On the solution of \(\mathcal{T}-\)controllable abstract fractional differential equations with impulsive effects

Ganga Ram Gautam; Sandra Pinelas; Manoj Kumar; Namrata Arya; Jaimala Bishnoi

doi:10.56754/0719-0646.2503.363

DOI:

https://doi.org/10.56754/0719-0646.2503.363

Abstract

In this research article, we delimitate the definition of mild solution for abstract fractional differential equations with state-dependent delay (AFDEw/SDD) of order \(\alpha\in(1,2)\) with impulsive effects and compare the solution to the second-order impulsive differential equations. Further, we obtain sufficient conditions of the existence of mild solution for instantaneous and non-instantaneous impulsive fractional functional differential inclusions with state-dependent delay (IFDIw/SDD) using the multi-valued fixed point theory and operator techniques. Furthermore, we study the trajectory controllability (\(\mathcal{T}-\)controllability) of the AFDEw/SDD. At last, we present some examples to illustrate the sufficient conditions involving partial and ordinary derivatives.

Keywords

Fractional differential equation , functional-differential equations with fractional derivatives , initial value problems , fixed point theorems , controllability

Mathematics Subject Classification:

34A08 , 34K37 , 34A12 , 37H10 , 93B05

Ganga Ram Gautam DST-Centre for Interdisciplinary Mathematical Sciences, Institute of Science, Banaras Hindu University, Varanasi-221005, India. https://orcid.org/0000-0002-2819-4076
Sandra Pinelas Department of Exact Sciences and Engineering, Academia Militar, Military Academy 2720-113 Amadora, Portugal. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal. https://orcid.org/0000-0002-0984-0159
Manoj Kumar Department of mathematics, Deshbandhu College, University of Delhi, Delhi-110019, India.
Namrata Arya DST-Centre for Interdisciplinary Mathematical Sciences, Institute of Science, Banaras Hindu University, Varanasi-221005, India.
Jaimala Bishnoi Department of mathematics, Chaudhary Charan Singh University Meerut, Meerut-250001, India.

Pages: 363–386
Date Published: 2023-12-20
Vol. 25 No. 3 (2023)

S. Abbas and M. Benchohra, “Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay”, Fract. Calc. Appl. Anal., vol. 13, no. 3, pp. 225–244, 2010.

S. Abbas and M. Benchohra, “Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses”, Appl. Math. Comput., vol. 257, pp. 190–198, 2015, doi: 10.1016/j.amc.2014.06.073.

R. Agarwal, S. Hristova, and D. O’Regan, “Mittag-Leffler stability for impulsive Caputo fractional differential equations”, Differ. Equ. Dyn. Syst., vol. 29, no. 3, pp. 689–705, 2021, doi: https://doi.org/10.1007/s12591-017-0384-4.

R. Agarwal, S. Hristova, and D. O’Regan, “Stability of solutions to impulsive Caputo fractional differential equations,” Electron. J. Differential Equations, 2016, Art. no. 58.

R. Agarwal, S. Hristova, and D. O’Regan, “Global mittag—leffler synchronization for neu- ral networks modeled by impulsive Caputo fractional differential equations with distributed delays,” Symmetry, vol. 10, no. 10, 2018, Art. no. 473.

R. P. Agarwal, B. de Andrade, and G. Siracusa, “On fractional integro-differential equations with state-dependent delay,” Comput. Math. Appl., vol. 62, no. 3, pp. 1143–1149, 2011, doi: 10.1016/j.camwa.2011.02.033.

D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,” Nonlinear Anal., vol. 69, no. 11, pp. 3692–3705, 2008, doi: 10.1016/j.na.2007.10.004.

M. Benchohra and F. Berhoun, “Impulsive fractional differential equations with state- dependent delay,” Commun. Appl. Anal., vol. 14, no. 2, pp. 213–224, 2010.

