Browsing by Author "Fedorov, Vladimir E."

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Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case
(Mdpi, 2019) Baleanu, Dumitru; Baleanu, Dumitru; Fedorov, Vladimir E.; Taş, Kenan; Gordievskikh, Dmitriy M.; Tas, Kenan; 4971; 56389
We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter-Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input.
Criterion of the approximate controllability of a class of degenerate distributed systems with the riemann--liouville derivative
(M. K. Ammosov North-Eastern Federal University, 2019) Baleanu, Dumitru; Taş, Kenan; Baleanu, Dumitru; Taş, Kenan; 56389; 4971
The issues of approximate controllability in fixed time and in free time of a class of distributed control systems whose dynamics are described by linear differential equations of fractional order in reflexive Banach spaces are investigated. It is assumed that the operator at the fractional Riemann-Liouville derivative has a non-trivial kernel, i. e., the equation is degenerate, and the pair of operators in the equation generates an analytic in a sector resolving family of operators of the corresponding homogeneous equation. The initial state of the control system is set by the Showalter-Sidorov type conditions. To obtain a criterion for the approximate controllability, the system is reduced to a set of two subsystems, one of which has a trivial form and the another is solved with respect to the fractional derivative. The equivalence of the approximate controllability of the system and of the approximate controllability of its two mentioned subsystems is proved. A criterion of the approximate controllability of the system is obtained in terms of the operators from the equation. The general results are used to find a criterion for the approximate controllability for a distributed control system, whose dynamics is described by the linearized quasistationary system of the phase field equations of a fractional order in time, as well as degenerate systems of the class under consideration with finite-dimensional input. © 2019 V. E. Fedorov, D. M. Gordievskikh, D. Baleanu, and K. Taş.