September  2017, 6(3): 471-486. doi: 10.3934/eect.2017024

Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions

1. 

Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

2. 

Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina 842 48, Bratislava, Slovakia

3. 

Mathematical Institute, Slovak Academy of Sciences, Śtefánikova 49,814 73 Bratislava, Slovakia

4. 

Faculty of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China

5. 

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

* Corresponding author: Michal Fečkan

Received  October 2015 Revised  April 2017 Published  July 2017

In this paper, we study the approximate controllability of Sobolev-type fractional evolution systems with non-local conditions in Hilbert spaces. Sufficient conditions of approximate controllability of the desired problem are presented by supposing an approximate controllability of the corresponding linear system. By constructing a control function involving Gramian controllability operator, we transform our problem to a fixed point problem of nonlinear operator. Then the Schauder Fixed Point Theorem is applied to complete the proof. An example is given to illustrate our theoretical results.

Citation: Jinrong Wang, Michal Fečkan, Yong Zhou. Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 471-486. doi: 10.3934/eect.2017024
References:
[1]

C. Atkinson and A. Osseiran, Rational solutions for the time-fractional diffusion equation, SIAM J. Appl. Math., 71 (2011), 92-106.  doi: 10.1137/100799307.  Google Scholar

[2]

D. Baleanu, J. A. T. Machado and A. C. Luo, Fractional Dynamics and Control, Springer, New York, 2012. doi: 10.1007/978-1-4614-0457-6.  Google Scholar

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D. Baleanu and O. Mustafa, Asymptotic Integration and Stability (Series on Complexity, Nonlinearity and Chaos) World Scientific, London, 2015. doi: 10.1142/9789814641104_fmatter.  Google Scholar

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A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442-1450.  doi: 10.1016/j.camwa.2011.03.075.  Google Scholar

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A. Debbouche and D. F. M. Torres, Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces, Internat. J. Control, 86 (2013), 1577-1585.  doi: 10.1080/00207179.2013.791927.  Google Scholar

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K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics 2004 Springer, New York, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

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E. H. DohaA. H. BhrawyD. Baleanu and R. M. Hafez, A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations, Appl. Numer. Math., 77 (2014), 43-54.  doi: 10.1016/j.apnum.2013.11.003.  Google Scholar

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M. 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. FečkanJ. 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

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R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747_0001.  Google Scholar

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M. KerbouaA. Debbouche and D. Baleanu, Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces, Electron. J. Qual. Theory Differ. Equ., 58 (2014), 1-16.  doi: 10.14232/ejqtde.2014.1.58.  Google Scholar

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M. Kerboua, A. Debbouche and D. Baleanu, Approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces Abstr. Appl. Anal. 2013 (2013), Art. ID 262191, 10pp. doi: 10.1155/2013/262191.  Google Scholar

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A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006. doi: 10.1016/S0304-0208(06)80002-2.  Google Scholar

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M. Li and J. Wang, Finite time stability of fractional delay differential equations, Appl. Math. Lett., 64 (2017), 170-176.  doi: 10.1016/j.aml.2016.09.004.  Google Scholar

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J. H. Lightbourne and S. M. Rankin, A partial functional differential equation of Sobolev type, J. Math. Anal. Appl., 93 (1983), 328-337.  doi: 10.1016/0022-247X(83)90178-6.  Google Scholar

[16]

J. T. MachadoV. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140-1153.  doi: 10.1016/j.cnsns.2010.05.027.  Google Scholar

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R. MaginX. Feng and D. Baleanu, Solving the fractional order Bloch equation, Conc. Magn. Reson. Part A, 34A (2009), 16-23.  doi: 10.1002/cmr.a.20129.  Google Scholar

[18]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622.  doi: 10.1137/S0363012901391688.  Google Scholar

[19]

N. I. Mahmudov, Approximate controllability of fractional Sobolev-type evolution equations in Banach spaces Abstr. Appl. Anal., 2013 (2013), Art. ID 502839, 9pp. doi: 10.1155/2013/502839.  Google Scholar

[20]

N. I. Mahmudov and S. Zorlu, Approximate controllability of fractional integro-differential equations involving nonlocal initial conditions Bound. Value Probl., 2013 (2013), 16pp. doi: 10.1186/1687-2770-2013-118.  Google Scholar

[21]

N. I. Mahmudov and S. Zorlu, On the approximate controllability of fractional evolution equations with compact analytic semigroup, J. Comput. Appl. Math., 259 (2014), 194-204.  doi: 10.1016/j.cam.2013.06.015.  Google Scholar

[22]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, Inc., New York, 1993.  Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[24]

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.  Google Scholar

[25]

R. Sakthivel and Y. Ren, Approximate controllability of fractional differential equations with state-dependent delay, Results Math., 63 (2013), 949-963.  doi: 10.1007/s00025-012-0245-y.  Google Scholar

[26]

V. E. Tarasov, Fractional Dynamics, Springer-HEP, Heidelberg, Beijing, 2010. doi: 10.1007/978-3-642-14003-7_1.  Google Scholar

[27]

J. Wang and Y. Zhou, Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear Anal., 74 (2011), 5929-5942.  doi: 10.1016/j.na.2011.05.059.  Google Scholar

[28]

J. WangM. 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

[29]

J. WangM. Fečkan and Y. Zhou, A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal., 19 (2016), 806-831.  doi: 10.1515/fca-2016-0044.  Google Scholar

[30]

J. WangA. G. Ibrahim and M. Fečkan, Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces, Appl. Math. Comput., 257 (2015), 103-118.  doi: 10.1016/j.amc.2014.04.093.  Google Scholar

