×

Approximate controllability of fractional Sobolev-type evolution equations in Banach spaces. (English) Zbl 1271.93021

Summary: We discuss the approximate controllability of semilinear fractional Sobolev-type differential system under the assumption that the corresponding linear system is approximately controllable. Using Schauder’s fixed point theorem, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional Sobolev-type differential equations, are formulated and proved. We show that our result has no analogue with the concept of complete controllability. The results of the paper are a generalization and a continuation of the recent results on this issue.

MSC:

93B05 Controllability
93C25 Control/observation systems in abstract spaces
35R11 Fractional partial differential equations
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, xxxvi+976 (1993), London, UK: Gordon and Breach Science Publishers, London, UK · Zbl 0818.26003
[2] Feckan, M.; Wang, J. R.; Zhou, Y., Controllability of fractional functional evolution equations of sobolev type via characteristic solution operators, Journal of Optimization Theory and Applications (2012) · Zbl 1263.93031 · doi:10.1007/s10957-012-0174-7
[3] El-Borai, M. M., Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons and Fractals, 14, 3, 433-440 (2002) · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9
[4] El-Borai, M. M., The fundamental solutions for fractional evolution equations of parabolic type, Journal of Applied Mathematics and Stochastic Analysis, 3, 197-211 (2004) · Zbl 1081.34053 · doi:10.1155/S1048953304311020
[5] Balachandran, K.; Park, J. Y., Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear Analysis: Hybrid Systems, 3, 4, 363-367 (2009) · Zbl 1175.93028 · doi:10.1016/j.nahs.2009.01.014
[6] Zhou, Y.; Jiao, F., Existence of mild solutions for fractional neutral evolution equations, Computers & Mathematics with Applications, 59, 3, 1063-1077 (2010) · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026
[7] Zhou, Y.; Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Analysis: Real World Applications, 11, 5, 4465-4475 (2010) · Zbl 1260.34017 · doi:10.1016/j.nonrwa.2010.05.029
[8] Hernández, E.; O’Regan, D.; Balachandran, K., On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Analysis: Theory, Methods & Applications, 73, 10, 3462-3471 (2010) · Zbl 1229.34004 · doi:10.1016/j.na.2010.07.035
[9] Wang, J.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear Analysis: Real World Applications, 12, 1, 262-272 (2011) · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013
[10] Wang, J.; Zhou, Y., Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Analysis: Real World Applications, 12, 6, 3642-3653 (2011) · Zbl 1231.34108 · doi:10.1016/j.nonrwa.2011.06.021
[11] Wang, J.; Zhou, Y., Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 74, 17, 5929-5942 (2011) · Zbl 1223.93059 · doi:10.1016/j.na.2011.05.059
[12] Sakthivel, R.; Ren, Y.; Mahmudov, N. I., On the approximate controllability of semilinear fractional differential systems, Computers & Mathematics with Applications, 62, 3, 1451-1459 (2011) · Zbl 1228.34093 · doi:10.1016/j.camwa.2011.04.040
[13] Sakthivel, R.; Mahmudov, N. I.; Nieto, J. J., Controllability for a class of fractional-order neutral evolution control systems, Applied Mathematics and Computation, 218, 20, 10334-10340 (2012) · Zbl 1245.93022 · doi:10.1016/j.amc.2012.03.093
[14] Debbouche, A.; Baleanu, D., Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Computers & Mathematics with Applications, 62, 3, 1442-1450 (2011) · Zbl 1228.45013 · doi:10.1016/j.camwa.2011.03.075
[15] Wang, J.; Zhou, Y.; Medved, M., On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay, Journal of Optimization Theory and Applications, 152, 1, 31-50 (2012) · Zbl 1357.49018 · doi:10.1007/s10957-011-9892-5
[16] Wang, J.; Zhou, Y., Mittag-Leffler-Ulam stabilities of fractional evolution equations, Applied Mathematics Letters, 25, 4, 723-728 (2012) · Zbl 1246.34012 · doi:10.1016/j.aml.2011.10.009
[17] Wang, J.; Zhou, Y.; Wei, W., Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Systems & Control Letters, 61, 4, 472-476 (2012) · Zbl 1250.49035 · doi:10.1016/j.sysconle.2011.12.009
[18] Wang, J.; Fan, Z.; Zhou, Y., Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces, Journal of Optimization Theory and Applications, 154, 1, 292-302 (2012) · Zbl 1252.93028 · doi:10.1007/s10957-012-9999-3
