Sensitivity interpretations of the costate function of optimal control

It is well known that, for optimal control problems with freeright endpoints, the costate function P() of the maximum principlehas a sensitivity interpretation in terms of the value functionV. Specifically, if () is an optimal state trajectory and h(t) is the hamiltonian function evaluated alongp(t), (t), and the optimal control, we have where V is the Clarke generalized gradient.Recently Subbotina has improved this result, replacing the Clarkegeneralized gradient by the tighter ‘strict differential’.We provide a simple proof of this refinement, weaken the hypothesesunder which it is known to apply, and extend it to cover alsoproblems involving right endpoint constraints.