A learning theory approach to the computation of reachable sets

We present a proof of convergence of a randomized algorithm for the computation of reachable sets for nonlinear control system. The algorithm uses neural networks to solve a partial differential equation associated with a formulation of the reachability problem as an optimal control problem. Using a recent developments in the learning theory. We prove that with a finite number of training points, our approximation scheme converges within chosen accuracy towards the solution. The number of training points grows polynomially with respect to the dimension of the state space, which gives us a hope to break the curse of dimensionality and a numerical example is presented.