doi: 10.3934/eect.2022028
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Controllability results for Sobolev type $ \psi - $Hilfer fractional backward perturbed integro-differential equations in Hilbert space

Laboratoire de Mathématiques Appliquées et de Modé lisation, Université 8 Mai 1945 Guelma, B.P. 401, 24000, Guelma, Algeria

* Corresponding author: Mourad Kerboua

Received  January 2022 Revised  March 2022 Early access May 2022

In this paper, the approximate controllability for Sobolev type $ \psi - $ Hilfer fractional backward perturbed integro-differential equations with $ \psi - $ fractional non local conditions in a Hilbert space are studied. A new set of sufficient conditions are established by using semigroup theory, $ \psi - $Hilfer fractional calculus and the Schauder's fixed point theorem. The results are obtained under the assumption that the associate backward $ \psi - $ fractional linear system is approximately controllable. Finally, an example is given to illustrate the obtained results.

Citation: Ichrak Bouacida, Mourad Kerboua, Sami Segni. Controllability results for Sobolev type $ \psi - $Hilfer fractional backward perturbed integro-differential equations in Hilbert space. Evolution Equations and Control Theory, doi: 10.3934/eect.2022028
References:
[1]

H. M. Ahmed, M. M. El-Borai, A. S. Okb El Bab and M. Elsaid Ramadan, Approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion, Bound. Value Probl., (2020), Paper No. 120, 25 pp. doi: 10.1186/s13661-020-01418-0.

[2]

R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481.  doi: 10.1016/j.cnsns.2016.09.006.

[3]

P. Bedi, A. Kumar, T. Abdeljawad, Z. A. Khan and A. Khan, Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators, Adv. Difference Equ., (2020), Paper No. 615, 15 pp. doi: 10.1186/s13662-020-03074-1.

[4]

S. N. Bora and B. Roy, Approximate controllability of a class of semilinear Hilfer fractional differential equations, Results Math., 76 (2021), Paper No. 197, 20 pp. doi: 10.1007/s00025-021-01507-1.

[5]

Y.-K. ChangA. Pereira and R. Ponce, Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators, Fract. Calc. Appl. Anal., 20 (2017), 963-987.  doi: 10.1515/fca-2017-0050.

[6]

A. Debbouche and V. Antonov, Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces, Chaos Solitons Fractals, 102 (2017), 140-148.  doi: 10.1016/j.chaos.2017.03.023.

[7]

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

[8]

C. DineshkumarK. Sooppy NisarR. Udhayakumar and V. Vijayakumar, A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions, Asian Journal of Control, 2021 (2021), 1-17. 

[9]

J. DuW. Jiang and A. U. K. Niazi, Approximate controllability of impulsive Hilfer fractional differential inclusions, J. Nonlinear Sci. Appl., 10 (2017), 595-611.  doi: 10.22436/jnsa.010.02.23.

[10]

S. Guechi, R. Dhayal, A. Debbouche and M. Malik, Analysis and optimal control of $\varphi $-Hilfer fractional semilinear equations involving nonlocal impulsive conditions, Symmetry, 13 (2021), 2084, 1–18.

[11]

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

[12]

M. D. Kassim and N.-E. Tatar, Well-posedness and stability for a differential problem with Hilfer–Hadamard fractional derivative, Abstr. Appl. Anal., 2013 (2013), Art. ID 605029, 12 pp. doi: 10.1155/2013/605029.

[13]

J. P. Kharade and K. D. Kucche, On the impulsive implicit $\Psi$-Hilfer fractional differential equations with delay, Math. Methods Appl. Sci., 43 (2020), 1938-1952.  doi: 10.1002/mma.6017.

[14]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.

[15]

V. Lakshmikantham, S. Leela and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, 2009.

[16]

J. Lv and X. Yang, Approximate controllability of Hilfer fractional neutral stochastic differential equations, Dynamic Systems and Applications, 27 (2018), 691-713. 

[17]

N. I. Mahmudov and M. A. McKibben, On the approximate controllability of fractional evolution equations with generalized Riemann–Liouville fractional derivative, J. Funct. Spaces, 2015 (2015), Art. ID 263823, 9 pp. doi: 10.1155/2015/263823.

[18]

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.

[19]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York, 1993.

[20]

G. Mophou, Controllability of a backward fractional semilinear differential equation, Appl. Math. Comput., 242 (2014), 168-178.  doi: 10.1016/j.amc.2014.05.042.

[21]

K. Mourad, Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion, Stoch. Anal. Appl., 36 (2018), 209-223.  doi: 10.1080/07362994.2017.1386570.

[22]

K. MouradE. Fateh and D. Baleanu, Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non local conditions, Collect. Math., 69 (2018), 283-296.  doi: 10.1007/s13348-017-0207-5.

