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doi: 10.3934/eect.2021016
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Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space

Ankit Kumar , Kamal Jeet , and  Ramesh Kumar Vats

Department of Mathematics & Scientific Computing, National Institute of Technology, Hamirpur (H.P.)-177005, India

* Corresponding author: Kamal Jeet (kamaljeetp2@gmail.com)

Received  January 2020 Revised  August 2020 Early access April 2021

Fund Project: The second and third author are supported by CSIR, India under Research Project no-25(0268)/17/EMR-II

This paper aims to establish sufficient conditions for the exact controllability of the nonlocal Hilfer fractional integro-differential system of Sobolev-type using the theory of propagation family $ \{P(t), \; t\geq0\} $ generated by the operators $ A $ and $ R $. For proving the main result we do not impose any condition on the relation between the domain of the operators $ A $ and $ R $. We also do not assume that the operator $ R $ has necessarily a bounded inverse. The main tools applied in our analysis are the theory of measure of noncompactness, fractional calculus, and Sadovskii's fixed point theorem. Finally, we provide an example to show the application of our main result.

Keywords: Exact controllability, Hilfer fractional derivative, Sadovskii's fixed point theorem, measure of noncompactness, Sobolev-type.
Mathematics Subject Classification: Primary: 34G20, 34K30, 93B05.
Citation: Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, doi: 10.3934/eect.2021016
References:
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S. Abbas, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electronic Journal of Differential Equations, 9 (2011), 1-11.   Google Scholar

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S. Agarwal and D. Bahuguna, Existence of solutions to Sobolev-type partial neutral differential equations, J. Appl. Math. Stoch. Anal., 2006 (2006), Art. ID 16308, 10 pp. doi: 10.1155/JAMSA/2006/16308.  Google Scholar

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K. Balachandran and S. Kiruthika, Existence of solutions of abstract fractional integrodifferential equations of Sobolev-type, Computers & Mathematics with Applications, 64 (2012), 3406-3413.  doi: 10.1016/j.camwa.2011.12.051.  Google Scholar

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K. Balachandran and J. P. Dauer, Controllability of functional differential systems of Sobolev-type in Banach spaces, Kybernetika, 34 (1998), 349-357.   Google Scholar

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G. Barenblat, J. Zheltor and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Optim. Theory Appl., 24 (1960), 1286-1303.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

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K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

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M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons & Fractals, 14 (2002), 433-440.  doi: 10.1016/S0960-0779(01)00208-9.  Google Scholar

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M. Fečkan, J. Wang and Y. Zhou, Controllability of fractional functional evolution equations of Sobolev-type via characteristic solution operators, J. Optim. Theory Appl., 156 (2013), 79-95.  doi: 10.1007/s10957-012-0174-7.  Google Scholar

[12]

K. M. Furati, M. D. Kassim and N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Computers & Mathematics with Applications, 64 (2012), 1616-1626.  doi: 10.1016/j.camwa.2012.01.009.  Google Scholar

[13]

H. Gou and B. Li, Study on Sobolev-type Hilfer fractional integro-differential equations with delay, J. Fixed Point Theory Appl., 20 (2018), 1-26.  doi: 10.1007/s11784-018-0523-8.  Google Scholar

[14]

H. Gu and J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257 (2015), 344-354.  doi: 10.1016/j.amc.2014.10.083.  Google Scholar

[15]

H. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector valued functions, Nonlinear Analysis, 7 (1983), 1351-1371.  doi: 10.1016/0362-546X(83)90006-8.  Google Scholar

[16]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.  Google Scholar

[17]

K. Jeet and N. Sukavanam, Approximate controllability of nonlocal and impulsive neutral integro-differential equations using the resolvent operator theory and an approximating technique, Appl. Math. Comput., 364 (2020), Art. ID 124690, 15pp. doi: 10.1016/j.amc.2019.124690.  Google Scholar

[18]

K. Jeet and D. Bahuguna, Approximate controllability of nonlocal neutral fractional integro-differential equations with finite delay, J. Dyn. Control Syst., 22 (2016), 485-504.  doi: 10.1007/s10883-015-9297-0.  Google Scholar

[19]

K. Jeet and D. Bahuguna, Controllability of the impulsive finite delay differential equations of fractional order with nonlocal conditions, Neural, Parallel, and Sci. Comp., 21 (2013), 517-532.   Google Scholar

[20]

A. Kumar, R. K. Vats, A. Kumar and D. N. Chalishajar, Numerical approach to the controllability of fractional order impulsive differential equations, Demonstratio Mathematica, 53 (2020), 193-207.  doi: 10.1515/dema-2020-0015.  Google Scholar

[21]

A. Kumar, R. K. Vats and A. Kumar, Approximate controllability of second order non-autonomous system with finite delay, J. of Dyn. and Control Syst., 26 (2020), 611-627.  doi: 10.1007/s10883-019-09475-0.  Google Scholar

