First-order necessary conditions in the problem of optimal control of a differential inclusion with phase constraints

Nondegenerate first-order necessary conditions for optimality are obtained for the problem (1.1)–(1.4) under different assumptions about controllability at the endpoints. These necessary conditions are obtained in the Hamiltonian form of Clarke [1]. With the help of a smoothing technique [2] the perturbation method in [3] is used to carry the main results in [4] (there the case when the support function $H(x,t,\psi)=\sup_{y\in F(x,t)}\langle y,\psi\rangle$ depends smoothly on the variable $x$ is considered) over to the more natural class of problems with locally Lipschitz support function $H$.