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. 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.

[edit] References

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