nonsmooth maximum principle for control problems in finite dimensional state space

In a standard free end nonsmooth control problem in finite dimensional state space, a nonsmooth maximum principle is proved, in which the adjoint inclusion is sharper than the usual one. For end constrained problems, the same result holds, provided conditions ensuring local controllability are satisfied. The adjoint inclusion is expressed by means of a type of generalized gradient of the pseudoHamiltonian smaller than the standard one (Clarke’s generalized gradient). From the results in this paper, one can recover the standard Pontryagin maximum principle in case of (not necessarily continuous) differentiability with respect to the state. (In end constrained problems, this still holds only if local controllability prevails.) Mathematics Subject Classification 1991: Primary 49K15;Secondary 49J52 Key word: Nonsmooth optimal control, nonsmooth maximum principle, smaller generalized gradients.