Convergence of an iterative algorithm for solving Hamilton-Jacobi type equations

Solutions of the optimal control and -control problems for nonlinear affine systems can be found by solving Hamilton-Jacobi equations. However, these first order nonlinear partial differential equations can, in general, not be solved analytically. This paper studies the rate of convergence of an iterative algorithm which solves these equations numerically for points near the origin. It is shown that the procedure converges to the stabilizing solution exponentially with respect to the iteration variable. Illustrative examples are presented which confirm the theoretical rate of convergence.