Hamilton–Jacobi–Bellman equation

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. 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', that is 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. 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.

References

• R. E. Bellman. Dynamic Programming. Princeton, NJ, 1957.