Mixed control for linear infinite-dimensional systems of fractional order. (Russian. English summary) Zbl 1471.93148

Summary: Problem with a mixed control, start and distributed simultaneously, are considered for time-fractional order evolution equations. The results on the solvability of the mixed control problems for linear non-degenerate and degenerate equations with the Gerasimov-Caputo fractional derivative are obtained. It is shown that at some additional conditions a solution of the considered problem is unique. General results are used for consideration of abstract problems with specific quality functionals. Abstract results of the work are illustrated by the example of a mixed control problem for the time-fractional order system of gravitational-gyroscopic waves.


93C35 Multivariable systems, multidimensional control systems
93C05 Linear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
26A33 Fractional derivatives and integrals
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[1] F. Mainardi, “The time fractional diffusion-wave equations”, Radiophysics and Quantum Electronics, 38 (1995), 13-24
[2] F. Mainardi, Y. F. Luchko, G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation”, Fractional Calculus and Applied Analysis, 4:2 (2001), 153-192 · Zbl 1054.35156
[3] Uchaikin V.V., Fractional derivatives method, Artishok, Ulyanovsk, 2008, 510 pp. (In Russ.)
[4] V. E. Tarasov, Fractional dynamics, Higher Education Press, Beijing, 2010, 504 pp. · Zbl 1214.81004
[5] Plekhanova M.V., Islamova A.F., “Solvability of mixed-type optimal control problems for distributed systems”, Russian Mathematics, 55:7 (2011), 30-39 · Zbl 1302.49005
[6] Plekhanova M.V., Islamova A.F., “Problems with a robust mixed control for the linearized Boussinesq equation”, Differential Equations, 48:4 (2012), 574-585 · Zbl 1250.49006
[7] Shuklina A.F., Plekhanova M.V., “Mixed control problems for the Sobolev system”, Chelyabinsk Physical and Mathematical Journal, 1:2 (2016), 78-84 (In Russ.) · Zbl 1464.49004
[8] Mixed control problem for the linearized quasi-stationary phase field system of equations, “Plekhanova M.V.”, Materials Science Forum, 845 (2016), 170-173
[9] Plekhanova M.V., “Start control problems for evolution equations of fractional order”, Chelyabinsk Physical and Mathematical Journal, 1:3 (2016), 16-37 (In Russ.)
[10] Plekhanova M.V., “Strong solutions to nonlinear degenerate fractional order evolution equations”, Journal of Mathematical Sciences, 230:1 (2018), 146-158 · Zbl 1413.34041
[11] M. V. Plekhanova, “Degenerate distributed control systems with fractional time derivative”, Ural Mathematical Journal, 2:2 (2016), 58-71 · Zbl 1424.49007
[12] M. V. Plekhanova, “Optimal control for quasilinear degenerate systems of higher order”, Journal of Mathematical Sciences, 219 (2016), 236-244 · Zbl 1354.49008
[13] Plekhanova M.V., “Solvability of control problems for degenerate evolution equations of fractional order”, Chelyabinsk Physical and Mathematical Journal, 2:1 (2017), 53-65 (In Russ.) · Zbl 1465.49006
[14] M. V. Plekhanova, “Distributed control problems for a class of degenerate semilinear evolution equations”, Journal of Computational and Applied Mathematics, 312 (2017), 39-46 · Zbl 1350.49004
[15] M. V. Plekhanova, G. D. Baybulatova, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), v. 11548 LNCS, 2019, Problems of hard control for a class of degenerate fractional order evolution equations
[16] Fursikov A.V., Optimal Control of Distributed Systems. Theory and Applications., American Mathematical Society, 2000, 305 pp. (In Russ.) · Zbl 1027.93500
[17] E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, PhD thesis, University Press Facilities, Eindhoven University of Technology, Eindhoven, 2001, 107 pp. · Zbl 0989.34002
[18] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht, Boston, 2003 · Zbl 1102.47061
[19] Demidenko G.V., Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative, Marcel Dekker, Inc., New York, Basel, 2003, 481 pp. · Zbl 1061.35001
[20] Sobolev S.L., “On a new problem for systems of partial differential equations”, Reports of the USSR Academy of Sciences, 81:6 (1951), 1007-1009 (In Russ.) · Zbl 0044.09501
[21] Sobolev S.L., “On a new problem of mathematical physics”, News of the USSR Academy of Sciences, 18 (1954), 3-50 (In Russ.) · Zbl 0055.08401
[22] Plekhanova M.V., Baybulatova G.D., “Optimal control problems for a class of degenerate evolution equations with delay”, Chelyabinsk Physical and Mathematical Journal, 3:3 (2018), 319-331 (In Russ.) · Zbl 1465.49008
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