Definitions

# 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

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

subject to

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

where $x\left(t\right)$ is the system state, $x\left(0\right)$ is assumed given, and $u\left(t\right)$ for $0leq tleq T$ 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{ leftlangle frac{partial}{partial x}V(x,t), F(x, u) rightrangle + C(x,u) right} = 0

subject to the terminal condition


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

The unknown $V\left(t, x\right)$ 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. 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.

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