Solvability and controllability of second-order non-autonomous impulsive neutral evolution hemivariational inequalities

Yong-Ki Ma; N. Valliammal; K. Jothimani; V. Vijayakumar; Yong-Ki Ma; N. Valliammal; K. Jothimani; V. Vijayakumar

doi:10.3934/math.20241288

AIMS Mathematics

2024, Volume 9, Issue 10: 26462-26482. doi: 10.3934/math.20241288

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Solvability and controllability of second-order non-autonomous impulsive neutral evolution hemivariational inequalities

1.
Department of Applied Mathematics, Kongju National University, Chungcheongnam-do 32588, Republic of Korea
2.
Department of Mathematics, Sri Eshwar College of Engineering, Coimbatore 641202, Tamil Nadu, India
3.
Department of Mathematics, Sri G.V.G. Visalakshi College for Women, Udumalpet 642128, Tamil Nadu, India
4.
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India

Received: 11 July 2024 Revised: 03 September 2024 Accepted: 05 September 2024 Published: 12 September 2024
MSC : 34K30, 35R10, 35R12, 35R70, 93B05

The primary aim of this article is to explore the approximate controllability of second-order impulsive hemivariational inequalities with initial conditions in Hilbert space. The mild solution was initially derived using the properties of the cosine and sine family of operators, Clarke's subdifferential, and the fact that the related linear equation has an evolution operator. The results of the approximate controllability of the considered systems are then taken into account using the fixed-point theorem method. An application is provided to support our theoretical findings.
- existence,
- approximate controllability,
- generalized Clarke's subdifferential,
- hemivariational inequalities
Citation: Yong-Ki Ma, N. Valliammal, K. Jothimani, V. Vijayakumar. Solvability and controllability of second-order non-autonomous impulsive neutral evolution hemivariational inequalities[J]. AIMS Mathematics, 2024, 9(10): 26462-26482. doi: 10.3934/math.20241288

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Abstract

The primary aim of this article is to explore the approximate controllability of second-order impulsive hemivariational inequalities with initial conditions in Hilbert space. The mild solution was initially derived using the properties of the cosine and sine family of operators, Clarke's subdifferential, and the fact that the related linear equation has an evolution operator. The results of the approximate controllability of the considered systems are then taken into account using the fixed-point theorem method. An application is provided to support our theoretical findings.

