Abstract
This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder’s fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we provide two concrete examples to validate our results.
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Ahmed, H.M., El-Borai, M.M., El Bab, A.O., Ramadan, M.E.: Approximate controllability of non-instantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion. Bound. Value Probl. 2020, 1–25 (2020)
Arora, S., Mohan, M.T., Dabas, J.: Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach space. Nonlinear Anal. Hybrid Syst. 39, 100989 (2020)
Arora, S., Singh, S., Dabas, J., Mohan, M.T.: Approximate controllability of semilinear impulsive functional differential system with nonlocal conditions. IMA J. Math. Control Inform. 37, 1070–1088 (2020)
Arora, S., Mohan, M.T., Dabas, J.: Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Math. Control. Relat. Fields (2020). https://doi.org/10.3934/mcrf.2020049
Arora, U., Sukavanam, N.: Approximate controllability of second order semilinear stochastic system with nonlocal conditions. Appl. Math. Comput. 258, 111–119 (2015)
Barbu, V.: Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, New York (1993)
Barbu, V.: Controllability and Stabilization of Parabolic Equations. Springer, New York (2018)
Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)
Bochenek, J.: Existence of the fundamental solution of a second order evolution equation. Ann. Polon. Math. 66, 15–35 (1997)
Byszewski, L., Lakshmikantham, V.: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40, 11–19 (1991)
Chen, F., Sun, D., Shi, J.: Periodicity in a food-limited population model with toxicants and state dependent delays. J. Math. Anal. Appl. 288, 136–146 (2003)
Colao, V., Muglia, L., Xu, H.K.: Existence of solutions for a second order differential equation with non-instantaneous impulses and delay. Ann. Mat. 195, 697–716 (2016)
Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes. Cambridge University Press, Cambridge (1996)
Ekeland, I., Turnbull, T.: Infinite-Dimensional Optimization and Convexity. The University of Chicago press, Chicago and London (1983)
Feckan, M., Wang, J.: A general class of impulsive evolution equations. Topol. Methods Nonlinear Anal 46, 915–933 (2015)
Fu, X.: Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evol. Equ. Control Theory 6, 517–534 (2017)
Fu, X., Rong, H.: Approximate controllability of semilinear non-autonomous evolutionary systems with nonlocal conditions. Autom. Remote Control 77, 428–442 (2016)
Grudzka, A., Rykaczewski, K.: On approximate controllability of functional impulsive evolution inclusions in a Hilbert space. J. Optim. Theory Appl. 166, 414–439 (2015)
Guedda, L.: Some remarks in the study of impulsive differential equations and inclusions with delay. Fixed Point Theory 12, 349–354 (2011)
Henriquez, H.R.: Existence of solutions of non-autonomous second order functional differential equations with infinite delay. Nonlinear Anal. 74, 3333–3352 (2011)
Hernández, E., Henriquez, H.R., McKibben, M.A.: Existence results for abstract impulsive second order neutral functional differential equations. Nonlinear Anal. 70, 2736–2751 (2009)
Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141, 1641–1649 (2013)
Kisyński, J.: On cosine operator functions and one parameter group of operators. Stud. Math. 49, 93–105 (1972)
Kozak, M.: An abstract second order temporally inhomogeneous linear differential equation II. Univ. lagel. Acta Math. 32, 263–274 (1995)
Kumar, A., Muslim, M., Sakthivel, R.: Controllability of the second order nonlinear differential equations with non-instantaneous impulses. J. Dyn. Control Syst. 24, 325–342 (2018)
Kumar, A., Vats, R.K., Kumar, A.: Approximate controllability of second order nonautonomous system with finite delay. J. Dyn. Control Syst. 26, 611–627 (2020)
Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
Li, X.J., Yong, J.M.: Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc, Boston (1995)
Lin, Y.: Time-dependent perturbation theory for abstract evolution equations of second order. Stud. Math. 130, 263–274 (1998)
Liang, J., Liu, J.H., Xiao, T.J.: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. Math. Comput. Model. 49, 798–804 (2009)
Liu, S., Wang, J., O’Regan, D.: Trajectory approximately controllability and optimal control for non-instantaneous impulsive inclusions without compactness Topol. Methods Nonlinear Anal. 58, 19–49 (2021)
Lunardi, A.: On the linear heat equation with fading memory. SIAM J. Math. Anal. 21, 1213–1224 (1990)
