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 a value function, which gives the optimal cost-to-go for a given dynamical system with an associated cost function. For example, the brachistochrone problem can be solved using this method.
The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. The corresponding discrete-time equation is usually referred to as the Bellman equation. In continuous time, the result can be seen as an extension of earlier work in classical physics by William Rowan Hamilton and Carl Gustav Jacob Jacobi. For this reason the equation is sometimes referred to simply as the Hamilton-Jacobi equation, when the system contains no stochastic term.
In Hamiltonian mechanics, the Hamilton-Jacobi-Bellman equation is very close to quantum mechanics, and in particular, the WKB approximation
where S is a function of position x and time t. The momentum p is given by