M. Benchohra, S. Litimein, and G. N’Guérékata, “On fractional integro-differential inclusions with state-dependent delay in Banach spaces,” Appl. Anal., vol. 92, no. 2, pp. 335–350, 2013, doi: 10.1080/00036811.2011.616496.

C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, ser. Lecture Notes in Mathematics. Berlin-New York: Springer-Verlag, 1977, vol. 580.

R. Chaudhary and S. Reich, “Existence and controllability results for Hilfer fractional evolution equations via integral contractors,” Fract. Calc. Appl. Anal., vol. 25, no. 6, pp. 2400–2419, 2022, doi: 10.1007/s13540-022-00099-z.

A. Chauhan and J. Dabas, “Existence of mild solutions for impulsive fractional-order semilinear evolution equations with nonlocal conditions,” Electron. J. Differential Equations, 2011, Art. no. 107.

A. Chauhan and J. Dabas, “Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition,” Commun. Nonlinear Sci. Numer. Simul., vol. 19, no. 4, pp. 821–829, 2014, doi: 10.1016/j.cnsns.2013.07.025.

A. Chauhan, J. Dabas, and M. Kumar, “Integral boundary-value problem for impulsive fractional functional differential equations with infinite delay,” Electron. J. Differential Equations, 2012, Art. no. 229.

H. Covitz and S. B. Nadler, Jr., “Multi-valued contraction mappings in generalized metric spaces,” Israel J. Math., vol. 8, pp. 5–11, 1970, doi: 10.1007/BF02771543.

J. Dabas and A. Chauhan, “Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay,” Math. Comput. Modelling, vol. 57, no. 3-4, pp. 754–763, 2013, doi: 10.1016/j.mcm.2012.09.001.

J. Dabas and G. R. Gautam, “Impulsive neutral fractional integro-differential equations with state dependent delays and integral condition”, Electron. J. Differential Equations, 2013, Art. no. 273.

J. P. C. dos Santos, M. M. Arjunan, and C. Cuevas, “Existence results for fractional neutral integro-differential equations with state-dependent delay,” Comput. Math. Appl., vol. 62, no. 3, pp. 1275–1283, 2011, doi: 10.1016/j.camwa.2011.03.048.

M. Feckan, Y. Zhou, and J. Wang, “On the concept and existence of solution for impulsive fractional differential equations,” Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 7, pp. 3050–3060, 2012, doi: 10.1016/j.cnsns.2011.11.017.

M. Feckan, Y. Zhou, and J. Wang, “Response to “Comments on the concept of existence of solution for impulsive fractional differential equations [Commun Nonlinear Sci Numer Simul 2014; 19:401–3.]”, Commun. Nonlinear Sci. Numer. Simul., vol. 19, no. 12, pp. 4213–4215, 2014, doi: 10.1016/j.cnsns.2014.04.014.

X. Fu and R. Huang, “Existence of solutions for neutral integro-differential equations with state-dependent delay,” Appl. Math. Comput., vol. 224, pp. 743–759, 2013, doi: 10.1016/j.amc.2013.09.010.

G. R. Gautam and J. Dabas, “Existence of mild solutions for impulsive fractional functional integro-differential equations,” Fract. Differ. Calc., vol. 5, no. 1, pp. 65–77, 2015, doi: 10.7153/fdc-05-06.

V. Govindaraj and R. K. George, “Trajectory controllability of fractional integro-differential systems in Hilbert spaces,” Asian J. Control, vol. 20, no. 5, pp. 1994–2004, 2018, doi: 10.1002/asjc.1685.

J. K. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcial. Ekvac., vol. 21, no. 1, pp. 11–41, 1978.

E. Hernández and D. O’Regan, “On a new class of abstract impulsive differential equations,” Proc. Amer. Math. Soc., vol. 141, no. 5, pp. 1641–1649, 2013, doi: 10.1090/S0002-9939-2012- 11613-2.

E. Hernández Morales, “A second-order impulsive Cauchy problem,” Int. J. Math. and Math. Sci., vol. 31, 2001, Art. no. 649632.

