Hamilton–Jacobi Theory for Optimal Control Problems with Data Measurable in Time
Richard Vinter, Peter R. Wolenski · SIAM Journal on Control and Optimization · 1990
Hamilton–Jacobi theory provides necessary and sufficient conditions on minimizing arcs in terms of solutions to the Hamilton–Jacobi equation or inequality. The hypotheses under which such results have previously been obtained typically require the data to be continuous in its time-dependence. The present paper lifts this restriction. The basic hypotheses are Carathéodory-type with measurable time and Lipschitz state dependence, and they incorporate the growth condition of Valadier’s existence theory. It is shown that the value function is a solution to the Hamilton–Jacobi equation in an extended sense defined in terms of lower Dini directional derivatives, and that solutions of the related inequality furnish verification functions. Moreover, a characterization of the value function is provided as the pointwise maximum of the family of all verification functions. The methods developed to take account of the measurable time-dependence are based on a “uniform” Lebesgue point theorem for integrably bounded set-valued functions.