SVM approximation of value function contours in target hitting problems

In a problem of target hitting, the capture basin at cost c is the set of states that can reach the target with a cost lower or equal than c, without breaking the viability constraints. The boundary of a c-capture basin is the c-contour of the problem value function. In this paper, we propose a new algorithm that solves target hitting problems, by iteratively approximating capture basins at successive costs. We show that, by a simple change of variables, minimising a cost may be reduced to the problem of time minimisation, and hence a recursive backward procedure can be set. Two variants of the algorithm are derived, one providing an approximation from inside (the approximation is included in the actual capture basin) and one providing a outer approximation, which allows one to assess the approximation error. We use a machine learning algorithm (as a particular case, we consider Support Vector Machines) trained on points of a grid with boolean labels, and we state the conditions on the machine learning procedure that guarantee the convergence of the approximations towards the actual capture basin when the resolution of the grid decreases to 0. Moreover, we define a control procedure which uses the set of capture basin approximations to drive a point into the target. When using the inner approximation, the procedure guarantees to hit the target, and when the resolution of the grid tends to 0, the controller tends to the optimal one (minimizing the cost to hit the target). We illustrate the method on two simple examples, Zermelo and car on the hill problems.