APPROXIMATE CONTROLLABILITY OF RIEMANN-LIOUVILLE FRACTIONAL EVOLUTION EQUATIONS WITH INTEGRAL CONTRACTOR ASSUMPTION

Shouguo Zhu; Zhenbin Fan; Gang Li

doi:10.11948/2018.532

2018 Volume 8 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(2): 532-548

Next Article Previous Article

Shouguo Zhu, Zhenbin Fan, Gang Li. APPROXIMATE CONTROLLABILITY OF RIEMANN-LIOUVILLE FRACTIONAL EVOLUTION EQUATIONS WITH INTEGRAL CONTRACTOR ASSUMPTION[J]. Journal of Applied Analysis & Computation, 2018, 8(2): 532-548. doi: 10.11948/2018.532

Citation:

Shouguo Zhu, Zhenbin Fan, Gang Li. APPROXIMATE CONTROLLABILITY OF RIEMANN-LIOUVILLE FRACTIONAL EVOLUTION EQUATIONS WITH INTEGRAL CONTRACTOR ASSUMPTION[J]. Journal of Applied Analysis & Computation, 2018, 8(2): 532-548. doi: 10.11948/2018.532

APPROXIMATE CONTROLLABILITY OF RIEMANN-LIOUVILLE FRACTIONAL EVOLUTION EQUATIONS WITH INTEGRAL CONTRACTOR ASSUMPTION

1 School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China;
2 School of Mathematics, Taizhou College of Nanjing Normal University, Taizhou 225300, China

Fund Project:

Abstract

Abstract

We propose and investigate an evolution system with a RiemannLiouville fractional derivative. With the aid of a resolvent method, we formulate a suitable notion of solutions to this system and demonstrate the corresponding existence and uniqueness of solutions under a regular integral contractor condition. Furthermore, by applying a space decomposition technique, we exhibit the approximate controllability result of the system. This paper closes with a simple example, which confirms our analytical findings.
- Approximate controllability /
- Riemann-Liouville fractional derivatives /
- resolvent /
- regular integral contractor
MSC: 26A33;47A10;93B05

FullText(HTML)

References

[1]	M. Altman, Contractors and Contractor Directions, Theory and Applications, Marcel Dekker, New York, 1977. Google Scholar
[2]	W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser Verlag, Basel, 2001. Google Scholar
[3]	A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel:theory and application to heat transfer model, Thermal. Sci., 2016, 20(2), 763-769. Google Scholar
[4]	P. Balasubramaniam and P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi's function, Appl. Math. Comput., 2015, 256, 232-246. Google Scholar
[5]	M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals, 2002, 14(3), 433-440. Google Scholar
[6]	M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2015, 1(2), 73-85. Google Scholar
[7]	Z. Fan, Existence and regularity of solutions for evolution equations with Riemann-Liouville fractional derivatives, Indag. Math., 2014, 25(3), 516-524. Google Scholar
[8]	Z. Fan, Characterization of compactness for resolvents and its applications, Appl. Math. Comput., 2014, 232, 60-67. Google Scholar
[9]	R. K. George, Approximate controllability of semilinear systems using integral contractors, Numer. Funct. Anal. Optim., 1995, 16(1-2), 127-138. Google Scholar
[10]	R. K. George, D. N. Chalishajar and A. K. Nandakumaran, Exact controllability of the nonlinear third-order dispersion equation, J. Math. Anal. Appl., 2007, 332(2), 1028-1044. Google Scholar
[11]	S. Kumar and N. Sukavanam, Controllability of fractional order system with nonlinear term having integral contractor, Fract. Calc. Appl. Anal., 2013, 16(4), 791-801. Google Scholar
[12]	S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Diff. Equ., 2012, 252(11), 6163-6174. Google Scholar
[13]	F. Li, J. Liang and H. K. Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl., 2012, 391, 510-525. Google Scholar
[14]	M. L. Li and J. L. Ma, Approximate controllability of second order impulsive functional differential system with infinite delay in Banach spaces, J. Appl. Anal. Comput., 2016, 6(2), 492-514. Google Scholar
[15]	K. Li and J. Peng, Fractional resolvents and fractional evolution equations, Appl. Math. Lett., 2012, 25(5), 808-812. Google Scholar
[16]	Z. Liu and X. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives, SIAM J. Control Optim., 2015, 53(4), 1920-1933. Google Scholar
[17]	N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equation in abstract spaces, SIAM J. Control Optim., 2003, 42(5), 1604-1622. Google Scholar
[18]	N. I. Mahmudov and N. Semi, Approximate controllability of semilinear control systems in Hilbert spaces, TWMS J. Appl. Eng. Math., 2012, 2(1), 67-74. Google Scholar
[19]	K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim., 1987, 25(3), 715-722. Google Scholar
[20]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. Google Scholar
[21]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. Google Scholar
[22]	C. Rajivganthi, P. Muthukumar and B. G. Priya, Approximate controllability of fractional stochastic integro-differential equations with infinite delay of order 1< α < 2, IMA J. Math. Control. Inform., 2016, 33(3), 685-699. Google Scholar
[23]	N. Sukavanam and M. Kumar, S-controllability of an abstract first order semilinear control system, Numer. Funct. Anal. Optim., 2010, 31(9), 1023-1034. Google Scholar
[24]	P. Tamilalagan and P. Balasubramaniam, Approximate controllability of fractional stochastic differential equations driven by mixed fractional Brownian motion via resolvent operators, Internat. J. Control, 2017, 90(8), 1713-1727. Google Scholar
[25]	J. R. Wang and Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal. Real World Appl., 2011, 12(6), 3642-3653. Google Scholar
[26]	Z. M. Yan and F. X. Lu, Existence results for a new class of fractional implusive partial neutral stochastic integro-differential equations with infinite delay, J. Appl. Anal. Comput., 2015, 5(3), 329-346. Google Scholar
[27]	H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 2007, 328(2), 1075-1081. Google Scholar
[28]	E. Zeidler, Nonlinear Functional Analysis and Its Application Ⅱ/A, SpringerVerlag, New York, 1990. Google Scholar
[29]	H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim., 1983, 21(4), 551-565. Google Scholar
[30]	Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 2010, 59(3), 1063-1077. Google Scholar
[31]	Y. Zhou, L. Zhang and X. H. Shen, Existence of mild solutions for fractional evolution equations, J. Integral Equ. Appl., 2013, 25(4), 557-586. Google Scholar