The Hamilton–Jacobi–Bellman (HJB) equation is a fundamental partial differential equation in optimal control theory and dynamic programming. It provides a necessary condition for an optimal control policy for a given dynamic optimization problem. ### Context In many control problems, we aim to find a control strategy that minimizes (or maximizes) a cost function over time.
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The Hamilton--Jacobi--Bellman equation is the continuous-time Bellman equation. For state dynamics , running cost and terminal cost , a smooth value function satisfies