The approximate controllability issues for a class of control systems, whose dynamics is described by an equation in a Banach space with a linear degenerate operator at the Riemann — Liouville fractional derivative, is investigated. Under the condition of p-boundedness of the pair of operators in the equation the control system is reduced to subsystems on two mutually complement subspaces. It was shown that the approximate controllability of the whole system is equivalent to the approximate controllability of the two subsystems. Criteria of the approximate controllability for the system and two subsystems are obtained. Analogous results are got on the approximate controllability for free time and for systems of the same form with a finite-dimensional input. The obtained criteria were applied to the investigation of the approximate controllability for a distributed system with polynomials of a differential with respect to the spatial variables self-adjoint elliptic operator and for a system of the Scott-Blair — Oskolkov type.
Analytic resolving families of operators for equations with discretely distributed fractional derivative We study the unique solvability of linear equations in Banach spaces with a discretely distributed Gerasimov -Caputo fractional derivative in terms of analytic resolving families of operators. Necessary and sufficient conditions for the existence of such a family of operators are obtained in terms of the resolvent of a closed operator from the right side of the equation, and their properties are studied. These results are used to prove the existence of a unique solution to the Cauchy problem for a linear inhomogeneous equation of the corresponding class.
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