Verification Theorems for Hamilton--Jacobi--Bellman Equations

We study an optimal control problem in Bolza form and we consider the value function associated to this problem. We prove two verification theorems which ensure that, if a function W satisfies some suitable weak continuity assumptions and a Hamilton-Jacobi-Bellman inequality outside a countably H-rectifiable set, then it is lower or equal to the value function. These results can be used for optimal synthesis approach.