Value function

Summary

The value function of an optimal control problem gives the optimal cost as a function of the initial time and the initial state. It is the central object of dynamic programming: it satisfies the Hamilton–Jacobi–Bellman equation, in the viscosity sense when it is not smooth. Its gradient, where it exists, is the costate of Pontryagin's maximum principle up to sign. In economics the same notion appears as the indirect utility or the optimal value of a maximization problem.