On the inverse Sturm-Liouville problem for spatially symmetric operators,III

In this note it is proved that x(·) a boundary trajectory of a Lipschitz-continuous differential inclusion ? ? F(t, x), x(0) = 0, the tangent cone to F(t, x(t)) at ?(t) that of attainable set E(t) at x(t) coincide for almost every t provided that ?F(t, x) is smooth (similar results with more stringent assumptions were obtained by H. Hermes (J. Differential Equations3 (1967), 256–270) and S. ?ojasiewicz, Jr. (Asterisque75–76 (1980), 187–197)). It is also proved that the outward normal to these cones along the trajectory is Lipschitz-continuous (in t). Moreover, using the lower, one-side, directional derivative instead of F. H. Clarke's generalised gradient, first-order necessary conditions are obtained, which can be stronger than those of Clarke (in “International Symposium on the Calculus of Variation and Optimal Control, University of Wisconsin, Madison, Wisconsin, September 1975”). The main ideas of this paper were presented in J. Hale's seminar at Brown University (March 1976).