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Hamilton-Jacobi-Bellman (HJB) equation is the sufficient and necessary condition for continuous-time optimal control problem (OCP). Different from HJB equation in infinite horizon, finite-horizon HJB equation contains a time-dependent value function, whose partial derivative with respect to time is an intractable unknown term. My study has found that the partial derivative exactly equals the terminal-time utility function by analyzing the initial-time equivalency between fixed time horizon OCP and fixed terminal time OCP. We also provide another proof, which uses the definition of partial derivative. This finding allows reusing traditional approximate dynamic programming (ADP) algorithm to approximate optimal policy with a parameterized function like neural network, thus solving the continuous-time finite-horizon OCP. The correctness of our finding is evaluated by analyzing a linear quadratic problem.
Lin et al. (Fri,) studied this question.