[22]

V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis. Theory, Methods & Applications, 69 (2008), 2677-2682.  doi: 10.1016/j.na.2007.08.042.  Google Scholar

[23]

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., 399 (2012), 510-525.   Google Scholar

[24]

Y. Li, Existence of solutions to initial value problems for abstract semilinear evolution equations, Acta Math. Sinica, 48 (2005), 1089-1094.   Google Scholar

[25]

J. Liang and T. Xiao, Abstract degenerate Cauchy problems in locally convex spaces, J. Math. Anal. Appl., 259 (2001), 398-412.  doi: 10.1006/jmaa.2000.7406.  Google Scholar

[26]

J. Machado, C. Ravichandran, M. Rivero and J. Trujillo, Controllability results for impulsive mixed–type functional integro–differential evolution equations with nonlocal conditions, Fixed Point Theory and Applications, 2013 (2013), 1-16.  doi: 10.1186/1687-1812-2013-66.  Google Scholar

[27]

N. I. Mahmudov, Existence and approximate controllability of Sobolev-type fractional stochastic evolution equations, Bulletin of the Polish Academy of Sciences Technical Sciences, 62 (2014), 205-215.  doi: 10.2478/bpasts-2014-0020.  Google Scholar

[28]

I. Podlubny, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[29]

Z. Tai and S. Lun, On controllability of fractional impulsive neutral infinite delay evolution integrodifferential systems in Banach spaces, Applied Mathematics Letters, 25 (2012), 104-110.  doi: 10.1016/j.aml.2011.07.002.  Google Scholar

[30]

J. R. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850-859.  doi: 10.1016/j.amc.2015.05.144.  Google Scholar

[31]

W. Wang and Y. Zhou, Complete controllability of fractional evolution systems, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 4346-4355.  doi: 10.1016/j.cnsns.2012.02.029.  Google Scholar

[32]

J. Wang, M. Fečkan and Y. Zhou, Approximate controllability of sobolev type fractional evolution systems with nonlocal conditions, Evolution Equations and Control Theory, 6 (2017), 471-486.  doi: 10.3934/eect.2017024.  Google Scholar

[33]

J. Wang, M. Fečkan and Y. Zhou, Controllability of Sobolev type fractional evolution systems, Dyn. Partial. Differ. Equ., 11 (2014), 71-87.  doi: 10.4310/DPDE.2014.v11.n1.a4.  Google Scholar

[34]

M. Yang and Q. Wang, Existence of mild solutions for a class of Hilfer fractional evolution eqautions with nonlocal conditions, Fract. Calc. Appl. Anal., 20 (2017), 679-705.  doi: 10.1515/fca-2017-0036.  Google Scholar

[35]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Computers & Mathematics with Applications, 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026.  Google Scholar

show all references

References:
[1]

S. Abbas, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electronic Journal of Differential Equations, 9 (2011), 1-11.   Google Scholar

[2]

S. Agarwal and D. Bahuguna, Existence of solutions to Sobolev-type partial neutral differential equations, J. Appl. Math. Stoch. Anal., 2006 (2006), Art. ID 16308, 10 pp. doi: 10.1155/JAMSA/2006/16308.  Google Scholar

[3]

K. Balachandran and J. Y. Park, Nonlocal Cauchy problem for abstract fractional semilinear evolution equation, Nonlinear Analysis. Theory, Methods & Applications, 71 (2009), 4471-4475.  doi: 10.1016/j.na.2009.03.005.  Google Scholar

[4]

K. Balachandran, S. Kiruthika and J. J. Trujillo, On fractional impulsive equations of Sobolev-type with nonlocal condition in Banach spaces, Computers & Mathematics with Applications, 62 (2011), 1157-1165.  doi: 10.1016/j.camwa.2011.03.031.  Google Scholar

[5]

K. Balachandran and S. Kiruthika, Existence of solutions of abstract fractional integrodifferential equations of Sobolev-type, Computers & Mathematics with Applications, 64 (2012), 3406-3413.  doi: 10.1016/j.camwa.2011.12.051.  Google Scholar

[6]

K. Balachandran and J. P. Dauer, Controllability of functional differential systems of Sobolev-type in Banach spaces, Kybernetika, 34 (1998), 349-357.   Google Scholar

[7]

G. Barenblat, J. Zheltor and I. Kochiva, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Optim. Theory Appl., 24 (1960), 1286-1303.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

[8]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[9]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[10]

M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons & Fractals, 14 (2002), 433-440.  doi: 10.1016/S0960-0779(01)00208-9.  Google Scholar

[11]

M. Fečkan, J. Wang and Y. Zhou, Controllability of fractional functional evolution equations of Sobolev-type via characteristic solution operators, J. Optim. Theory Appl., 156 (2013), 79-95.  doi: 10.1007/s10957-012-0174-7.  Google Scholar

[12]

