This section briefly describes a problem for which the HJB equation can be directly solved to yield a closed-form expression, as opposed to an algorithm that computes numerical approximations. Suppose that a linear system is given by (13.37), which requires specifying the matrices and . The task is to design a feedback plan that asymptotically stabilizes the system from any initial state. This is an infinite-horizon problem, and no termination action is applied.
An optimal solution is requested with respect to a cost functional based on matrix quadratic forms. Let be a nonnegative definite^{15.4} matrix, and let be a positive definite matrix. The quadratic cost functional is defined as
Although it is not done here, the HJB equation can be used to derive the algebraic Riccati equation,
(15.21) |
The linear vector field
(15.22) |
(15.23) |
(15.24) |
However, many variations and extensions of the solutions given here do exist, but only for other problems that are expressed as linear systems with quadratic cost. In every case, some variant of Riccati equations is obtained by application of the HJB equation. Solutions to time-varying systems are derived in [28]. If there is Gaussian uncertainty in predictability, then the linear-quadratic Gaussian (LQG) problem is obtained [564]. Linear-quadratic problems and solutions even exist for differential games of the form (13.204) [59].
Steven M LaValle 2012-04-20