AIMS Mathematics, 2017, 2(3): 422-436. doi: 10.3934/Math.2017.3.422.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

On the approximate controllability for some impulsive fractional evolution hemivariational inequalities

Mathematics Department, Zunyi Normal College, 563006, Guizhou, P. R. China

In this paper, we study the approximate controllability for some impulsive fractional evolution hemivariational inequalities. We show the concept of mild solutions for these problems. The approximate controllability results are formulated and proved by utilizing fractional calculus, fixed points theorem of multivalued maps and properties of generalized Clarke subgradient under some certain conditions.
  Figure/Table
  Supplementary
  Article Metrics

Keywords Approximate controllability; hemivariational inequalities; fractional di erential; mild solutions; generalized Clarke subdi erential

Citation: Yanfang Li, Yanmin Liu, Xianghu Liu, He Jun. On the approximate controllability for some impulsive fractional evolution hemivariational inequalities. AIMS Mathematics, 2017, 2(3): 422-436. doi: 10.3934/Math.2017.3.422

References

  • 1. A.E. Bashirov and N.I. Mahmudov, On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim, 37 (1999), 1808-1821.    
  • 2. H.F. Bohnenblust and S. Karlin, On a Theorem of Ville, in: Contributions to the Theory of Games,Princeton University Press, Princeton, NJ, 1950, 155-160.
  • 3. P. Cannarsa, G. Fragnelli, and D. Rocchetti, Controllability results for a class of one-dimensionaldegenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616.    
  • 4. S. Carl and D. Motreanu, Extremal solutions of quasilinear parabolic inclusions with generalizedClarke0s gradient, J. Differ. Equa., 191 (2003), 206-233.    
  • 5. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
  • 6. Z. Denkowski, S. Migórski, and N.S. Papageorgiou, An Introduction to Non-linear Analysis: Theory,Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York, 2003.
  • 7. S. Hu and N.S. Papageorgiou, Handbook of multivalued Analysis (Theory), Kluwer AcademicPublishers, Dordrecht Boston, London, 1997.
  • 8. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional DifferentialEquations, North-Holland Math. Studies, Elservier Science B.V. Amsterdam, 204, 2006.
  • 9. M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands,1991.
  • 10. A. Kulig, Nonlinear evolution inclusions and hemivariational inequalities for nonsmooth problemsin contact mechanics. PhD thesis, Jagiellonian University, Krakow, Poland, 2010.
  • 11. S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systemswith bounded delay, J. Differ. Equa., 252 (2012), 6163-6174.    
  • 12. Z.H. Liu, A class of evolution hemivariational inequalities, Nonlinear Anal. Theory Methods Appl.,36 (1999), 91-100.    
  • 13. Z.H. Liu, Anti-periodic solutions to nonlinear evolution equations, J. Funct. Anal., 258 (2010),2026-2033.    
  • 14. Z.H. Liu and X.W. Li, Existence and uniqueness of solutions for the nonlinear impulsive fractionaldifferential equations, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 1362-1373.    
  • 15. Z.H. Liu and X.W. Li, On the Controllability of Impulsive Fractional Evolution Inclusions in BanachSpaces, J. Optim. Theory Appl., 156 (2013), 167-182.    
  • 16. Z.H. Liu and J.Y. Lv, R. Sakthivel, Approximate controllability of fractional functional evolutioninclusions with delay in Hilbert spaces, IMA. J. Math. Control Info., 31 (2014), 363-383.    
  • 17. N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolutionequations in abstract spaces, SIMA. J. Control Optim., 42 (2003), 1604-1622.    
  • 18. S. Migórski and A. Ochal, Existence of solutions for second order evolution inclusions with applicationto mechanical contact problems, Optimization, 55 (2006), 101-120.    
  • 19. S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach,SIAM J. Math. Anal., 41 (2009), 1415-1435.    
  • 20. S. Migórski, A. Ochal, and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities.Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, Springer,New York, 26, 2013.
  • 21. S. Migorski, Existence of Solutions to Nonlinear Second Order Evolution Inclusions without andwith Impulses, Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 18 (2011),493-520.
  • 22. S. Migorski and A. Ochal, Nonlinear Impulsive Evolution Inclusions of Second Order, Dynam.Syst. Appl., 16 (2007), 155-174.
  • 23. P.D. Panagiotopoulos, Nonconvex superpotentials in sense of F.H. Clarke and applications, Mech.Res. Comm., 8 (1981), 335-340.    
  • 24. P.D. Panagiotopoulos, Hemivariational inequalities, Applications in Mechanics and Engineering,Springer, Berlin, 1993.
  • 25. P.D. Panagiotopoulos, Hemivariational inequality and Fan-variational inequality, New Applicationsand Results, Atti. Sem. Mat. Fis. Univ. Modena XLIII, (1995), 159-191.
  • 26. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • 27. K. Rykaczewski, Approximate conrtollability of differential inclusions in Hilbert spaces, NonlinearAnalysis, 75 (2012), 2701-2712.
  • 28. Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Compu.Math. Appl., 59 (2010), 1063-1077.    
  • 29. E. Zuazua, Controllability of a system of linear thermoelasticity, J. Math. Pures Appl.,74 (1995),291-315.
  • 30. Jinrong Wang, M. Fe˜ckan, and Y. Zhou, On the new concept of solutions and existence results forimpulsive fractional evolution equations, Dynamics of PDE, 8 (2011), 345-361.

 

Reader Comments

your name: *   your email: *  

Copyright Info: © 2017, Yanfang Li, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved