APPROXIMATE CONTROLLABILITY OF SECOND-ORDER SEMILINEAR EVOLUTION SYSTEMS WITH STATE-DEPENDENT INFINITE DELAY

Xiaofeng Su; Xianlong Fu

doi:10.11948/20190217

2020 Volume 10 Issue 3

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Xiaofeng Su, Xianlong Fu. APPROXIMATE CONTROLLABILITY OF SECOND-ORDER SEMILINEAR EVOLUTION SYSTEMS WITH STATE-DEPENDENT INFINITE DELAY[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 1118-1148. doi: 10.11948/20190217

Citation:

Xiaofeng Su, Xianlong Fu. APPROXIMATE CONTROLLABILITY OF SECOND-ORDER SEMILINEAR EVOLUTION SYSTEMS WITH STATE-DEPENDENT INFINITE DELAY[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 1118-1148. doi: 10.11948/20190217

APPROXIMATE CONTROLLABILITY OF SECOND-ORDER SEMILINEAR EVOLUTION SYSTEMS WITH STATE-DEPENDENT INFINITE DELAY

Xiaofeng Su¹,
Xianlong Fu^1, ,

1.
School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

Corresponding author: Email address: xlfu@math.ecnu.edu.cn (X. Fu)
Fund Project: The authors were supported by National Natural Science Foundation of China (Nos. 11671142 and 11771075) and Science and Technology Commission of Shanghai Municipality (STCSM) (No. 18dz2271000)

Abstract

Abstract

In this article, we study the problem of approximate controllability for a class of semilinear second-order control systems with state-dependent delay. We establish some sufficient conditions for approximate controllability for this kind of systems by constructing fundamental solutions and using the resolvent condition and techniques on cosine family of linear operators. Particularly, theory of fractional power operators for cosine families is also applied to discuss the problem so that the obtained results can be applied to the systems involving derivatives of spatial variables. To illustrate the applications of the obtained results, two examples are presented in the end.
- Second-order evolution equation /
- approximate controllability /
- cosine operator /
- fundamental solution /
- fractional power operator
MSC: 34K30, 34K35, 35R10, 93B05, 93C10

