The Method of Reflection for Solving the Time-Optimal Hamilton-Jacobi-Bellman Equation on the Interval Using Wavelets

In this paper we use the antiderivatives of wavelets to efficiently represent functions which are defined over a bounded interval and which satisfy a boundary condition in the interior of this interval. The functions we want to approximate are typically nonsmooth at the origin. Such functions appear as solutions to Hamilton-Jacobi-Bellman (HJB) equations for time-optimal control problems. We first give the degree of approximation of the antiderivatives and then propose a wavelet reflection algorithm (WRA) to solve numerically the time-optimal HJB equation on the interval. Several numerical examples demonstrate the advantages of the technique developed in this paper over polynomial expansions