[19] Wang, J.; Zhou, Y., Complete controllability of fractional evolution systems, Communications in Nonlinear Science and Numerical Simulation, 17, 4346-4355 (2012) · Zbl 1248.93029
[20] Wang, J.; Zhou, Y.; Wei, W., Fractional Schrödinger equations with potential and optimal controls, Nonlinear Analysis: Real World Applications, 13, 6, 2755-2766 (2012) · Zbl 1253.35205 · doi:10.1016/j.nonrwa.2012.04.004
[21] Wang, R.-N.; Chen, D.-H.; Xiao, T.-J., Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252, 1, 202-235 (2012) · Zbl 1238.34015 · doi:10.1016/j.jde.2011.08.048
[22] Kumar, S.; Sukavanam, N., Approximate controllability of fractional order semilinear systems with bounded delay, Journal of Differential Equations, 252, 11, 6163-6174 (2012) · Zbl 1243.93018 · doi:10.1016/j.jde.2012.02.014
[23] Balachandran, K.; Dauer, J. P., Controllability of functional-differential systems of Sobolev type in Banach spaces, Kybernetika, 34, 3, 349-357 (1998) · Zbl 1274.93031
[24] Ahmed, H., Controllability for Sobolev type fractional integro-differential systems in a Banach space, Advances in Difference Equations, 2012, article 167 (2012) · Zbl 1377.34098
[25] Bashirov, A. E.; Mahmudov, N. I., On concepts of controllability for deterministic and stochastic systems, SIAM Journal on Control and Optimization, 37, 6, 1808-1821 (1999) · Zbl 0940.93013 · doi:10.1137/S036301299732184X
[26] Bashirov, A. E.; Mahmudov, N.; Şemi, N.; Etikan, H., Partial controllability concepts, International Journal of Control, 80, 1, 1-7 (2007) · Zbl 1115.93013 · doi:10.1080/00207170600885489
[27] Mahmudov, N. I., Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM Journal on Control and Optimization, 42, 5, 1604-1622 (2003) · Zbl 1084.93006 · doi:10.1137/S0363012901391688
[28] Dauer, J. P.; Mahmudov, N. I., Approximate controllability of semilinear functional equations in Hilbert spaces, Journal of Mathematical Analysis and Applications, 273, 2, 310-327 (2002) · Zbl 1017.93019 · doi:10.1016/S0022-247X(02)00225-1
[29] Mahmudov, N. I., Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Analysis: Theory, Methods & Applications, 68, 3, 536-546 (2008) · Zbl 1129.93004 · doi:10.1016/j.na.2006.11.018
[30] Sakthivel, R.; Mahmudov, N. I.; Kim, J. H., On controllability of second order nonlinear impulsive differential systems, Nonlinear Analysis: Theory, Methods & Applications, 71, 1-2, 45-52 (2009) · Zbl 1177.34080 · doi:10.1016/j.na.2008.10.029
[31] Mahmudov, N. I.; Zorlu, S., Controllability of semilinear stochastic systems, International Journal of Control, 78, 13, 997-1004 (2005) · Zbl 1097.93034 · doi:10.1080/00207170500207180
[32] Sakthivel, R.; Suganya, S.; Anthoni, S. M., Approximate controllability of fractional stochastic evolution equations, Computers & Mathematics with Applications, 63, 3, 660-668 (2012) · Zbl 1238.93099 · doi:10.1016/j.camwa.2011.11.024
[33] Sakthivel, R.; Ganesh, R.; Suganya, S., Approximate controllability of fractional neutral stochastic system with infinite delay, Reports on Mathematical Physics, 70, 3, 291-311 (2012) · Zbl 1263.93039 · doi:10.1016/S0034-4877(12)60047-0
[34] Sakthivel, R.; Nieto, J. J.; Mahmudov, N. I., Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay, Taiwanese Journal of Mathematics, 14, 5, 1777-1797 (2010) · Zbl 1220.93011
[35] Sakthivel, R.; Ren, Y.; Mahmudov, N. I., Approximate controllability of second-order stochastic differential equations with impulsive effects, Modern Physics Letters B, 24, 14, 1559-1572 (2010) · Zbl 1211.93026 · doi:10.1142/S0217984910023359
[36] Sakthivel, R.; Anandhi, E. R., Approximate controllability of impulsive differential equations with state-dependent delay, International Journal of Control, 83, 2, 387-393 (2010) · Zbl 1184.93021 · doi:10.1080/00207170903171348
[37] Li, X. J.; Yong, J. M., Optimal Control Theory for Infinite-Dimensional Systems, xii+448 (1995), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · doi:10.1007/978-1-4612-4260-4
[38] Barbu, V.; Precupanu, Th., Convexity and Optimization in Banach Spaces. Convexity and Optimization in Banach Spaces, Mathematics and Its Applications (East European Series), 10, xviii+397 (1986), Dordrecht, The Netherlands: D. Reidel Publishing, Dordrecht, The Netherlands · Zbl 0594.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.