Time-parametric control : uniform convergence of the optimal value functions of discretized problems

The problem of time-optimal control of linear hyperbolic systems is equivalent to the computation of the root of the optimal value function of a time-parametric program, whose feasible set is described by a countable system of moment equations. To compute this root, discretized problems with a finite number of equality constraints can be used. In this paper, we show that on a certain time-interval, the optimal value functions of the discretized problems converge uniformly to the optimal value function of the original problem. We also give sufficient conditions fot Lipschitz and Hoelder continuity of the optimal value function of the original problem.