I, where (1) the net of sets {Xi}I converges to X in the sense of Kuratowski and (2) the net {ci}I of functions converges to c uniformly on X. We show that for large i, any optimal solution xi* to the approximating problem (Pi) arbitrarily well approximates some optimal solution x* to (P). It follows that if (P) is well-posed, i.e., lim sup Xi* is a singleton {x*}, then any net {xi*}I of (Pi)-optimal solutions converges in C(T, A) to x*. For this case, we construct a finite algorithm with the following property: given any prespecified error δ and any compact subset Q of T, our algorithm computes an i in I and an associated xi* in Xi* which is within δ of x* on Q. We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon.
