Finite-dimensional approximations in the derivation of necessary optimality conditions in nonsmooth constrained optimal control

Traditional proofs of the Pontryagin Maximum Principle (PMP) require the continuous differentiability of the dynamics with respect to the state variable on a neighborhood of the minimizing state trajectory, when arbitrary values of control variable are inserted into the dynamic equations. Sussman has drawn attention to the fact that the PMP remains valid when the dynamics are differentiable with respect to the state variable, merely when the minimizing control is inserted into the dynamic equations. This weakening of earlier hypotheses has been referred to as the Lojasiewicz refinement. Arutyunov and Vinter showed that these extensions of early versions of the PMP can be simply proved by finite approximations, application of a Lagrange Multiplier Rule in finite dimensions and passage to the limit. This paper generalizes the finite approximations technique to a problem with state constraints, where the use of needle variations of the optimal control had not been successful. Moreover, the cost function and endpoint constraints are not assumed to be differentiable, but merely Lipschitz continuous around the optimal trajectory. The Maximum Principle is expressed in terms of Michel-Penot subdifferential.