References

[1]	Y. Huang, Z. Liu, B. Zeng, Optimal control of feedback control systems governed by hemivariational inequalities, Comput. Math. Appl., 70 (2015), 2125–2136. https://doi.org/10.1016/j.camwa.2015.08.029 doi: 10.1016/j.camwa.2015.08.029
[2]	P. D. Panagiotopoulos, Hemivariational inequalities: applications in mechanics and engineering, Berlin: Springer, 1993. https://doi.org/10.1007/978-3-642-51677-1
[3]	P. D. Panagiotopoulos, Non-convex superpotentials in the sense of F.H. Clarke and applications, Mech. Res. Commun., 8 (1981), 335–340. https://doi.org/10.1016/0093-6413(81)90064-1 doi: 10.1016/0093-6413(81)90064-1
[4]	S. Carl, D. Motreanu, Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient, J. Differ. Equations, 191 (2003), 206–233. https://doi.org/10.1016/S0022-0396(03)00022-6 doi: 10.1016/S0022-0396(03)00022-6
[5]	S. Carl, Existence of extremal solutions of boundary hemivariational inequalities, J. Differ. Equations, 171 (2001), 370–396. https://doi.org/10.1006/jdeq.2000.3845 doi: 10.1006/jdeq.2000.3845
[6]	F. H. Clarke, Optimization and nonsmooth analysis, Wiley, 1990. https://doi.org/10.1137/1.9781611971309
[7]	Z. Denkowski, S. Migórski, N. S. Papageorgiou, An introduction to non-linear analysis: theory, New York: Springer, 2003. https://doi.org/10.1007/978-1-4419-9158-4
[8]	Z. Liu, N. S. Papageorgiou, Double phase Dirichlet problems with unilateral constraints, J. Differ. Equations, 316 (2022), 249–269. https://doi.org/10.1016/j.jde.2022.01.040 doi: 10.1016/j.jde.2022.01.040
[9]	Z. H. Liu, X. Li, Approximate controllability for a class of hemivariational inequalities, Nonlinear Anal., 22 (2015), 581–591. https://doi.org/10.1016/j.nonrwa.2014.08.010 doi: 10.1016/j.nonrwa.2014.08.010
[10]	F. S. Acharya, N. C. Dimplekumar, Controllability of neutral impulsive differential inclusions with nonlocal conditions, Appl. Math., 2 (2011), 1486–1496. https://doi.org/10.4236/am.2011.212211 doi: 10.4236/am.2011.212211
[11]	M. Benchohra, J. Henderson, S. K. Ntouyas, Existence results for impulsive multivalued semilinear neutral functional differential inclusions in Banach spaces, J. Math. Anal. Appl., 263 (2001), 763–780. https://doi.org/10.1006/jmaa.2001.7663 doi: 10.1006/jmaa.2001.7663
[12]	X. Hao, L. Liu, Y. Wu, Mild solutions of impulsive semilinear neutral evolution equations in Banach spaces, J. Nonlinear Sci. Appl., 9 (2016), 6183–6194. https://doi.org/10.22436/jnsa.009.12.23 doi: 10.22436/jnsa.009.12.23
[13]	N. Valliammal, K. Jothimani, S. K. Panda, V. Vijayakumar, An investigation on the existence and approximate controllability of neutral stochastic hemivariational inequalities, Rend. Circ. Mat. Palermo, II. Ser., 73 (2024), 941–958. https://doi.org/10.1007/s12215-023-00967-4 doi: 10.1007/s12215-023-00967-4
[14]	J. Haslinger, P. D. Panagiotopoulos, Optimal control of systems governed by hemivariational inequalities. Existence and approximation results, Nonlinear Anal., 24 (1995), 105–119. https://doi.org/10.1016/0362-546X(93)E0022-U doi: 10.1016/0362-546X(93)E0022-U
[15]	Y. Liu, Z. Liu, N. S. Papageorgiou, Sensitivity analysis of optimal control problems driven by dynamic history-dependent variational-hemivariational inequalities, J. Differ. Equations, 342 (2023), 559–595. https://doi.org/10.1016/j.jde.2022.10.009 doi: 10.1016/j.jde.2022.10.009
[16]	S. Migórski, On existence of solutions for parabolic hemivariational inequalities, J. Comput. Appl. Math., 129 (2001), 77–87. https://doi.org/10.1016/S0377-0427(00)00543-4 doi: 10.1016/S0377-0427(00)00543-4
[17]	J. Zhao, L. Liu, E. Vilches, C. Wen, J. C. Yao, Optimal control of an evolution hemivariational inequality involving history-dependent operators, Commun. Nonlinear Sci. Numer. Simul., 103 (2021), 105992. https://doi.org/10.1016/j.cnsns.2021.105992 doi: 10.1016/j.cnsns.2021.105992
[18]	J. Y. Park, S. H. Park, Optimal control problems for anti-periodic quasi-linear hemivariational inequalities, Optim. Control Appl. Met., 28 (2007), 275–287. https://doi.org/10.1002/oca.803 doi: 10.1002/oca.803
