On the strong convergence of derivatives in a time optimal problem

We consider a time optimal problem for a system described by a differential inclusion, whose right hand side is not necessarily convex valued. Under the assumption of strict convexity of the map obtained by convexifying the original, non-convex valued map, we obtain the strong convergence of the derivatives of any uniformly converging minimizing sequence. The assumptions required by this result are satisfied, for instance, by the classical brachystochrone problem and by Fermat's principle. © 2007 Elsevier Ltd. All rights reserved.