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References

[1]	W. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 1992, 52,855-869. doi: 10.1137/0152048 CrossRef Google Scholar
[2]	F. Andradea, C. Cuevasa and H. R. Henr$\acute{i}$quez, Periodic solutions of abstract functional differential equations with state-dependent delay, Math. Meth. Appl. Sci., 2016, 39, 3897-3909. Google Scholar
[3]	A. Baliki, M. Benchohra and J. R. Graef, Global existence and stability for second order functional evolution equations with infinite delay, Electr. J. Qual. Theory Diff. Equ., 2016, 1-122, 1-10. Google Scholar
[4]	A. E. Bashirov and N. I. Mahmudov, On concepts of controllability for linear deterministic and stochastic systems, SIAM J. Control Optim., 1999, 37(6), 1808-1821. Google Scholar
[5]	M. Benchohra and S. K. Ntouyas, Controllability of second-order differential inclusion in Banach spaces with nonlocal conditions, J. Optim. Theory Appl., 2000,107(3), 559-571. Google Scholar
[6]	M. Buger and M. R. W. Martin, The escaping disaster: a problem related to state-dependent delay, J. Appl. Math. Phys., 2004, 55 (4), 547-574. Google Scholar
[7]	Y. Chang and W. Li, Controllability of second-order differential and integrodifferential inclusions in Banach spaces, J. Optim. Theory Appl., 2006,126, 77-87. Google Scholar
[8]	F. Chen, D. Sun and J. Shi, Periodicity in a food-limited population model with toxicants and state-dependent delays, J. Math. Anal. Appl., 2003,288 (1), 136-146. Google Scholar
[9]	I. Chueshovand A. Rezounenko, Dynamics of second order in time evolution equations with state-dependent delay, Nonl. Anal. (TMA), 2015,123,126-149. Google Scholar
[10]	R. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995. Google Scholar
[11]	S. Das, D. N. Pandey and N. Sukavanam, Approximate Controllability of a Second Order Neutral Differential Equation with State Dependent Delay, Diff. Equ. Dyn. Syst., 2016, 24 (2), 201-214. doi: 10.1007/s12591-014-0218-6 CrossRef Google Scholar
[12]	H. O. Fattorini, Ordinary differential equations in linear topological space I, J. Diff. Equ., 1969, 5 (1), 72-105. doi: 10.1016/0022-0396(69)90105-3 CrossRef Google Scholar
[13]	H. O. Fattorini, Second-Order Linear Differential Equations in Banach Space, North Holland Mathematics Studies 108, North Holland, 1985. Google Scholar
[14]	X. Fu and J. Zhang, Approximate Controllability of Neutral Functional Differential Systems with State-Dependent Delay, Chin. Ann. Math., 2016, 37B, 291-308. Google Scholar
[15]	J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funk. Ekvac., 1978, 21 (1), 11-41. Google Scholar
[16]	F. Hartung, T. Krisztin, H. O. Walther and J. Wu, Functional differential equations with state-dependent delays: theory and applications, Handbook of differential equations: ODE (Ch 5), 2006, 3,435-545. Google Scholar
[17]	E. M. Hernández, Existence of solutions for a second-order abstract functional differential equation with state-dependent delay, Electr. J. Diff. Equ., 2007, 21, 1-10. Google Scholar
[18]	E. Hernández, K. Azevedo and V. Rolnik, Well-posedness of abstract differential equations with state-dependent delay, Math. Nachr., 2018,291 (13), 2045-2056. Google Scholar
[19]	E. Hernández and M. A. Mckibben, On state-dependent delay partial neutral functional differential equations, Appl. Math. Comp., 2007,186 (1), 294-301. Google Scholar
[20]	E. Hernández, D. O'Regan and K. Azevedo, On second order differential equations with state-dependent delay, Appl. Anal., 2018, 97(15), 2610-2617. doi: 10.1080/00036811.2017.1382685 CrossRef Google Scholar
[21]	E. Hernández, M. Pierri and J. Wu, $C.{1+\alpha}$-strict solutions and wellposedness of abstract differential equations with state dependent delay, J. Diff. Equ., 2016,261 (12), 6856-6882. doi: 10.1016/j.jde.2016.09.008 CrossRef $C.{1+\alpha}$-strict solutions and wellposedness of abstract differential equations with state dependent delay" target="_blank">Google Scholar
[22]	H. R. Henr$\acute{i}$quez and E. Hernández M, Approximate controllability of second-order distributed implicit functional systems, Nonl. Anal., 2009, 70 (2), 1023-1039. Google Scholar
[23]	Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer, Berlin, 1991. Google Scholar
[24]	Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Diff. Equ., 2010,248 (12), 2801-2840. doi: 10.1016/j.jde.2010.03.020 CrossRef Google Scholar
