CONVEX DUALITY AND NONLINEAR OPTIMAL-CONTROL

Problems in nonlinear optimal control can be reformulated as convex optimization problems over a vector space of linear functionals. In this way, methods of convex analysis can be brought to bear on the task of characterizing solutions to such problems. The result is a necessary and sufficient condition of optimality that generalizes well-known sufficient conditions, referred to as verification theorems, in dynamic programming; as a byproduct, we obtain a representation of the minimum cost in terms of the upper envelope of subsolutions to the Hamilton-Jacobi equation. It is a striking illustration of the wide range of problems to which convex analysis, and, in particular, convex duality, is applicable. The approach, applied to parametric problems in the calculus of variations, was pioneered by L. C. Young [Lectures on the Calculus of Variations and Optimal Control Theory, W. B. Saunders, Philadelphia, PA, 1969]. As recent work has shown, however, it is equally fruitful when applied in optimal control. This paper, which is expository, offers a self-contained treatment of the application of methods of convex duality to general nonlinear problems in deterministic optimal control. At the same time, it provides extensions of previously published results in several directions. A simple proof is given of the main ''convex closure'' theorem relating generalized flows and relaxed arcs; this is based on mollification techniques recently developed by Fleming and Vermes [SIAM J. Control Optim., 27 (1989), pp. 1136-1155] for constructing smooth subsolutions to the Hamilton-Jacobi equation.