A Necessary and Sufficient Condition for Optimality of Dynamic Programming Type, Making No a Priori Assumptions on the Controls
Richard Vinter, Richard M. Lewis · SIAM Journal on Control and Optimization · 1978
A well-known sufficient condition for optimality in control theory, given in terms of a solution to the Bellman partial differential equation, is considered. It is shown that if the equation is relaxed to inequality, and if the resulting inequality is required to be satisfied only in a limiting sense, then the condition becomes also necessary for optimality. In contrast to previous results of a similar nature, this is accomplished without making regularity assumptions about optimal feedback controls. The results here are obtained through application of convex analysis to the weak version of the control problem studied in a companion paper.