Criterion of the approximate controllability of a class of degenerate distributed systems with the Riemann - Liouville derivative
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.
 Fedorov V. E. and Avilovich A. S., “A Cauchy type problem for a degenerate equation with the Riemann–Liouville derivative in the sectorial case,” Sib. Math. J, 60, No. 2, 359–372 (2019).
 Curtain R. F., “The Salamon–Weiss class of well-posed infinite dimensional linear systems: a survey,” IMA J. Math. Control Inform., 14, No. 2, 207–223 (1997).
 Sholokhovich F. A., “On controllability of linear dynamical systems [in Russian],” Izv. Uralsk. Gos. Univ., 10, No. 1, 103–126 (1998).
 Debbouche A. and Baleanu D., “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,” Comput. Math. Appl., 62, No. 3, 1442–1450 (2011).
 Chalishajar D. N., Malar K., and Karthikeyan K., “Approximate controllability of abstract impulsive fractional neutral evolution equations with infinite delay in Banach spaces,” Electron. J. Differ. Equ., 2013, No. 275, 1–21 (2013).
 Fedorov V. E. and Ruzakova O. A., “Controllability of linear Sobolev type equations with relatively p-radial operators,” Russ. Math., 46, No. 7, 54–57 (2002).
 Fedorov V. E. and Ruzakova O. A., “One-dimensional controllability of Sobolev linear equations in Hilbert spaces,” Differ. Equ., 38, iss. 8, 1216–1218 (2002).
 Fedorov V. E. and Ruzakova O. A., “Controllability in dimensions of one and two of Sobolevtype equations in Banach spaces,” Math. Notes, 74, No. 4, 583–592 (2003).
 Ruzakova O. A. and Fedorov V. E., “On ε-controllability of linear equations, not solved with respect to the derivative, in Banach spaces [in Russian],” Vychisl. Tekhnol., 10, No. 5, 90–102 (2005).
 Fedorov V. E. and Shklyar B., “Exact null controllability of degenerate evolution equations with scalar control,” Sb. Math., 203, No. 12, 1817–1836 (2012).
 Plekhanova M. V. and Fedorov V. E., Optimal Control for Degenerate Distributed Systems, Chelyabinsk: Izdat. Tsentr Yuzhno-Uralsk. Gos. Univ., 2013. (In Russian).
 Plekhanova M. V. and Fedorov V. E., “On controllability of degenerate distributed systems [in Russian],” Ufa. Math. J., 6, No. 2, 77–96 (2014).
 Fedorov V. E., Gordievskikh D. M., and Baybulatova G. D., “Controllability of a class of weakly degenerate fractional order evolution equations,” in: AIP Conf. Proc., 1907, 020009-1–020009-14 (2017).
 Fedorov V. E., Gordievskikh D. M., and Turov M. M., “Infinite-dimensional and finitedimensional ε-controllability for a class of fractional order degenerate evolution equations [in Russian],” Chelyab. Fiz.-Mat. Zh., 3, No. 1, 5–26 (2018).
 Fedorov V. E. and Gordievskikh D. M., “Approximate controllability of strongly degenerate fractional order system of distributed control,” in: IFAC-PapersOnLine, 17th IFAC Workshop on Control Applications of Optimization (CAO 2018, Yekaterinburg, Russia, Oct. 15–19, 2018), 51, No. 32, 675–680 (2018).
 Fedorov V. E., Romanova E. A., and Debbouche A., “Analytic in a sector resolving families of operators for degenerate evolution fractional equations,” J. Math. Sci., 228, No. 4, 380–394 (2018).
 Romanova E. A. and Fedorov V. E., “Resolving operators of linear degenerate evolution equation with the Caputo derivative. The sectorial case [in Russian],” Math. Zamet. SVFU, 23, No. 4, 58–72 (2016).
 Fedorov V. E. and Romanova E. A., “Inhomogeneous evolution equations of fractional order in the sectorial case [in Russian],” Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obzory, 149, 103–112 (2018).
 Bajlekova E. G., Fractional Evolution Equations in Banach Spaces: PhD Thes., Eindhoven Univ. Technology, Univ. Press Facilities, Eindhoven (2001).
 Solomyak M. Z., “Applications of the theory of semigroups to the study of differential equations in Banach spaces,” Dokl. Math., 122, No. 5, 766–769 (1958).
 Yosida K., Functional Analysis, Springer-Verl., Berlin (1965).
 Fedorov V. E., “A class of fractional order semilinear evolutions in Banach spaces,” in: Integral Equations and Their Applications, Proc. Univ. Network Seminar on the occasion of the 3rd Mongolia–Russia–Vietnam Workshop (NSIDE 2018, Hanoi Math. Soc.), pp. 11–20, Hung Yen Univ. Tech. Edu., Hung Yen (2018).
 Plotnikov P. I. and Starovoitov V. N., “The Stefan problem with surface tension as the limit of a phase field model,” Differ. Equ., 29, No. 3, 395–404 (1993).
 Plotnikov P. I. and Klepacheva A. V., “The phase field equations and gradient flows of marginal functions,” 42, No. 3, 651–669 (2001).
 Bateman H. and Erdel´yi A., Higher Transcendental Functions, V. 3, McGraw-Hill Book Co., New York (1953).
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