Truncation and approximation errors in the max-plus algorithm for H∞ value function computation

We consider the H_∞. problem for a nonlinear system. The corresponding dynamic programming equation (DPE) is a fully nonlinear, first-order, steady-state partial differential equation (PDE), possessing a term which is quadratic in the gradient. The solutions are typically nonsmooth, and further, there are multiple viscosity solutions. The computation of the solution of a nonlinear, steady-state, first-order PDE is typically quite difficult. However, the semi-group associated with such a PDE is linear over the max-plus algebra. Combining this with a basis for the space of semi-convex functions over the max-plus algebra leads to a new class of numerical methods for such problems. Here we examine the errors associated with such a numerical method, and obtain a (conservative) bound on the convergence rate.