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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 minimum cost for a given dynamical system with an associated cost function.
When solved locally, the HJB is a necessary condition, but when solved over the whole of state space, the HJB equation is a necessary and sufficient condition for an optimum. The solution is open loop, but it also permits the solution of the closed loop problem. The HJB method can be generalized to stochastic systems as well.
Classical variational problems, 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 on the Hamilton–Jacobi equation by William Rowan Hamilton and Carl Gustav Jacob Jacobi.
Optimal control problems
Consider the following problem in deterministic optimal control over the time period :
where C[ ] is the scalar cost rate function and D[ ] is a function that gives the economic value or utility at the final state, x(t) is the system state vector, x(0) is assumed given, and u(t) for 0 ≤ t ≤ T is the control vector that we are trying to find.
The system must also be subject to
where F[ ] gives the vector determining physical evolution of the state vector over time.
The partial differential equation
For this simple system, the Hamilton–Jacobi–Bellman partial differential equation is
subject to the terminal condition
The unknown scalar in the above partial differential equation is the Bellman value function, which represents the cost incurred from starting in state at time and controlling the system optimally from then until time .
Deriving the equation
Intuitively HJB can be derived as follows. If is the optimal cost-to-go function (also called the 'value function'), then by Richard Bellman's principle of optimality, going from time t to t + dt, we have
Note that the Taylor expansion of the first term is
where denotes the terms in the Taylor expansion of higher order than one in little-o notation. Then if we cancel on both sides, divide by dt, and take the limit as dt approaches zero, we obtain the HJB equation defined above.
Solving the equation
The HJB equation is usually solved backwards in time, starting from and ending at .
When solved over the whole of state space and is continuously differentiable, the HJB equation is a necessary and sufficient condition for an optimum when the terminal state is unconstrained. If we can solve for then we can find from it a control that achieves the minimum cost.
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.
Extension to stochastic problems
The idea of solving a control problem by applying Bellman's principle of optimality and then working out backwards in time an optimizing strategy can be generalized to stochastic control problems. Consider similar as above
now with the stochastic process to optimize and the steering. By first using Bellman and then expanding with Itô's rule, one finds the stochastic HJB equation
where represents the stochastic differentiation operator, and subject to the terminal condition
Note that the randomness has disappeared. In this case a solution of the latter does not necessarily solve the primal problem, it is a candidate only and a further verifying argument is required. This technique is widely used in Financial Mathematics to determine optimal investment strategies in the market (see for example Merton's portfolio problem).
Application to LQG Control
As an example, we can look at a system with linear stochastic dynamics and quadratic cost. If the system dynamics is given by
and the cost accumulates at rate , the HJB equation is given by
- Bellman equation, discrete-time counterpart of the Hamilton–Jacobi–Bellman equation
- Pontryagin's minimum principle, necessary but not sufficient condition for optimum, by minimizing a Hamiltonian, but this has the advantage over HJB of only needing to be satisfied over the single trajectory being considered.
- Bellman, R. E. (1957). Dynamic Programming. Princeton, NJ.
- Bertsekas, Dimitri P. (2005). Dynamic Programming and Optimal Control. Athena Scientific.
- Bellman, R. E. (1954). "Dynamic Programming and a new formalism in the calculus of variations". Proc. Natl. Acad. Sci. 40 (4): 231–235. doi:10.1073/pnas.40.4.231. PMC . PMID 16589462.
- Bellman, R. E. (1957). Dynamic Programming. Princeton.
- Bellman, R.; Dreyfus, S. (1959). "An Application of Dynamic Programming to the Determination of Optimal Satellite Trajectories". J. Brit. Interplanet. Soc. 17: 78–83.
- Bertsekas, Dimitri P. (2005). Dynamic programming and optimal control. Athena Scientific.