The Hamilton–Jacobi–Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. The solution of the HJB equation is the 'value function' which gives the minimum cost for a given dynamical system with an associated cost function. When solved locally, the HJB is a necessary condition, but when solved over the whole of state space, the HJB equation is a necessary and sufficient condition for an optimum. The solution is open loop, but it also permits the solution of the closed loop problem. The HJB method can be generalized to stochastic systems as well...