  In optimal control,sensitivity relations are usually understood as inclusions
that identify the pair formed by the dual arc and the Hamiltonian, evaluated
along the associated minimizing trajectory, as a suitable generalized gradient
of the value function. In this paper, sensitivity relations are obtained for
the Mayer problem associated with the differential inclusion $\dot x\in F(x)$
and applied to derive optimality conditions. Our first application concerns the
maximum principle and consists in showing that a dual arc can be constructed
for every element of the superdifferential of the final cost. As our second
application, with every nonzero limiting gradient of the value function at some
point $(t,x)$ we associate a family of optimal trajectories at $(t,x)$ with the
property that families corresponding to distinct limiting gradients have empty
intersection.