K. Jothimani, C. Ravichandran, V. Kumar, M. Djemai, and K. S. Nisar, “Interpretation of trajectory control and optimization for the nondense fractional system,” Int. J. Appl. Comput. Math., vol. 8, 2022, Art. no. 273.

R. E. Kalman, “A new approach to linear filtering and prediction problems”, Trans. ASME Ser. D. J. Basic Engrg., vol. 82, no. 1, pp. 35–45, 1960.

K. Karthikeyan, G. S. Murugapandian, and Z. Hammouch, “On mild solutions of fractional impulsive differential systems of Sobolev type with fractional nonlocal conditions,” Math. Sci. (Springer), vol. 17, no. 3, pp. 285–295, 2023, doi: 10.1007/s40096-022-00469-x.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, ser. North-Holland Mathematics Studies. Amsterdam, Netherlands: Elsevier Science B.V., 2006, vol. 204.

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

Y. Liu and B. Ahmad, “A study of impulsive multiterm fractional differential equations with single and multiple base points and applications,” The Scientific World Journal, 2014, Art. no. 194346.

W. X. Ma, T. Huang, and Y. Zhang, “A multiple exp-function method for nonlinear differential equations and its application”, Physica Scripta, no. 82, 2010, Art. no. 065003.

K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, ser. A Wiley-Interscience Publication. New York, USA: John Wiley & Sons, Inc., 1993.

M. Muslim and A. Kumar, “Trajectory controllability of fractional differential systems of order α ∈ (1,2] with deviated argument,” J. Anal., vol. 28, no. 1, pp. 295–304, 2020, doi: 10.1007/s41478-018-0081-x.

B. O. Onasanya, S. Wen, Y. Feng, W. Zhang, and J. Xiong, “Fuzzy coefficient of impulsive intensity in a nonlinear impulsive control system”, Neural Processing Letters, vol. 53, pp. 4639–4657, 2021, doi: 10.1007/s11063-021-10614-7.

M. Pierri, D. O’Regan, and V. Rolnik, “Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses,” Appl. Math. Comput., vol. 219, no. 12, pp. 6743–6749, 2013, doi: 10.1016/j.amc.2012.12.084.

I. Podlubny, Fractional differential equations. San Diego, CA, USA: Academic Press, Inc., 1999, vol. 198.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives. Yverdon, Switzerland: Gordon and Breach Science Publishers, 1993.

X.-B. Shu and Q. Wang, “The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1 < α < 2,” Comput. Math. Appl., vol. 64, no. 6, pp. 2100–2110, 2012, doi: 10.1016/j.camwa.2012.04.006.

J. Wang, M. Feckan, and Y. Zhou, “On the new concept of solutions and existence results for impulsive fractional evolution equations,” Dyn. Partial Differ. Equ., vol. 8, no. 4, pp. 345–361, 2011, doi: 10.4310/DPDE.2011.v8.n4.a3.

J. Wang, A. G. Ibrahim, and M. Feckan, “Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces”, Appl. Math. Comput., vol. 257, pp. 103–118, 2015, doi: 10.1016/j.amc.2014.04.093.

J. Wang, X. Li, and W. Wei, “On the natural solution of an impulsive fractional differential equation of order q ∈ (1,2)”, Commun. Nonlinear Sci. Numer. Simul., vol. 17, no. 11, pp. 4384–4394, 2012, doi: 10.1016/j.cnsns.2012.03.011.

Y. Z, F. H, and Z. X, “Existence of mild solutions for a class of fractional neutral evolution equations”, Pure Mathematics, vol. 12, no. 8, pp. 1333–1340, 2022, doi: 10.12677/PM.2022.128146.

J. Zeng, X. Yang, L. Wang, and X. Chen, “Robust asymptotical stability and stabilization of fractional-order complex-valued neural networks with delay”, Discrete Dyn. Nat. Soc., vol. 2021, 2021, Art. no. 5653791.