# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [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. [7] E. H. Doha, A. H. Bhrawy, D. 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. [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. [9] 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. [10] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747_0001. [11] M. Kerboua, A. 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. [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. [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. [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. [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. [16] J. T. Machado, V. 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. [17] R. Magin, X. 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. [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. [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. [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. [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. [22] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, Inc., New York, 1993. [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. [24] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. [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. [26] V. E. Tarasov, Fractional Dynamics, Springer-HEP, Heidelberg, Beijing, 2010. doi: 10.1007/978-3-642-14003-7_1. [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. [28] 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. [29] J. Wang, M. 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. [30] J. Wang, A. 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. [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. [32] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientifc, Singapore, 2014. doi: 10.1142/9789814579902_0001. [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.

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. [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. [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. [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. [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. [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. [7] E. H. Doha, A. H. Bhrawy, D. 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. [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. [9] 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. [10] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747_0001. [11] M. Kerboua, A. 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. [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. [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. [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. [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. [16] J. T. Machado, V. 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. [17] R. Magin, X. 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. [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. [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. [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. [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. [22] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, Inc., New York, 1993. [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. [24] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. [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. [26] V. E. Tarasov, Fractional Dynamics, Springer-HEP, Heidelberg, Beijing, 2010. doi: 10.1007/978-3-642-14003-7_1. [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. [28] 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. [29] J. Wang, M. 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. [30] J. Wang, A. 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. [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. [32] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientifc, Singapore, 2014. doi: 10.1142/9789814579902_0001. [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.
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