Abstract
We study the approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay under the assumptions that the corresponding linear system is approximately controllable. Using fractional calculus, stochastic analysis theory, and the fixed-point technique with the properties of analytic \(\alpha \)-resolvent operators, a new set of sufficient conditions for approximate controllability of fractional stochastic functional integro-differential inclusions are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result.
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References
Triggiani, R.: A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control Optim. 15, 407–411 (1977)
Mahmudov, N.I., Denker, A.: On controllability of linear stochastic systems. Int. J. Control 73, 144–151 (2000)
Dauer, J.P., Mahmudov, N.I.: Controllability of stochastic semilinear functional differential equations in Hilbert spaces. J. Math. Anal. Appl. 290, 373–394 (2004)
Anguraj, A., Vinodkumar, A.: Existence, uniqueness and stability results of impulsive stochastic semilinear neutral functional differential equations with infinite delays. Electron. J. Qual. Theory Differ. Equ. 2009(67), 1–13 (2009)
Ren, Y., Sakthivel, R.: Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps. J. Math. Phys. 53, 073517 (2012)
Yan, Z., Yan, X.: Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces. Z. Angew. Math. Phys. 64, 573–590 (2013)
Mahmudov, N.I.: Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim. 42, 1604–1622 (2003)
Klamka, J.: Stochastic controllability of linear systems with state delays. Int. J. Appl. Math. Comput. Sci. 17, 5–13 (2007)
Balasubramaniam, P., Park, J.Y., Muthukumar, P.: Approximate controllability of neutral stochastic functional differential systems with infinite delay. Stoch. Anal. Appl. 28, 389–400 (2010)
Sakthivel, R., Ren, Y., Mahmudov, N.I.: Approximate controllability of second-order stochastic differential equations with impulsive effects. Modern Phys. Lett. B 24, 1559–1572 (2010)
Lakshmikantham, V.: Theory of fractional differential equations. Nonlinear Anal. 60, 3337–3343 (2008)
Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008)
Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional functional differential inclusions with infinite delay and application to control theory. Fract. Calc. Appl. Anal. 11, 35–56 (2008)
El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433–440 (2002)
Balachandran, K., Trujillo, J.J.: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Anal. 72, 4587–4593 (2010)
Agarwal, R.P., Santos, J.P.C.: Cuevas, analytic resolvent operator and existence results for fractional order evolutionary integral equations. J. Abstr. Differ. Equ. Appl. 2, 26–47 (2012)
de Andrade, B., Santos, J.P.C.: Existence of solutions for a fractional neutral integro-differential equation with unbounded delay. Electron. J. Differ. Equ. 2012(90), 1–13 (2012)
Darwish, M.A., Ntouyas, S.K.: Functional differential equations of fractional order with state-dependent delay. Dynam. Syst. Appl. 18, 539–550 (2009)
Agarwal, R.P., De Andrade, B., Siracusa, G.: On fractional integro-difierential equations with state-dependent delay. Comput. Math. Appl. 62, 1143–1149 (2011)
Santos, J.P.C., Arjunan, M.M., Cuevas, C.: Existence results for fractional neutral integro-differential equations with state-dependent delay. Comput. Math. Appl. 62, 1275–1283 (2011)
Wang, J.R., Fan, Z., Zhou, Y.: Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. J. Optim. Theory Appl. 154, 292–302 (2012)
Debbouchea, A., Baleanu, D.: Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62, 1442–1450 (2011)
Yan, Z.: Controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay. J. Franklin Inst. 348, 2156–2173 (2011)
Liu, Z., Li, X.: On the controllability of impulsive fractional evolution inclusions in Banach spaces. J. Optim. Theory Appl. 156, 167–182 (2013)
Sakthivel, R., Ren, Y., Mahmudov, N.I.: On the approximate controllability of semilinear fractional differential systems. Comput. Math. Appl. 62, 1451–1459 (2011)
Yan, Z.: Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay. Int. J. Control 85, 1051–1062 (2012)
Mahmudov, N.I.: Approximate controllability of fractional Sobolev-type evolution equations in Banach spaces. Abstr. Appl. Anal. 2013, 1–9, Article ID 502839 (2013)
Debbouche, A., Delfim, F.M.: Torres, approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces. Int. J. Control 86, 1577–1585 (2013)
El-Borai, M.M., EI-Said EI-Nadi, K., Fouad, H.A.: On some fractional stochastic delay differential equations. Comput. Math. Appl. 59, 1165–1170 (2010)
Cui, J., Yan, L.: Existence result for fractional neutral stochastic integro-differential equations with infinite delay. J. Phys. A 44, 335201 (2011)
Sakthivel, R., Revathi, P., Ren, Y.: Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. 81, 70–86 (2013)
Sakthivel, R., Ganesh, R., Suganya, S.: Approximate controllability of fractional neutral stochastic system with infinite delay. Rep. Math. Phys. 70, 291–311 (2012)
Sakthivel, R., Suganya, S., Anthoni, S.M.: Approximate controllability of fractional stochastic evolution equations. Comput. Math. Appl. 63, 660–668 (2012)
Zang, Y., Li, J.: Approximate controllability of fractional impulsive neutral stochastic differential equations with nonlocal conditions. Bound. Value Probl. 193, 1–14 (2013)
Muthukumar, P., Rajivganthi, C.: Approximate controllability of fractional order neutral stochastic integro-differential system with nonlocal conditions and infinite delay. Taiwan. J. Math. 17, 1693–1713 (2013)
Agarwal, R.P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann–Liouville fractional derivative. Adv. Differ. Equ. 2009 1–47, Article ID 981728 (2009)
Yan, Z.: On a nonlocal problem for fractional integrodifferential inclusions in Banach spaces. Ann. Pol. Math. 101, 87–103 (2011)
Balasubramaniam, P., Ntouyas, S.K.: Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space. J. Math. Anal. Appl. 324, 161–176 (2006)
Yan, Z., Zhang, H.: Existence of impulsive fractional partial neutral stochastic integro-differential inclusions with state-dependent delay in Hilbert spaces. Electron. J. Differ. Equ. 2013(81), 1–21 (2013)
Dhage, B.C.: Fixed-point theorems for discontinuous multi-valued operators on ordered spaces with applications. Comput. Math. Appl. 51, 589–604 (2006)
Deimling, K.: Multi-Valued Differential Equations. De Gruyter, Berlin (1992)
Hu, S., Papageorgiou, N.: Handbook of Multivalued Analysis. Kluwer Academic Publishers, Dordrecht, Boston (1997)
Hale, J.K., Kato, J.: Phase spaces for retarded equations with infinite delay. Funkcial. Ekvac. 21, 11–41 (1978)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Lasota, A., Opial, Z.: An application of the KakutaniCKy Fan theorem in the theory of ordinary differential equations. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13, 781–786 (1965)
Hino, Y., Murakami, S., Naito, T.: Functional-Differential Equations with Infinite Delay. Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991)
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Yan, Z., Jia, X. Approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay. Collect. Math. 66, 93–124 (2015). https://doi.org/10.1007/s13348-014-0109-8
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DOI: https://doi.org/10.1007/s13348-014-0109-8
Keywords
- Approximate controllability
- Fractional neutral stochastic functional integro-differential inclusions
- Multi-valued map
- Analytic \(\alpha \)-resolvent operator
- State-dependent delay