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-n-rectifiable set, then it is lower than or equal to the value function. These results can be used for optimal synthesis approach.