FIRST-ORDER NECESSARY CONDITIONS IN THE PROBLEM OF OPTIMAL-CONTROL OFA DIFFERENTIAL INCLUSION WITH PHASE CONSTRAINTS

Nondegenerate first-order necessary conditions for optimality are obta ined for the problem (1.1)-(1.4) under different assumptions about con trollability at the endpoints. These necessary conditions are obtained in the Hamiltonian form of Clarke [1] With the help of a smoothing te chnique [2] the perturbation method in [3] is used to carry the main r esults in [4] (there the case when the support function H(x, t, psi) = sup(gamma is an element of F(x,t))[gamma, psi] depends smoothly on th e variable x is considered) over to the more natural class of problems with locally Lipschitz support function H.