APPROXIMATE CONTROLLABILITY OF NONLINEAR EVOLUTION FRACTIONAL CONTROL SYSTEM WITH DELAY

doi:10.1007/s10473-025-0216-5

Abstract

Abstract: This article studies the existence and uniqueness of the mild solution of a family of control systems with a delay that are governed by the nonlinear fractional evolution differential equations in Banach spaces. Moreover, we establish the controllability of the considered system. To do so, first, we investigate the approximate controllability of the corresponding linear system. Subsequently, we prove the nonlinear system is approximately controllable if the corresponding linear system is approximately controllable. To reach the conclusions, the theory of resolvent operators, the Banach contraction mapping principle, and fixed point theorems are used. While concluding, some examples are given to demonstrate the efficacy of the proposed results.

Key words: nonlinear fractional differential equation, Caputo fractional derivative, mild solution, existence and uniqueness theorems, approximate controllability

CLC Number:

47J35

Kamla Kant Mishra, Shruti Dubey. APPROXIMATE CONTROLLABILITY OF NONLINEAR EVOLUTION FRACTIONAL CONTROL SYSTEM WITH DELAY[J].Acta mathematica scientia,Series B, 2025, 45(2): 553-568.

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References

[1] Podlubny I. Fractional Differential Equations.New York: Academic Press, 1998[] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. New York: Elsevier, 2006
[2] Zhou Y, Wang J, Zhang L.Basic Theory of Fractional Differential Equations. Singapore: World Scientific, 2016
[3] Hilfer R.Applications of Fractional Calculus in Physics. Singapore: World Scientific, 2000
[4] Mishra K K, Dubey S.On space-fractional diffusion equations with conformable derivative// Filipe J, Ghosh A, Prates R O, Zhou L. Computational Sciences-Modelling, Computing and Soft Computing. Singapore: Springer, 2020: 79-90
[5] Raja M M, Vijayakumar V, Udhayakumar R, Zhou Y. A new approach on the approximate controllability of fractional differential evolution equations of order 1 $<r <$ 2 in Hilbert spaces. Chaos, Solitons & Fractals, 2020, 141: 110310
[6] Li M, Wang J. Exploring delayed mittag-leffler type matrix functions to study finite time stability of fractional delay differential equations. Applied Mathematics and Computation, 2018, 324: 254-265
[7] Alam M M, Dubey S, Baleanu D. New interpolation spaces and strict Hölder regularity for fractional abstract Cauchy problem. Boundary Value Problems, 2021, 2021(1): 1-18
[8] Du F, Lu J G. Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities. Applied Mathematics and Computation, 2020, 375: 125079
[9] Vinodbhai C D, Dubey S. Investigation to analytic solutions of modified conformable time-space fractional mixed partial differential equations. Partial Differential Equations in Applied Mathematics, 2022, 5: 100294
[10] Sakthivel R, Ren Y, Mahmudov N I. On the approximate controllability of semilinear fractional differential systems. Computers & Mathematics with Applications, 2011, 62(3): 1451-1459
[11] Kumar S, Sukavanam N. Approximate controllability of fractional order semilinear systems with bounded delay. Journal of Differential Equations, 2012, 252(11): 6163-6174
[12] Fan Z, Dong Q, Li G. Approximate controllability for semilinear composite fractional relaxation equations. Fractional Calculus and Applied Analysis, 2016, 19(1): 267-284
[13] Ge F D, Zhou H C, Kou C H. Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique. Applied Mathematics and Computation, 2016, 275: 107-120
[14] Chang Y K, Pereira A, Ponce R. Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators. Fractional Calculus and Applied Analysis, 2017, 20(4): 963-987
[15] Kalman R E. Controllability of linear dynamical systems. Contributions to Differential Equations, 1963, 1: 190-213
[16] Monje C A, Chen Y, Vinagre B M, et al.Fractional-order Systems and Controls: Fundamentals and Applications. London: Springer, 2010
[17] Zhou H X. Approximate controllability for a class of semilinear abstract equations. SIAM Journal on Control and Optimization, 1983, 21(4): 551-565
[18] Mahmudov N I. Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM Journal on Control and Optimization, 2003, 42(5): 1604-1622
[19] Amann H. Parabolic evolution equations and nonlinear boundary conditions. Journal of Differential Equations, 1988, 72(2): 201-269
[20] Chen P, Zhang X, Li Y. Study on fractional non-autonomous evolution equations with delay. Computers & Mathematics with Applications, 2017, 73(5): 794-803
[21] Hilfer R.Functional Analytic Methods for Partial Differential Equations. New York: Marcel Dekker, 1997
[22] Weiss C J, Bloemen Waanders B G, Antil H. Fractional operators applied to geophysical electromagnetics. Geophysical Journal International, 2020, 220(2): 1242-1259
[23] George R K. Approximate controllability of nonautonomous semilinear systems. Nonlinear Analysis: Theory, Methods & Applications, 1995, 24(9): 1377-1393
[24] Fu X, Rong H. Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions. Automation and Remote Control, 2016, 77(3): 428-442
[25] Fu X. Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evolution Equations & Control Theory, 2017, 6(4): 517-534
[26] Chen P, Zhang X, Li Y. Approximate controllability of non-autonomous evolution system with nonlocal conditions. Journal of Dynamical and Control Systems, 2020, 26(1): 1-16
[27] Dubey S, Sharma M. Solutions to fractional functional differential equations with nonlocal conditions. Fractional Calculus and Applied Analysis, 2014, 17(3): 654-673
[28] Sharma M, Dubey S. Asymptotic behavior of solutions to nonlinear nonlocal fractional functional differential equations. Journal of Nonlinear Evolution Equations and Applications, 2015, 2015(2): 21-30
[29] Mishra K K, Dubey S, Baleanu D. Existence and controllability of a class of non-autonomous nonlinear evolution fractional integrodifferential equations with delay. Qualitative Theory of Dynamical Systems, 2022, 21(4): Art 165
[30] Sharma M, Dubey S. Controllability of Sobolev type nonlinear nonlocal fractional functional integrodifferential equations. Progress in Fractional Differentiation and Applications, 2015, 1(4): 281-293
[31] Zhu B, Han B. Approximate controllability for mixed type non-autonomous fractional differential equations. Qualitative Theory of Dynamical Systems, 2022, 21(4): Art 111
[32] Zhu B, Han B, Yu W. Existence of mild solutions for a class of fractional non-autonomous evolution equations with delay. Acta Mathematicae Applicatae Sinica English Series, 2020, 36(4): 870-878
[33] Kumaresan S.Topology of Metric Spaces. New Delhi: Narosa Publishing House, 2011
[34] Zhu B, Liu L, Wu Y. Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay. Computers & Mathematics with Applications2019, 78(6): 1811-1818
[35] Jain P K, Ahmad K, Ahuja O P. Functional Analysis. New Delhi: New Age International, 1995
[36] Sakthivel R, Choi Q, Anthoni S M. Controllability of nonlinear neutral evolution integrodifferential systems. Journal of Mathematical Analysis and Applications, 2002, 275(1): 402-417