[19]	J. Y. Park, S. H. Park, Existence of solutions and optimal control problems for hyperbolic hemivariational inequalities, ANZIAM J., 47 (2005), 51–63. https://doi.org/10.1017/S1446181100009767 doi: 10.1017/S1446181100009767
[20]	X. Li, Z. Liu, N. S. Papageorgiou, Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications, Nonlinearity, 36 (2023), 1323–1348. https://doi.org/10.1088/1361-6544/acb191 doi: 10.1088/1361-6544/acb191
[21]	Y. K. Ma, J. Pradeesh, A. Shukla, V. Vijayakumar, K. Jothimani, An analysis on the approximate controllability of neutral impulsive stochastic integrodifferential inclusions via resolvent operators, Heliyon, 9 (2023), e20837. https://doi.org/10.1016/j.heliyon.2023.e20837 doi: 10.1016/j.heliyon.2023.e20837
[22]	J. Chen, Z. Liu, F. E. Lomovtsev, V. Obukhovskii, Optimal feedback control for a class of second-order evolution differential inclusions with Clarke's subdifferential, J. Nonlinear Var. Anal., 6 (2022), 551–565. https://doi.org/10.23952/jnva.6.2022.5.08 doi: 10.23952/jnva.6.2022.5.08
[23]	Y. Liu, Z. Liu, S. Peng, C. Wen, Optimal feedback control for a class of fractional evolution equations with history-dependent operators, Fract. Calc. Appl. Anal., 25 (2022), 1108–1130. https://doi.org/10.1007/s13540-022-00054-y doi: 10.1007/s13540-022-00054-y
[24]	J. Pradeesh, V. Vijayakumar, Investigating the existence results for Hilfer fractional stochastic evolution inclusions of order $1 < \mu < 2$, Qual. Theory Dyn. Syst., 23 (2024), 46. https://doi.org/10.1007/s12346-023-00899-5 doi: 10.1007/s12346-023-00899-5
[25]	K. Rykaczewski, Approximate controllability of differential inclusions in Hilbert spaces, Nonlinear Anal., 75 (2012), 2701–2712. https://doi.org/10.1016/j.na.2011.10.049 doi: 10.1016/j.na.2011.10.049
[26]	V. Vijayakumar, Approximate controllability for a class of second-order stochastic evolution inclusions of Clarke's subdifferential type, Results Math., 73 (2018), 1–17. https://doi.org/10.1007/s00025-018-0807-8 doi: 10.1007/s00025-018-0807-8
[27]	J. Pradeesh, V. Vijayakumar, A new approach on the approximate controllability results for Hilfer fractional stochastic hemivariational inequalities of order $1 < \mu < 2$, Qual. Theory Dyn. Syst., 23 (2024), 158. https://doi.org/10.1007/s12346-024-01012-0 doi: 10.1007/s12346-024-01012-0
[28]	C. J. K. Batty, R. Chill, S. Srivastava, Maximal regularity for second order non-autonomous Cauchy problems, Studia Math., 189 (2008), 205–223.
[29]	F. Faraci, A. Iannizzotto, A multiplicity theorem for a perturbed second-order non-autonomous system, Proc. Edinburgh Math. Soc., 49 (2006), 267–275. https://doi.org/10.1017/S001309150400149X doi: 10.1017/S001309150400149X
[30]	E. Obrecht, Evolution operators for higher order abstract parabolic equations, Czech. Math. J., 36 (1986), 210–222. https://doi.org/10.21136/CMJ.1986.102085 doi: 10.21136/CMJ.1986.102085
[31]	M. Kozak, A fundamental solution of a second order differential equation in a Banach space, Univ. Lagel. Acta Math., 32 (1995), 275–289.
[32]	S. Hu, N. S. Papageorgiou, Handbook of multivalued analysis, volume I: theory, New York: Springer, 1997.
[33]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, New York: Springer-Verlag, 1983. https://doi.org/10.1007/978-1-4612-5561-1
[34]	T. W. Ma, Topological degrees of set-valued compact fields in locally convex spaces, Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1972.
[35]	S. Migórski, A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Control Optim., 41 (2009), 1415–1435. https://doi.org/10.1137/080733231 doi: 10.1137/080733231
[36]	S. Migórski, A. Ochal, M. Sofonea, Nonlinear inclusions and hemivariational inequalities: models and analysis of contact problems, Advances in Mechanics and Mathematics, Vol. 26, New York: Springer, 2013. https://doi.org/10.1007/978-1-4614-4232-5

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