Y. Feng, B.D.O. Anderson, and M. Rotkowitz
A Game Theoretic Algorithm to Compute Local Stabilizing Solutions to HJBI Equations in Nonlinear H Control
Automatica, vol. 45, no. 4, pp. 881-888, April 2009.

Abstract

In this paper, an iterative algorithm to solve Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations for a broad class of nonlinear control systems is proposed. By constructing two series of nonnegative functions, we replace the problem of solving an HJBI equation by the problem of solving a sequence of Hamilton-Jacobi-Bellman (HJB) equations whose solutions can be approximated recursively by existing methods. The local convergence of the algorithm and local quadratic rate of convergence of the algorithm are guaranteed and a proof is given. Numerical examples are also provided to demonstrate the effectiveness of the proposed algorithm. A game theoretical interpretation of the algorithm is given.