Hamilton–Jacobi–Bellman equation

Summary

The Hamilton–Jacobi–Bellman equation is a first-order partial differential equation for the value function of an optimal control problem. It is the continuous-time form of the principle of dynamic programming: an optimal strategy stays optimal from every later state. A solution gives the optimal cost from every starting point and, through a pointwise minimization, an optimal feedback control. In general the value function is not differentiable, and the equation is understood in the sense of viscosity solutions.