Analysis of the Asymptotic Behavior of Optimal Control Trajectories: The Implicit Programming Problem

The asymptotic behavior of the optimal trajectories of the infinite horizon control problem with discounting, is characterized by a static optimization problem. In the undiscounted case, the limit point of the optimal dynamic trajectory is the steady-state that minimizes the kernel of the objective functional. The corresponding static characterization of the limit point in the discounted case, called the implicit programming problem, is derived. The implicit programming problem is a mathematical programming problem with the special feature that part of the solution is contained in the definition of the problem. All results are achieved in the context of a sufficient maximum principle, which is shown to be equivalent to the other approaches taken in the literature to perform the dynamic analysis. The equivalence is based on convexity conditions assumed in the current dynamic theory. The class of problems that satisfy such convexity conditions is characterized in terms of a property of vector-valued mapping...