[31]

J. Wang and Y. Zhang, On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives, Appl. Math. Lett., 39 (2015), 85-90.  doi: 10.1016/j.aml.2014.08.015.  Google Scholar

[32]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientifc, Singapore, 2014. doi: 10.1142/9789814579902_0001.  Google Scholar

[33]

Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. Real World Appl., 11 (2010), 4465-4475.  doi: 10.1016/j.nonrwa.2010.05.029.  Google Scholar

show all references

References:
[1]

C. Atkinson and A. Osseiran, Rational solutions for the time-fractional diffusion equation, SIAM J. Appl. Math., 71 (2011), 92-106.  doi: 10.1137/100799307.  Google Scholar

[2]

D. Baleanu, J. A. T. Machado and A. C. Luo, Fractional Dynamics and Control, Springer, New York, 2012. doi: 10.1007/978-1-4614-0457-6.  Google Scholar

[3]

D. Baleanu and O. Mustafa, Asymptotic Integration and Stability (Series on Complexity, Nonlinearity and Chaos) World Scientific, London, 2015. doi: 10.1142/9789814641104_fmatter.  Google Scholar

[4]

A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442-1450.  doi: 10.1016/j.camwa.2011.03.075.  Google Scholar

[5]

A. Debbouche and D. F. M. Torres, Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces, Internat. J. Control, 86 (2013), 1577-1585.  doi: 10.1080/00207179.2013.791927.  Google Scholar

[6]

K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics 2004 Springer, New York, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[7]

E. H. DohaA. H. BhrawyD. Baleanu and R. M. Hafez, A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations, Appl. Numer. Math., 77 (2014), 43-54.  doi: 10.1016/j.apnum.2013.11.003.  Google Scholar

[8]

M. 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

[9]

M. FečkanJ. 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

[10]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747_0001.  Google Scholar

[11]

M. KerbouaA. Debbouche and D. Baleanu, Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces, Electron. J. Qual. Theory Differ. Equ., 58 (2014), 1-16.  doi: 10.14232/ejqtde.2014.1.58.  Google Scholar

[12]

M. Kerboua, A. Debbouche and D. Baleanu, Approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces Abstr. Appl. Anal. 2013 (2013), Art. ID 262191, 10pp. doi: 10.1155/2013/262191.  Google Scholar

[13]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006. doi: 10.1016/S0304-0208(06)80002-2.  Google Scholar

[14]

M. Li and J. Wang, Finite time stability of fractional delay differential equations, Appl. Math. Lett., 64 (2017), 170-176.  doi: 10.1016/j.aml.2016.09.004.  Google Scholar

[15]

J. H. Lightbourne and S. M. Rankin, A partial functional differential equation of Sobolev type, J. Math. Anal. Appl., 93 (1983), 328-337.  doi: 10.1016/0022-247X(83)90178-6.  Google Scholar

[16]

J. T. MachadoV. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140-1153.  doi: 10.1016/j.cnsns.2010.05.027.  Google Scholar

[17]

R. MaginX. Feng and D. Baleanu, Solving the fractional order Bloch equation, Conc. Magn. Reson. Part A, 34A (2009), 16-23.  doi: 10.1002/cmr.a.20129.  Google Scholar

[18]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622.  doi: 10.1137/S0363012901391688.  Google Scholar

[19]

N. I. Mahmudov, Approximate controllability of fractional Sobolev-type evolution equations in Banach spaces Abstr. Appl. Anal., 2013 (2013), Art. ID 502839, 9pp. doi: 10.1155/2013/502839.  Google Scholar

[20]

N. I. Mahmudov and S. Zorlu, Approximate controllability of fractional integro-differential equations involving nonlocal initial conditions Bound. Value Probl., 2013 (2013), 16pp. doi: 10.1186/1687-2770-2013-118.  Google Scholar

[21]

N. I. Mahmudov and S. Zorlu, On the approximate controllability of fractional evolution equations with compact analytic semigroup, J. Comput. Appl. Math., 259 (2014), 194-204.  doi: 10.1016/j.cam.2013.06.015.  Google Scholar

[22]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, Inc., New York, 1993.  Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[24]

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.  Google Scholar

[25]

R. Sakthivel and Y. Ren, Approximate controllability of fractional differential equations with state-dependent delay, Results Math., 63 (2013), 949-963.  doi: 10.1007/s00025-012-0245-y.  Google Scholar

[26]

V. E. Tarasov, Fractional Dynamics, Springer-HEP, Heidelberg, Beijing, 2010. doi: 10.1007/978-3-642-14003-7_1.  Google Scholar

[27]

J. Wang and Y. Zhou, Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear Anal., 74 (2011), 5929-5942.  doi: 10.1016/j.na.2011.05.059.  Google Scholar

[28]

J. WangM. 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

[29]

J. WangM. Fečkan and Y. Zhou, A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal., 19 (2016), 806-831.  doi: 10.1515/fca-2016-0044.  Google Scholar

[30]

J. WangA. G. Ibrahim and M. Fečkan, Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces, Appl. Math. Comput., 257 (2015), 103-118.  doi: 10.1016/j.amc.2014.04.093.  Google Scholar

[31]

J. Wang and Y. Zhang, On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives, Appl. Math. Lett., 39 (2015), 85-90.  doi: 10.1016/j.aml.2014.08.015.  Google Scholar

[32]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientifc, Singapore, 2014. doi: 10.1142/9789814579902_0001.  Google Scholar

[33]

Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. Real World Appl., 11 (2010), 4465-4475.  doi: 10.1016/j.nonrwa.2010.05.029.  Google Scholar

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