[25]	J. Jeong, Y. Kwun and J. Park, Approximate controllability for semilinear retarded functional differential equations, J. Dyn. Control Syst., 1999, 5 (3), 329-346. Google Scholar
[26]	H. Khatibzadeh and G. Morosanu, Asymptotically periodic solutions to some second-order evolution and difference equations, Appl. Anal., 2015, 94 (5), 1042-1050. Google Scholar
[27]	J. Kisynski, On cosine operator functions and one parameter group of operetors, Studia Math., 1972, 49, 93-105. Google Scholar
[28]	N Kosovalic, Y. Chen and J. Wu, Algebraic-delay differential systems:$C_0$-extendable submanifolds and linearization, Trans. Amer. Math. Soc., 2017,369(5), 3387-3419. doi: 10.1090/tran/6760 CrossRef $C_0$-extendable submanifolds and linearization" target="_blank">Google Scholar
[29]	N. Kosovalic, F. M. G. Magpantay, Y. Chen and J. Wu, Abstract algebraic-delay differential systems and age structured population dynamics, J. Diff. Equ., 2015,255(3), 593-609. Google Scholar
[30]	T. Krisztin and A. Rezounenkob, Parabolic partial differential equations with discrete state-dependent delay: Classical solutions and solution manifold, J. Diff. Equ., 2016,260(5), 4454-4472. doi: 10.1016/j.jde.2015.11.018 CrossRef Google Scholar
[31]	M. Li and C. Kou, Existence results for second-order impulsive neutral functional differential equations with nonlocal conditions, J. Discr. Dyn. Nat. Soc., 2009, 2, 1-11. Google Scholar
[32]	F. Liang and Z. Guo, Asymptotic behavior for second order stochastic evolution equations with memory, J. Math. Anal. Appl., 2014,419 (2), 1333-1350. Google Scholar
[33]	J. Mahaffy, J. Belair and M. Mackey, Hematopoietic model with moving boundary condition and state-dependent delay: Applications in Erythropoiesis, J. Theor. Biol., 1998,190(2), 135-146. Google Scholar
[34]	F. Z. Mokkedem and X. Fu, Approximate controllability of semi-linear neutral integro-differential systems with finite delay, Appl. Math. Comp., 2014,242,202-215. doi: 10.1016/j.amc.2014.05.055 CrossRef Google Scholar
[35]	F. Z. Mokkedem and X. Fu, Approximate controllability of a semi-linear neutral evolution system with infinite delay, Int. J. Rob. Nonl. Control, 2017, 27, 1122-1146. doi: 10.1002/rnc.3619 CrossRef Google Scholar
[36]	K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim., 1987, 25,715-722. doi: 10.1137/0325040 CrossRef Google Scholar
[37]	S. Nicaise and C. Pignotti, Stability results for second-order evolution equations with switching time-delay, J. Dyn. Diff. Equ., 2014, 26,781-803. doi: 10.1007/s10884-014-9382-1 CrossRef Google Scholar
[38]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. Google Scholar
[39]	R. Sakthivel and E. R. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Int. J. Control, 2010, 83 (2), 387-393. Google Scholar
[40]	R. Sakthivel, E. R. Anandhi and N. I. Mahmudov, Approximate controllability of second-order systems with state-dependent delay, Numer. Funct. Anal. Optim., 2008, 29 (11-12), 1347-1362. 1158-1164. Google Scholar
[41]	R. Sakthivel and Y. Ren, Approximate controllability of fractional differential equations with state-dependent delay, Result. Math., 2013, 63(3-4), 949-963. doi: 10.1007/s00025-012-0245-y CrossRef Google Scholar
[42]	R. Sakthivel, Y. Ren and N. I. Mahmudov, Approximate controllability of second-order stochastic differential equations with impulsive effects, Modern Phys. Lett. B, 2010, 24(14), 1559-1572. doi: 10.1142/S0217984910023359 CrossRef Google Scholar
[43]	D. Sanjukta, D. N. Pandey and N. Sukavanam, Existence of solution and approximate controllability for neutral differential equation with state dependent delay, Int. J. Part. Diff. Equ., 2014, 3, 1-12. Google Scholar
[44]	C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar., 1978, 32(1), 75-96. Google Scholar
[45]	C. C. Travis and G. F. Webb, Second order differential equations in Banach space, Nonlinear Equations in Abstract Spaces, 1987,331-361. Google Scholar
[46]	L. Wang, Approximate controllability for integrodifferential equations with multiple delays, J. Optim. Theory Appl., 2009,143 (1), 185-206. Google Scholar
[47]	Z. Yan, Approximate controllability of fractional neutral integrodifferential inclusions with state-dependent delay in Hilbert spaces, IMA J. Math. Control Inf., 2013, 30,443-462. doi: 10.1093/imamci/dns033 CrossRef Google Scholar