[23]

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

[24] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. 
[25]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 1993.

[26]

J. Vanterler da C. SousaM. Benchohra and G. M. N'Guérékata, Attractivity for differential equations of fractional order and $\psi $-Hilfer type, Fract. Calc. Appl. Anal., 23 (2020), 1188-1207.  doi: 10.1515/fca-2020-0060.

[27]

J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the $\psi $-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91.  doi: 10.1016/j.cnsns.2018.01.005.

[28]

J. Vanterler da C. SousaK. D. Kucche and E. Capelas de Oliveira, On the Ulam–Hyers stabilities of the solutions of $\psi $ Hilfer fractional differential equation with abstract Volterra operator, Math. Methods Appl. Sci., 42 (2019), 3021-3032.  doi: 10.1002/mma.5562.

[29]

J. Vanterler da C. SousaK. D. Kucche and E. Capelas de Oliveira, Stability of $\psi $ Hilfer impulsive fractional differential equations, Appl. Math. Lett., 88 (2019), 73-80.  doi: 10.1016/j.aml.2018.08.013.

[30]

C. S. Varun Bose and R. Udhayakumar, A note on the existence of Hilfer fractional differential inclusions with almost sectorial operators, Math. Methods Appl. Sci., 45 (2022), 2530-2541.  doi: 10.1002/mma.7938.

[31]

V. Vijayakumar and R. Udhayakumar, Results on approximate controllability for non-densely defined Hilfer fractional differential system with infinite delay, Chaos Solitons Fractals, 139 (2020), 110019, 11 pp. doi: 10.1016/j.chaos.2020.110019.

[32]

V. Vijayakumar and R. Udhayakumar, A new exploration on existence of Sobolev-type Hilfer fractional neutral integro-differential equations with infinite delay, Numerical Methods for Partial Differential Equations, 37 (2021), 750-766.  doi: 10.1002/num.22550.

[33]

V. VijayakumarR. Udhayakumar and K. Kavitha, On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay, Evol. Equ. Control Theory, 10 (2021), 271-296.  doi: 10.3934/eect.2020066.

[34]

V. VijayakumarR. UdhayakumarY. Zhou and N. Sakthivel, Approximate controllability results for Sobolev-type delay differential system of fractional order without uniqueness, Numerical Methods for Partial Differential Equations, 2020 (2020), 1-20.  doi: 10.1002/num.22642.

[35]

J. WangM. Fěckan and Y. Zhou, Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions, Evol. Equ. Control Theory, 6 (2017), 471-486.  doi: 10.3934/eect.2017024.

[36]

M. Yang, Existence uniqueness of mild solutions for $\psi -$Caputo fractional stochastic evolution equations driven by fBm, J. Inequal. Appl., 2021 (2021), Paper No. 170, 18 pp. doi: 10.1186/s13660-021-02703-x.

[37]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.

show all references

References:
[1]

H. M. Ahmed, M. M. El-Borai, A. S. Okb El Bab and M. Elsaid Ramadan, Approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion, Bound. Value Probl., (2020), Paper No. 120, 25 pp. doi: 10.1186/s13661-020-01418-0.

[2]

R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481.  doi: 10.1016/j.cnsns.2016.09.006.

[3]

P. Bedi, A. Kumar, T. Abdeljawad, Z. A. Khan and A. Khan, Existence and approximate controllability of Hilfer fractional evolution equations with almost sectorial operators, Adv. Difference Equ., (2020), Paper No. 615, 15 pp. doi: 10.1186/s13662-020-03074-1.

[4]

S. N. Bora and B. Roy, Approximate controllability of a class of semilinear Hilfer fractional differential equations, Results Math., 76 (2021), Paper No. 197, 20 pp. doi: 10.1007/s00025-021-01507-1.

[5]

Y.-K. ChangA. Pereira and R. Ponce, Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators, Fract. Calc. Appl. Anal., 20 (2017), 963-987.  doi: 10.1515/fca-2017-0050.

[6]

A. Debbouche and V. Antonov, Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces, Chaos Solitons Fractals, 102 (2017), 140-148.  doi: 10.1016/j.chaos.2017.03.023.

[7]

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

[8]

C. DineshkumarK. Sooppy NisarR. Udhayakumar and V. Vijayakumar, A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions, Asian Journal of Control, 2021 (2021), 1-17. 

[9]

J. DuW. Jiang and A. U. K. Niazi, Approximate controllability of impulsive Hilfer fractional differential inclusions, J. Nonlinear Sci. Appl., 10 (2017), 595-611.  doi: 10.22436/jnsa.010.02.23.

