Approximate Controllability of Fractional Differential Equations with State-Dependent Delay

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• 作者：Sakthivel Rathinasamy (1)
  Ren Yong (2)
  
• 关键词：34A08 ; 93B05 ; 47H10 ; Nonlinear fractional systems ; state ; dependent delay ; approximate controllability
• 刊名：Results in Mathematics
• 出版年：2013
• 出版时间：June 2013
• 年：2013
• 卷：63
• 期：3-4
• 页码：949-963
• 全文大小：250KB
• 参考文献：1. Benchohra M., Ouahab A.: Controllability results for functional semilinear differential inclusions in Frechet spaces. Nonlinear Anal. Theor. 61, 405鈥?23 (2005) CrossRef
  2. Abada N., Benchohra M., Hammouche H.: Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differ. Equ. 246, 3834鈥?863 (2009) CrossRef
  3. Klamka J.: Constrained controllability of semilinear systems with delays. Nonlinear Dynam. 56, 169鈥?77 (2009) CrossRef
  4. Klamka J.: Constrained controllability of semilinear systems with delayed controls. Bull. Pol. Ac. Tech. 56, 333鈥?37 (2008)
  5. G贸rniewicz L., Ntouyas S.K., O鈥橰egan D.: Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces. Rep. Math. Phys. 56, 437鈥?70 (2005) CrossRef
  6. G贸rniewicz L., Ntouyas S.K., O鈥橰egan D.: Existence and controllability results for first and second order functional semilinear differential inclusions with nonlocal conditions. Numer. Funct. Anal. Optim. 28, 53鈥?2 (2007) CrossRef
  7. G贸rniewicz L., Ntouyas S.K., O鈥橰egan D.: Controllability results for first and second order evolution inclusions with nonlocal conditions. Ann. Polon. Math. 89, 65鈥?01 (2006) CrossRef
  8. Mahmudov N.I.: Approximate controllability of evolution systems with nonlocal conditions. Nonlinear Anal. Theor. 68, 536鈥?46 (2008) CrossRef
  9. Sakthivel R., Ren Y., Mahmudov N.I.: Approximate controllability of second-order stochastic differential equations with impulsive effects. Modern Phys. Lett. B 24, 1559鈥?572 (2010) CrossRef
  10. Sakthivel R., Juan J.Nieto, Mahmudov N.I.: Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay. Taiwan J. Math. 14, 1777鈥?797 (2010)
  11. Sakthivel R.: Approximate controllability of impulsive stochastic evolution equations. Funkcialaj Ekvacioj 52, 381鈥?93 (2009) CrossRef
  12. Klamka J.: Constrained approximate controllability. IEEE. T. Automat. Contr. 45, 1745鈥?749 (2000) CrossRef
  13. Mahmudov N.I.: Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim. 42, 1604鈥?622 (2003) CrossRef
  14. Mophou G.M.: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. Theor. 72, 1604鈥?615 (2010) CrossRef
  15. Podlubny I.: Fractional differential equations. San Diego Academic Press, San Diego (1999)
  16. Hilfer, R. (eds): Applications of Fractional Calculus in Physics. World Scientific, River Edge (2000)
  17. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204 (2006)
  18. Zhou Y., Jiao F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063鈥?077 (2010) CrossRef
  19. Wang J., Zhou Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. Real. 12, 262鈥?72 (2011) CrossRef
  20. Wang J., Zhou Y.: Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal. Real World Appl. 12, 3642鈥?653 (2011) CrossRef
  21. Wang, R.N., Yang, Y.H.: On the cauchy problems of fractional evolution equations with nonlocal initial conditions. Results Math. doi:10.1007/s00025-011-0142-9 (2012)
  22. Mahmudov N.I.: Controllability of semilinear stochastic systems in Hilbert spaces. J. Math. Anal. Appl. 288, 197鈥?11 (2001) CrossRef
  23. Wang J., Zhou Y., Wei W., Xu H.: Nonlocal problems for fractional integro-differential equations via fractional operators and optimal controls. Comput. Math. Appl. 62, 1427鈥?441 (2011) CrossRef
  24. Debbouche A., Baleanu D.:Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62, 1442鈥?450 (2011) CrossRef
  25. Sakthivel R., Ren Y., Mahmudov N.I.: On the approximate controllability of semilinear fractional differential systems. Comput. Math. Appl. 62, 1451鈥?459 (2011) CrossRef
  26. Chang Y.K., Li W.S.: Solvability for impulsive neutral integro-differential equations with state-dependent delay via fractional operators. J. Optim. Theory Appl. 144, 445鈥?59 (2010) CrossRef
  27. Agarwal R.P., Andrade B., Siracusa G.: On fractional integro-differential equations with state-dependent delay. Comput. Math. Appl. 62, 1143鈥?149 (2011) CrossRef
  28. Lizama C.: On approximation and representation of / k-regularized resolvent families. Integr. Equat. Oper. Theory 41, 223鈥?29 (2001) CrossRef
  29. Cuesta, E.: Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst.(Suppl), 277鈥?85 (2007)
  30. Hino Y., Murakami S., Naito T.: Functional differential equations with infinite delay, In Lecture Notes in Mathematics, vol1473. Springer, Berlin (1991)
  31. Mahmudov I.N., Denker A.: On controllability of linear stochastic systems. Int. J. Control 73, 144鈥?51 (2000) CrossRef
  32. dos Dantos, J.P.C., Cuevas, C., de Andrade, B.: Existence results for a fractional equation with state-dependent delay, Adv. Differ. Equ. (2010). Article ID 642013
  33. dos Dantos J.P.C., Mallika Arjunan M., Cuevas C.: Existence results for fractional neutral integro-differential equations with state-dependent delay. Comput. Math. Appl. 62, 1275鈥?283 (2011) CrossRef
  
• 作者单位：Sakthivel Rathinasamy (1)
  Ren Yong (2)
  
  1. Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, South Korea
  2. Department of Mathematics, Anhui Normal University, Wuhu, 241000, China
  
• ISSN：1420-9012

文摘

The systems governed by delay differential equations come up in different fields of science and engineering but often demand the use of non-constant or state-dependent delays. The corresponding model equation is a delay differential equation with state-dependent delay as opposed to the standard models with constant delay. The concept of controllability plays an important role in physics and mathematics. In this paper, first we study the approximate controllability for a class of nonlinear fractional differential equations with state-dependent delays. Then, the result is extended to study the approximate controllability fractional systems with state-dependent delays and resolvent operators. A set of sufficient conditions are established to obtain the required result by employing semigroup theory, fixed point technique and fractional calculus. In particular, the approximate controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is approximately controllable. Also, an example is presented to illustrate the applicability of the obtained theory.