Hamilton-Jacobi-Bellman equation

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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 optimal cost-to-go for a given dynamical system with an associated cost function. Classical variational problems, for example, the brachistochrone problem can be solved using this method as well.

The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers.^[1] 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 on the Hamilton-Jacobi equation by William Rowan Hamilton and Carl Gustav Jacob Jacobi.

Consider the following problem in deterministic optimal control

$\min \int_0^T C[x(t),u(t)]\,dt + D[x(T)]$

subject to

$\dot{x}(t)=F[x(t),u(t)]$

where $x (t)$ is the system state, $x (0)$ is assumed given, and $u (t)$ for $0\leq t\leq T$ is the control that we are trying to find. For this simple system, the Hamilton Jacobi Bellman partial differential equation is

$\frac{\partial}{\partial t} V(x,t) + \min_u \left\{ \left\langle \frac{\partial}{\partial x}V(x,t), F(x, u) \right\rangle + C(x,u) \right\} = 0$

subject to the terminal condition

$V(x,T) = D(x).\,$

The unknown $V (t, x)$ in the above PDE is the Bellman 'value function', which represents the cost incurred from starting in state $x$ at time $t$ and controlling the system optimally from then until time $T$ . The HJB equation needs to be solved backwards in time, starting from $t = T$ and ending at $t = 0$ . (The notation $\langle a,b \rangle$ means the inner product of the vectors a and b).

The HJB equation is a sufficient condition for an optimum.^{[citation needed]} If we can solve for $V$ then we can find from it a control $u$ that achieves the minimum cost.

The HJB method can be generalized to stochastic systems as well.

In general case, the HJB equation does not have a classical (smooth) solution. Several notions of generalized solutions have been developed to cover such situations, including viscosity solution (Pierre-Louis Lions and Michael Crandall), minimax solution (Andrei Izmailovich Subbotin), and others.