[10]

S. Guechi, R. Dhayal, A. Debbouche and M. Malik, Analysis and optimal control of $\varphi $-Hilfer fractional semilinear equations involving nonlocal impulsive conditions, Symmetry, 13 (2021), 2084, 1–18.

[11]

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

[12]

M. D. Kassim and N.-E. Tatar, Well-posedness and stability for a differential problem with Hilfer–Hadamard fractional derivative, Abstr. Appl. Anal., 2013 (2013), Art. ID 605029, 12 pp. doi: 10.1155/2013/605029.

[13]

J. P. Kharade and K. D. Kucche, On the impulsive implicit $\Psi$-Hilfer fractional differential equations with delay, Math. Methods Appl. Sci., 43 (2020), 1938-1952.  doi: 10.1002/mma.6017.

[14]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.

[15]

V. Lakshmikantham, S. Leela and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, 2009.

[16]

J. Lv and X. Yang, Approximate controllability of Hilfer fractional neutral stochastic differential equations, Dynamic Systems and Applications, 27 (2018), 691-713. 

[17]

N. I. Mahmudov and M. A. McKibben, On the approximate controllability of fractional evolution equations with generalized Riemann–Liouville fractional derivative, J. Funct. Spaces, 2015 (2015), Art. ID 263823, 9 pp. doi: 10.1155/2015/263823.

[18]

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.

[19]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, Wiley, New York, 1993.

[20]

G. Mophou, Controllability of a backward fractional semilinear differential equation, Appl. Math. Comput., 242 (2014), 168-178.  doi: 10.1016/j.amc.2014.05.042.

[21]

K. Mourad, Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion, Stoch. Anal. Appl., 36 (2018), 209-223.  doi: 10.1080/07362994.2017.1386570.

[22]

K. MouradE. Fateh and D. Baleanu, Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non local conditions, Collect. Math., 69 (2018), 283-296.  doi: 10.1007/s13348-017-0207-5.

[23]

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

[24] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. 
[25]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Yverdon, 1993.

[26]

J. Vanterler da C. SousaM. Benchohra and G. M. N'Guérékata, Attractivity for differential equations of fractional order and $\psi $-Hilfer type, Fract. Calc. Appl. Anal., 23 (2020), 1188-1207.  doi: 10.1515/fca-2020-0060.

[27]

J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the $\psi $-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91.  doi: 10.1016/j.cnsns.2018.01.005.

[28]

J. Vanterler da C. SousaK. D. Kucche and E. Capelas de Oliveira, On the Ulam–Hyers stabilities of the solutions of $\psi $ Hilfer fractional differential equation with abstract Volterra operator, Math. Methods Appl. Sci., 42 (2019), 3021-3032.  doi: 10.1002/mma.5562.

[29]

J. Vanterler da C. SousaK. D. Kucche and E. Capelas de Oliveira, Stability of $\psi $ Hilfer impulsive fractional differential equations, Appl. Math. Lett., 88 (2019), 73-80.  doi: 10.1016/j.aml.2018.08.013.

[30]

C. S. Varun Bose and R. Udhayakumar, A note on the existence of Hilfer fractional differential inclusions with almost sectorial operators, Math. Methods Appl. Sci., 45 (2022), 2530-2541.  doi: 10.1002/mma.7938.

[31]

V. Vijayakumar and R. Udhayakumar, Results on approximate controllability for non-densely defined Hilfer fractional differential system with infinite delay, Chaos Solitons Fractals, 139 (2020), 110019, 11 pp. doi: 10.1016/j.chaos.2020.110019.

[32]

V. Vijayakumar and R. Udhayakumar, A new exploration on existence of Sobolev-type Hilfer fractional neutral integro-differential equations with infinite delay, Numerical Methods for Partial Differential Equations, 37 (2021), 750-766.  doi: 10.1002/num.22550.

[33]

V. VijayakumarR. Udhayakumar and K. Kavitha, On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay, Evol. Equ. Control Theory, 10 (2021), 271-296.  doi: 10.3934/eect.2020066.

[34]

V. VijayakumarR. UdhayakumarY. Zhou and N. Sakthivel, Approximate controllability results for Sobolev-type delay differential system of fractional order without uniqueness, Numerical Methods for Partial Differential Equations, 2020 (2020), 1-20.  doi: 10.1002/num.22642.

[35]

J. WangM. Fěckan and Y. Zhou, Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions, Evol. Equ. Control Theory, 6 (2017), 471-486.  doi: 10.3934/eect.2017024.

[36]

M. Yang, Existence uniqueness of mild solutions for $\psi -$Caputo fractional stochastic evolution equations driven by fBm, J. Inequal. Appl., 2021 (2021), Paper No. 170, 18 pp. doi: 10.1186/s13660-021-02703-x.

[37]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.

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