×

Approximate controllability of fractional integrodifferential evolution equations. (English) Zbl 1266.93019

Summary: This paper addresses the issue of approximate controllability for a class of control system which is represented by nonlinear fractional integrodifferential equations with nonlocal conditions. By using semigroup theory, \(p\)-mean continuity and fractional calculations, a set of sufficient conditions, are formulated and proved for the nonlinear fractional control systems. More precisely, the results are established under the assumption that the corresponding linear system is approximately controllable and functions satisfy non-Lipschitz conditions. The results generalize and improve some known results.

MSC:

93B05 Controllability
34A08 Fractional ordinary differential equations
45J05 Integro-ordinary differential equations
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. P. Agarwal, Y. Zhou, and Y. He, “Existence of fractional neutral functional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1095-1100, 2010. · Zbl 1189.34152 · doi:10.1016/j.camwa.2009.05.010
[2] J. Tenreiro Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140-1153, 2011. · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027
[3] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[4] J. R. Wang, Y. Zhou, W. Wei, and H. Xu, “Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1427-1441, 2011. · Zbl 1228.45015 · doi:10.1016/j.camwa.2011.02.040
[5] Z. Liu and X. Li, “On the controllability of impulsive fractional evolution inclusions in Banach spaces,” Journal of Optimization Theory and Applications, vol. 156, no. 1, pp. 167-182, 2013. · Zbl 1263.93035 · doi:10.1007/s10957-012-0236-x
[6] L. Shen and J. Sun, “Relative controllability of stochastic nonlinear systems with delay in control,” Nonlinear Analysis: Real World Applications, vol. 13, no. 6, pp. 2880-2887, 2012. · Zbl 1254.93029 · doi:10.1016/j.nonrwa.2012.04.017
[7] J. Klamka, “Stochastic controllability of systems with multiple delays in control,” International Journal of Applied Mathematics and Computer Science, vol. 19, no. 1, pp. 39-47, 2009. · Zbl 1169.93005 · doi:10.2478/v10006-009-0003-9
[8] J. Klamka, “Constrained exact controllability of semilinear systems,” Systems & Control Letters, vol. 47, no. 2, pp. 139-147, 2002. · Zbl 1003.93005 · doi:10.1016/S0167-6911(02)00184-6
[9] X. J. Wan, Y. P. Zhang, and J. T. Sun, “Controllability of impulsive neutral functional differential inclusions in Banach spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 861568, 8 pages, 2013. · Zbl 1271.93027 · doi:10.1155/2013/861568
[10] J. Klamka, “Constrained controllability of semilinear systems with delayed controls,” Bulletin of the Polish Academy of Sciences, vol. 56, no. 4, pp. 333-337, 2008.
[11] J. Klamka, “Constrained controllability of semilinear systems with delays,” Nonlinear Dynamics, vol. 56, no. 1-2, pp. 169-177, 2009. · Zbl 1170.93009 · doi:10.1007/s11071-008-9389-4
[12] Z. Tai and X. Wang, “Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1760-1765, 2009. · Zbl 1181.34078 · doi:10.1016/j.aml.2009.06.017
[13] S. Kumar and N. Sukavanam, “Approximate controllability of fractional order semilinear systems with bounded delay,” Journal of Differential Equations, vol. 252, no. 11, pp. 6163-6174, 2012. · Zbl 1243.93018 · doi:10.1016/j.jde.2012.02.014
[14] A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1442-1450, 2011. · Zbl 1228.45013 · doi:10.1016/j.camwa.2011.03.075
[15] J. Klamka, “Local controllability of fractional discrete-time semilinear systems,” Acta Mechanica et Automatica, vol. 5, no. 2, pp. 55-58, 2011.
[16] J. Klamka, “Controllability and minimum energy control problem of fractional discrete-time systems,” in New Trends in Nanotechnology and Fractional Calculus Applications, D. Baleanu, Z. B. Guvenc, and J. A. Tenreiro Machado, Eds., pp. 503-509, Springer, New York, NY, USA, 2010. · Zbl 1222.93030 · doi:10.1007/978-90-481-3293-5_45
[17] R. Sakthivel, N. I. Mahmudov, and J. J. Nieto, “Controllability for a class of fractional-order neutral evolution control systems,” Applied Mathematics and Computation, vol. 218, no. 20, pp. 10334-10340, 2012. · Zbl 1245.93022 · doi:10.1016/j.amc.2012.03.093
[18] N. I. Mahmudov and A. Denker, “On controllability of linear stochastic systems,” International Journal of Control, vol. 73, no. 2, pp. 144-151, 2000. · Zbl 1031.93033 · doi:10.1080/002071700219849
[19] P. Muthukumar and P. Balasubramaniam, “Approximate controllability of mixed stochastic Volterra-Fredholm type integrodifferential systems in Hilbert space,” Journal of the Franklin Institute, vol. 348, no. 10, pp. 2911-2922, 2011. · Zbl 1254.93028 · doi:10.1016/j.jfranklin.2011.10.001
[20] R. Sakthivel, Y. Ren, and N. I. Mahmudov, “On the approximate controllability of semilinear fractional differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1451-1459, 2011. · Zbl 1228.34093 · doi:10.1016/j.camwa.2011.04.040
[21] R. Sakthivel, J. J. Nieto, and N. I. Mahmudov, “Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay,” Taiwanese Journal of Mathematics, vol. 14, no. 5, pp. 1777-1797, 2010. · Zbl 1220.93011
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.