Convergence Results for Smooth Regularizations of Hybrid Nonlinear Optimal Control Problems

We consider a class of hybrid nonlinear optimal control problems having a discontinuous dynamics ruled by a partition of the state space. For this class of problem, some hybrid versions of the usual Pontryagin Maximum Principle are known. We introduce general regularization procedures, parameterized by a small parameter, smoothing the previous hybrid problems to standard smooth optimal control problems, for which we can apply the usual Pontryagin Maximum Principle. We investigate the question of the convergence of the resulting extremals as the regularization parameter tends to zero. Under some general assumptions, we prove that smoothing regularization procedures converge, in the sense that the solution of the regularized problem (as well as its extremal lift) converges to the solution of the initial hybrid problem. To illustrate our convergence result, we apply our approach to the minimal time low-thrust coplanar orbit transfer with eclipse constraint.