Hamilton Jacobi theory for optimal control problems with data measurable in time

Dynamic programming provides necessary and sufficient conditions on minimizing arcs in terms of solutions to the hamiltonian 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. Results in which this restriction is lifted are presented. The right side of the differential inclusion is merely required to be measurable in time and Lipshitz continuous in the state variable and to satisfy the growth conditions of Valadier´s existence theory. In this setting it is found that verification functions can be defined in terms of lower Dini solutions of the Hamilton Jacobi inequality. The value function is the upper envelope of the set of all verification functions