K. M. Furati, M. D. Kassim and N. E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Computers & Mathematics with Applications, 64 (2012), 1616-1626.  doi: 10.1016/j.camwa.2012.01.009.  Google Scholar

[13]

H. Gou and B. Li, Study on Sobolev-type Hilfer fractional integro-differential equations with delay, J. Fixed Point Theory Appl., 20 (2018), 1-26.  doi: 10.1007/s11784-018-0523-8.  Google Scholar

[14]

H. Gu and J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257 (2015), 344-354.  doi: 10.1016/j.amc.2014.10.083.  Google Scholar

[15]

H. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector valued functions, Nonlinear Analysis, 7 (1983), 1351-1371.  doi: 10.1016/0362-546X(83)90006-8.  Google Scholar

[16]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.  Google Scholar

[17]

K. Jeet and N. Sukavanam, Approximate controllability of nonlocal and impulsive neutral integro-differential equations using the resolvent operator theory and an approximating technique, Appl. Math. Comput., 364 (2020), Art. ID 124690, 15pp. doi: 10.1016/j.amc.2019.124690.  Google Scholar

[18]

K. Jeet and D. Bahuguna, Approximate controllability of nonlocal neutral fractional integro-differential equations with finite delay, J. Dyn. Control Syst., 22 (2016), 485-504.  doi: 10.1007/s10883-015-9297-0.  Google Scholar

[19]

K. Jeet and D. Bahuguna, Controllability of the impulsive finite delay differential equations of fractional order with nonlocal conditions, Neural, Parallel, and Sci. Comp., 21 (2013), 517-532.   Google Scholar

[20]

A. Kumar, R. K. Vats, A. Kumar and D. N. Chalishajar, Numerical approach to the controllability of fractional order impulsive differential equations, Demonstratio Mathematica, 53 (2020), 193-207.  doi: 10.1515/dema-2020-0015.  Google Scholar

[21]

A. Kumar, R. K. Vats and A. Kumar, Approximate controllability of second order non-autonomous system with finite delay, J. of Dyn. and Control Syst., 26 (2020), 611-627.  doi: 10.1007/s10883-019-09475-0.  Google Scholar

[22]

V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis. Theory, Methods & Applications, 69 (2008), 2677-2682.  doi: 10.1016/j.na.2007.08.042.  Google Scholar

[23]

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., 399 (2012), 510-525.   Google Scholar

[24]

Y. Li, Existence of solutions to initial value problems for abstract semilinear evolution equations, Acta Math. Sinica, 48 (2005), 1089-1094.   Google Scholar

[25]

J. Liang and T. Xiao, Abstract degenerate Cauchy problems in locally convex spaces, J. Math. Anal. Appl., 259 (2001), 398-412.  doi: 10.1006/jmaa.2000.7406.  Google Scholar

[26]

J. Machado, C. Ravichandran, M. Rivero and J. Trujillo, Controllability results for impulsive mixed–type functional integro–differential evolution equations with nonlocal conditions, Fixed Point Theory and Applications, 2013 (2013), 1-16.  doi: 10.1186/1687-1812-2013-66.  Google Scholar

[27]

N. I. Mahmudov, Existence and approximate controllability of Sobolev-type fractional stochastic evolution equations, Bulletin of the Polish Academy of Sciences Technical Sciences, 62 (2014), 205-215.  doi: 10.2478/bpasts-2014-0020.  Google Scholar

[28]

I. Podlubny, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[29]

Z. Tai and S. Lun, On controllability of fractional impulsive neutral infinite delay evolution integrodifferential systems in Banach spaces, Applied Mathematics Letters, 25 (2012), 104-110.  doi: 10.1016/j.aml.2011.07.002.  Google Scholar

[30]

J. R. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850-859.  doi: 10.1016/j.amc.2015.05.144.  Google Scholar

[31]

W. Wang and Y. Zhou, Complete controllability of fractional evolution systems, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 4346-4355.  doi: 10.1016/j.cnsns.2012.02.029.  Google Scholar

[32]

J. Wang, M. Fečkan and Y. Zhou, Approximate controllability of sobolev type fractional evolution systems with nonlocal conditions, Evolution Equations and Control Theory, 6 (2017), 471-486.  doi: 10.3934/eect.2017024.  Google Scholar

[33]

J. Wang, M. Fečkan and Y. Zhou, Controllability of Sobolev type fractional evolution systems, Dyn. Partial. Differ. Equ., 11 (2014), 71-87.  doi: 10.4310/DPDE.2014.v11.n1.a4.  Google Scholar

[34]

M. Yang and Q. Wang, Existence of mild solutions for a class of Hilfer fractional evolution eqautions with nonlocal conditions, Fract. Calc. Appl. Anal., 20 (2017), 679-705.  doi: 10.1515/fca-2017-0036.  Google Scholar

[35]

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Computers & Mathematics with Applications, 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026.  Google Scholar

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