DEGENERATE DISTRIBUTED CONTROL SYSTEMS WITH FRACTIONAL TIME DERIVATIVE

Marina V. Plekhanova     (Computational Mechanics Department, South Ural State Universit and Laboratory of Quantum Topology, Chelyabinsk State University, Chelyabinsk, Russian Federation)

Abstract


The existence of a unique strong solution for the Cauchy problem to semilinear nondegenerate fractional differential equation and for the generalized Showalter–Sidorov problem to semilinear fractional differential equation with degenerate operator at the Caputo derivative in Banach spaces is proved. These results are used for search of solution existence conditions for a class of optimal control problems to a system described by the degenerate semilinear fractional evolution equation. Abstract results are applied to the research of an optimal control problem solvability for the equations system of Kelvin–Voigt fractional viscoelastic fluids.


Keywords


Fractional dierential calculus, Caputo deivative, Mittag-Leer function, Partial dierential equation, Degenerate evolution equation, Optimal control, Fractional viscoelastic fluid

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References


  1. Bajlekova E.G. Fractional Evolution Equations in Banach Spaces // PhD thesis, Eindhoven University of Technology, University Press Facilities, 2001.
  2. Mainardi F., Spada G. Creep, relaxation and viscosity properties for basic fractional models in rheology // The European Physics Journal, Special Topics, 2011. Vol. 193. P. 133–160.
  3. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. VSP, Utrecht, Boston, 2003.
  4. Fedorov V.E., Davydov P.N. On nonlocal solutions of semilinear equations of the Sobolev type // Differential Equations. 2013. Vol. 49, no. 3. P. 338–347. DOI: 10.1134/S0012266113030087
  5. Fedorov V.E., Gordievskikh D.M. Resolving operators of degenerate evolution equations with fractional derivative with respect to time // Russian Math., 2015. Vol. 59. P. 60–70. DOI: 10.3103/S1066369X15010065
  6. Fedorov V.E., Gordievskikh D.M., Plekhanova M.V. Equations in Banach spaces with a degenerate operator under a fractional derivative // Differential Equations, 2015. Vol. 51. P. 1360–1368.
  7. Fedorov V.E., Debbouche A. A class of degenerate fractional evolution systems in Banach spaces // Differential Equations, 2013. Vol. 49, no. 12. P. 1569–1576.
  8. DOI: 10.1134/S0012266113120112
  9. Gordievskikh D.M., Fedorov V.E. Solutions for initial boundary value problems for some degenerate equations systems of fractional order with respect to the time // The Bulletin of Irkutsk State University. Series Mathematics, 2015. Vol. 12. P. 12–22.
  10. Plekhanova M.V. Quasilinear equations that are not solved the higher-order time derivative // Siberian Mathematical Journal, 2015. Vol. 56. P. 725–735.
  11. Plekhanova M.V. Nonlinear equations with degenerate operator at fractional Caputo derivative // Mathematical Methods in the Applied Sciences. 2016. In press. DOI: 10.1002/mma.3830
  12. Kostic M. Abstract time-fractional equations: Existence and growth of solutions // Fractional Calculus and Applied Analysis. 2011. Vol. 14, no. 2. P. 301–316.
  13. Glushak A.V. Correctness of Cauchy-type problems for abstract differential equations with fractional derivatives // Russian Mathematics. 2009. Vol. 53, no. 9. P. 10–19.
  14. Vorob'eva S.A., Glushak A.V. An abstract Euler–Poisson–Darboux equation containing powers of an unbounded operator // Differential Equations. 2001. Vol. 37, no. 5. P. 743–746.
  15. Fursikov, A.V. Optimalnoe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya. Optimal Control of Distributed Systems. Theory and Applications, Novosibirsk, 1999. [In Russian]
  16. Fedorov V.E., Plekhanova M.V. Optimal control of Sobolev type linear equations // Differential equations, 2004. Vol. 40. P. 1548–1556.
  17. Fedorov V.E., Plekhanova M.V. The problem of start control for a class of semilinear distributed systems of Sobolev type // Proceeding of the Steklov institute if mathematics. 2011. Vol. 275. P. 40–48.
  18. Plekhanova M.V. Distributed control problems for a class of degenerate semilinear evolution equations // J. of Computational and Applied Mathematics, 2017. Vol. 312. P. 39–46. DOI: 10.1016/j.cam.2015.09.034
  19. Kochubei N. Fractional-parabolic systems // Potential Anal. 2012. Vol. 37. P. 1–30.
  20. Kamocki R. On the existence of optimal solutions to fractional optimal control problems // Applied Mathematics and Computation. 2014. Vol. 235. P. 94–104.
  21. Kerboua M., Debbouche A., Baleanu D. Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces // Electronic J. of Qualitative Theory of Differential Equations, 2014. Vol. 58. P. 1–16.
  22. Showalter R.E. Nonlinear degenerate evolution equations and partial differential equations of mixed type // SIAM J. Math. Anal., 1975. Vol. 6. P. 25–42. DOI: 10.1137/0506004
  23. Sidorov N.A. A class of degenerate differential equations with convergence // Math. Notes., 1984. Vol. 35. P. 300–305.
  24. Ladyzhenskaya O.A. The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, Science Publishers, New York, London, Paris, 1969.



DOI: http://dx.doi.org/10.15826/umj.2016.2.006

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