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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[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. 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.

[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.

[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.

[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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