SOME NECESSARY OPTIMALITY CONDITIONS FOR SYSTEMS WITH FRACTIONAL CAPUTO DERIVATIVES

In this paper, we consider an optimal control problem in which a dynamical system is controlled by a nonlinear Caputo fractional state equation. First, an analogue of the Pontryagin maximum principle is obtained, and in the case of the degeneration of the Pontryagin maximum principle, a high-order necessary optimality condition is obtained. Further, if the control under study lies inside the set of restrictions on the control, then we obtain an analogue of the Euler equation, an analogue of the Legendre-Clebsch condition, and when the Legendre-Clebsch condition degenerates, we obtain the necessary high-order optimality condition.