Sufficient conditions for relative minima of broken extremals in optimal control theory

We formulate a version of the method of characteristics based on parametrizations of extremals by their terminal values. Sufficient conditions are given for imbedding a reference trajectory into a local field of broken extremals. For a problem with terminal manifold of codimension 1 it is shown that a broken extremal is a relative minimum if (i) the restrictions of the flow to intervals where the control is continuous have nonsingular partial derivatives with respect to the parameter and (ii) the switching surfaces are crossed transversally.