Approximate controllability for some retarded integrodifferential inclusions using a version of the Leray-Schauder fixed point theorem for the multivalued maps

Jaouad El Matloub; Khalil Ezzinbi

doi:10.3934/mcrf.2023026

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2024, Volume 14, Issue 3: 821-847. Doi: 10.3934/mcrf.2023026

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Approximate controllability for some retarded integrodifferential inclusions using a version of the Leray-Schauder fixed point theorem for the multivalued maps

Jaouad El Matloub^1, , and
Khalil Ezzinbi^{1, 2,}

1.
Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, B.P: 2390 Marrakech, Morocco
2.
Sorbonne University, EDITE (ED130), IRD, UMMISCO, F-93143, Bondy, Paris, France

^*Corresponding author: Jaouad El Matloub
^*Corresponding author: Jaouad El Matloub

Received: January 2023

Revised: July 2023

Early access: August 7, 2023

Published online: August 7, 2023

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Abstract

The main goal of this work is to investigate the approximate controllability for a class of non-autonomous delayed integrodifferential inclusions with unbounded delay. Our technique starts with the search for the optimal control for a linear quadratic regulator problem. The existence of such an optimal control aids to establish sufficient conditions insuring our inclusion problem's approximate controllability. The findings we acquired represent a generalization and extension of previous results on this topic. Finally, we present an example to illustrate the abstract theory.

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Mathematics Subject Classification: 45K05, 93B05, 35D30, 35R70.

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[1]	M. Adimy, H. Bouzahir and K. Ezzinbi, Existence for a class of partial functional differential equations with infinite delay, Nonlinear Anal., 46 (2001), 91-112. doi: 10.1016/S0362-546X(99)00447-2.
[2]	M. Adimy, H. Bouzahir and K. Ezzinbi, Local existence for a class of partial neutral functional differential equations with infinite delay, Differ. Equ. Dyn. Syst., 12 (2004), 353-370.
[3]	S. Arora, M. T. Mohan and J. Dabas, Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces, J. Differential Equations, 307 (2022), 83-113. doi: 10.1016/j.jde.2021.10.049.
[4]	E. Asplund, Averaged norms, Isr. J. Math., 5 (1967), 227-233. doi: 10.1007/BF02771611.
[5]	M. Benchohra and S. K. Ntouyas, Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces, J. Math. Anal. Appl., 258 (2001), 573-590. doi: 10.1006/jmaa.2000.7394.
[6]	J. M. Borwein and J. Vanderwerff, Fréchet-Legendre functions and reflexive Banach spaces, J. Convex Anal., 173 (2010), 915-924.
[7]	Y. K. Chang and W. T. Li, Controllability of functional integro-differential inclusions with an unbounded delay, J. Optim. Theory Appl., 132 (2007), 125-142. doi: 10.1007/s10957-006-9088-6.
[8]	J. P. Dauer, N. I. Mahmudov and M. M. Matar, Approximate controllability of backward stochastic evolution equations in Hilbert spaces, J. Math. Anal. Appl., 323 (2006), 42-56. doi: 10.1016/j.jmaa.2005.09.089.
[9]	K. Deimling, Multivalued differential equations, De Gruyter Ser. Nonlinear Anal. Appl., 1. Walter de Gruyter & Co., Berlin, 1992. doi: 10.1515/9783110874228.
[10]	A. Diop, M. Dieye and B. Hazarika, Random integrodifferential equations of Volterra type with delay: attractiveness and stability, Appl. Math. Comput., 430 (2022), 127301, 18 pp. doi: 10.1016/j.amc.2022.127301.
[11]	K. Ezzinbi, S. Ghnimi and M. A. Taoudi, Existence results for some partial integrodifferential equations with nonlocal conditions, Glasnik matematički, 51 (2016), 413-430. doi: 10.3336/gm.51.2.09.
[12]	I. Ekeland and T. Turnbull, Infinite-dimensional Optimization and Convexity, Chicago Lectures in Math., The University of Chicago press, Chicago and London, 1983.
[13]	C. A. Gallegos and H. R. Henríquez, Fixed points of multivalued maps under local Lipschitz conditions and applications, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 150 (2020), 1467-1494. doi: 10.1017/prm.2018.151.
[14]	A. Granas and J. Dugundji, Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003. doi: 10.1007/978-0-387-21593-8.
[15]	R. C. Grimmer and W. Schappacher, Weak solutions of integrodifferential equations and resolvent operators, Journal of Integral Equations, 6 (1984), 205-229.
[16]	R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Transactions of the American Mathematical Society, 273 (1982), 333-349. doi: 10.1090/S0002-9947-1982-0664046-4.
[17]	J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.
[18]	Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., vol. 1473. Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.
[19]	S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Math. Appl., 419. Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.
[20]	M. I. Kamenskii, V. V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl., 7. Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110870893.
[21]	S. G. Kreǐn, Linear differential equations in Banach space, Translations of Mathematical Monographs, Vol. 29. American Mathematical Society, Providence, R.I., 1971.
[22]	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.
[23]	J. W. Nunziato, On heat conduction in materials with memory, Quarterly of Applied Mathematics, 29 (1971), 187-204. doi: 10.1090/qam/295683.
[24]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[25]	K. Ravikumar, M. T. Mohan and A. Anguraj, Approximate controllability of a non-autonomous evolution equation in Banach spaces, Numer. Algebra Control Optim., 11 (2021), 461–485, arXiv: 2004.10460. doi: 10.3934/naco.2020038.
[26]	K. Rykaczewski, Approximate controllability of differential inclusions in Hilbert spaces, Nonlinear Anal., 75 (2012), 2701-2712. doi: 10.1016/j.na.2011.10.049.
[27]	R. Sakthivel and E. R. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Int. J. Control, 83 (2010), 387-393. doi: 10.1080/00207170903171348.
[28]	H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM Journal on Control and Optimization, 21 (1983), 551-565. doi: 10.1137/0321033.
[29]	E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ. Elsevier/North-Holland, Amsterdam, 3 (2007), 527-621. doi: 10.1016/S1874-5717(07)80010-7.