Criterion of the approximate controllability of a class of degenerate distributed systems with the Riemann - Liouville derivative

  • Fedorov Vladimir E., kar@csu.ru Mathematical Analysis Department, Chelyabinsk State University, 129 Kashirin Brothers Street, Chelyabinsk, Russia 454001; Functional Materials Laboratory, South Ural State University (National Research University), 76 Lenin Avenue, Chelyabinsk, Russia 454080
  • Gordievskikh Dmitriy M., dm_gordiev@mail.ru Shadrinsk State Pedagogical University, 3 Karl Liebknecht Street, Shadrinsk, Russia 641870
  • Baleanu Dumitru, dumitru.baleanu@gmail.com Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, Çukurambar Mah. Öǧretmenler Cad. No: 14, 06530 Çankaya, Ankara, Turkey; Institute of Space Science, R-077125 Măgurle-Bucharest, Romania
  • Taş Kenan, kenantas@gmail.com Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, Çukurambar Mah. Öǧretmenle Cad. No: 14, 06530 Çankaya, Ankara, Turkey
Keywords: approximate controllability, degenerate evolution equation, fractional Riemann-Liouville derivative, analytic in a sector resolving family of operators

Abstract

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.

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How to Cite
Fedorov, V., Gordievskikh, D., Baleanu, D. and Taş, K. (2019) “Criterion of the approximate controllability of a class of degenerate distributed systems with the Riemann - Liouville derivative”, Mathematical notes of NEFU, 26(2), pp. 41-59. doi: https://doi.org/10.25587/SVFU.2019.102.31511.
Section
Mathematics