Necessary Conditions for Optimality in Problems of Optimal Control of Systems with Discontinuous Right-Hand Side

We consider the problems of optimal control of a dynamical system whose right-hand side is discontinuous in the state variable and is linear in the control with sufficiently smooth coefficients in each of the half-spaces into which the space is divided by the switching hyperplane. The main attention is paid to the situation where there exist intervals on which the optimal trajectory lies on the switching surface. New nondegenerate necessary conditions for optimality are stated and proved in the maximum principle form. The obtained optimality conditions are compared with the already known conditions.