Mahmudov, N.I.: Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control. Optim. 42, 1604–1622 (2003)
Mahmudov, N.I., Vijayakumar, V., Murugesu, R.: Approximate controllability of second order evolution differential inclusions in Hilbert spaces. Mediterr. J. Math. 13, 3433–3454 (2016)
Malik, M., Kumar, A., Feckan, M.: Existence, uniqueness, and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses. J. King Saud Univ. Sci. 30, 204–213 (2018)
Ntouyas, S.K.: Nonlocal initial and boundary value problems: a survey. In: Handbook of Differential Equations: Ordinary Differential Equations, vol. 2, pp. 461–557. Elsevier (2006)
Nunziato, J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–204 (1971)
Obrecht, E.: Evolution operators for higher order abstract parabolic equations. Czechoslov. Math. J. 36, 210–222 (1986)
Obrecht, E.: The Cauchy problem for time-dependent abstract parabolic equations of higher order. J. Math. Anal. Appl. 125, 508–530 (1987)
Perelson, A.S., Neumann, A.U., Markowitz, M., Leonard, J.M., Ho, D.D.: HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science 271, 1582–1586 (1996)
Pierri, M., Henríquez, H.R., Prokopczyk, A.: Global solutions for abstract differential equations with non-instantaneous impulses. Mediterr. J. Math. 49, 1685–1708 (2016)
Ravikumar, K., Mohan, M.T., Anguraj, A.: Approximate controllability of a nonautonomous evolution equation in Banach spaces. Numer. Algebra Control Optim. (2020)
Sakthivel, R., Anandhi, E.R., Mahmudov, N.I.: Approximate controllability of second order systems with state-dependent delay. Numer. Funct. Anal. Optim. 29, 1347–1362 (2008)
Serizawa, H., Watanabe, M.: Time-dependent perturbation for cosine families in Banach spaces. Houst. J. Math. 2, 579–586 (1986)
Singh, S., Arora, S., Mohan, M.T., Dabas, J.: Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evol. Equ. Control Theory (2020). https://doi.org/10.3934/eect.2020103
Shu, L., Shu, X.B., Mao, J.: Approximate controllability and existence of mild solutions for Riemann–Liouville fractional stochastic evolution equations with nonlocal conditions of order \(1<\alpha <2\). Fract. Calc. Appl. Anal. 22, 1086–1112 (2019)
Travis, C.C., Webb, G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Hungar. 32, 75–96 (1978)
Travis, C.C., Webb, G.F.: Compactness, regularity, and uniform continuity properties of strongly continuous cosine families. Houst. J. Math. 3, 555–567 (1977)
Travis, C.C., Webb, G.F.: Second order differential equations in Banach space. In: Nonlinear Equations in Abstract Spaces, pp. 331–361. Academic Press (1978)
Triggiani, R.: Addendum: a note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control. Optim. 18, 98 (1980)
Triggiani, R.: A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control Optim. 15, 407–411 (1977)
Vijaykumar, V., Udhayakumar, R., Dineshkumar, C.: Approximate controllability of second order nonlocal neutral differential evolution inclusions. IMA J. Math. Control Inform. (2020). https://doi.org/10.1093/imamci/dnaa001
Wei, W., Xiang, X., Peng, Y.: Nonlinear impulsive integro-differential equations of mixed type and optimal controls. Optimization 55, 141–156 (2006)
Yan, Z.: Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay. Int. J. Control 85, 1051–1062 (2012)
Zuazua, E.: Controllability and observability of partial differential equations: some results and open problems. In: Handbook of Differential Equations: Evolutionary Equations, vol. 3, pp. 527–621 (2007)
Acknowledgements
S. Arora would like to thank Council of Scientific and Industrial Research, New Delhi, Government of India (File No. 09/143(0931)/2013 EMR-I), for financial support to carry out his research work and Department of Mathematics, Indian Institute of Technology Roorkee (IIT Roorkee), for providing stimulating scientific environment and resources. J. Dabas would like to thank the Department of Atomic Energy (DAE), Mumbai, Government of India, project (file no-02011/12/2021 NBHM(R.P)/R &D II/7995).
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Arora, S., Singh, S., Mohan, M.T. et al. Approximate Controllability of Non-autonomous Second Order Impulsive Functional Evolution Equations in Banach Spaces. Qual. Theory Dyn. Syst. 22, 31 (2023). https://doi.org/10.1007/s12346-022-00718-3
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DOI: https://doi.org/10.1007/s12346-022-00718-3
Keywords
- Abstract functional evolution equations
- Non-instantaneous impulses
- Approximate controllability
- Evolution operator
- Cosine family