Mixed control for degenerate nonlinear equations with fractional derivatives

A class of problems of mixed control of systems is considered, the state of which is described by equations in Banach spaces that are not solvable with respect to the Gerasimov — Caputo highest fractional derivative and depend nonlinearly on lower-order fractional derivatives. The condition of 0-boundedness of a pair of operators in the linear part of the equation is used, which allows one to set the Schwalter — Sidorov initial conditions for the differential equation under study. The nonlinear operator is assumed to depend only on the elements of the subspace without degeneration. The objective functional in the mixed control problem is assumed to be convex, lower semicontinuous, and coercive, while the set of admissible controls is assumed to be non-empty, convex, and closed. A theorem on the existence of an optimal control is obtained. Abstract results are used in the study of the mixed control problem for the modified Sobolev equation of fractional order in time.