Some results on the approximate controllability of  impulsive stochastic integro-differential equations with nonlocal conditions and  state-dependent delay

Fall, M.; Mane, A.; Dehigbe, B.; Diop, M. A.

doi:10.22436/jnsa.015.04.04

Geodesic

Parcourir par

Some results on the approximate controllability of impulsive stochastic integro-differential equations with nonlocal conditions and state-dependent delay

Fall, M.¹ ; Mane, A.¹ ; Dehigbe, B.² ; Diop, M. A. ³

¹ UFR SAT Departement de Mathematiques, Universite Gaston Berger de Saint-Louis, B.P 234, Saint-Louis, Senegal
² Institut de Math'ematiques et de Sciences Physiques , URMPM B.P. 613, Porto-Novo, Benin
³ UFR SAT Departement de Mathematiques, Universite Gaston Berger de Saint-Louis, B.P 234, Saint-Louis, Senegal;UMMISCO UMI 209 IRD/UPMC, Bondy, France

Journal of nonlinear sciences and its applications, Tome 15 (2022) no. 4, p. 284-300.

Voir la notice de l'article provenant de la source International Scientific Research Publications

Résumé

This paper presents approximate controllability results for impulsive stochastic integro-differential systems with state-dependent delay in a Hilbert space. The use of the resolvent operator in the sense of Grimmer, as well as stochastic analysis techniques, yields a new set of results. Finally, an example is given to show how the theory that has been worked out can be put into practice.

DOI : 10.22436/jnsa.015.04.04

Classification : 93B05, 34A08, 34A37, 34F05
Keywords: Impulsive stochastic integrodifferential equations, state-dependent delay, mild solution, approximate controllability, semigroup theory, resolvent operator, fixed point theorem, nonlocal conditions

Affiliations des auteurs :

Fall, M. ¹ ; Mane, A. ¹ ; Dehigbe, B. ² ; Diop, M. A. ³

@article{JNSA_2022_15_4_a3,
     author = {Fall, M. and Mane, A. and Dehigbe, B. and Diop, M. A. },
     title = {Some results on the approximate controllability of  impulsive stochastic integro-differential equations with nonlocal conditions and  state-dependent delay},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {284-300},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {2022},
     doi = {10.22436/jnsa.015.04.04},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.015.04.04/}
}

TY  - JOUR
AU  - Fall, M.
AU  - Mane, A.
AU  - Dehigbe, B.
AU  - Diop, M. A. 
TI  - Some results on the approximate controllability of  impulsive stochastic integro-differential equations with nonlocal conditions and  state-dependent delay
JO  - Journal of nonlinear sciences and its applications
PY  - 2022
SP  - 284
EP  - 300
VL  - 15
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.015.04.04/
DO  - 10.22436/jnsa.015.04.04
LA  - en
ID  - JNSA_2022_15_4_a3
ER  -

%0 Journal Article
%A Fall, M.
%A Mane, A.
%A Dehigbe, B.
%A Diop, M. A. 
%T Some results on the approximate controllability of  impulsive stochastic integro-differential equations with nonlocal conditions and  state-dependent delay
%J Journal of nonlinear sciences and its applications
%D 2022
%P 284-300
%V 15
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.015.04.04/
%R 10.22436/jnsa.015.04.04
%G en
%F JNSA_2022_15_4_a3

Fall, M.; Mane, A.; Dehigbe, B.; Diop, M. A. . Some results on the approximate controllability of  impulsive stochastic integro-differential equations with nonlocal conditions and  state-dependent delay. Journal of nonlinear sciences and its applications, Tome 15 (2022) no. 4, p. 284-300. doi : 10.22436/jnsa.015.04.04. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.015.04.04/

Bibliographie
Cité par

[1] Ahmed, H. M. Approximate controllability of impulsive neutral stochastic differential equations with fractional Brownian motion in a Hilbert space, Adv. Difference Equ., Volume 2014 (2014), pp. 1-11

[2] Ahmed, H. M.; El-Borai, M. M.; Bab, A. S. Okb El; Ramadan, M. Elsaid Approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion, Bound. Value Probl., Volume 2020 (2020), pp. 1-25

[3] ouali, N. Ait; Kandouci, A. Existence of mild solution result for fractional neutral stochastic integro-differential equations with nonlocal conditions and infinite delay, Malaya J. Mat., Volume 3 (2015), pp. 1-13

[4] Arino, O.; Habid, M. L.; Parra, R. Bravo de la A mathematical model of growth of population of fish in the larval stage: density dependence effects, Math. Biosci., Volume 150 (1998), pp. 1-20

[5] Balachandran, K.; Kumar, R. R. Existence of solutions of integrodifferential evolution equations with time varying delays, Appl. Math. E-Notes, Volume 7 (2007), pp. 1-8

[6] Bashirov, A. E.; Mahmudov, N. I. On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim., Volume 37 (1999), pp. 1808-1821 | DOI

[7] Belmekki, M.; Benchohra, M.; Ezzinbi, K. Existence results for some partial functional differential equations with statedependent delay, Appl. Math. Lett., Volume 24 (2011), pp. 1810-1816

[8] Benchohra, M.; Henderson, J.; Ntouyas, S. K. Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, Hindawi Publishing Corporation, New York, 2006

[9] Boudaoui, A.; Slama, A. Approximate controllability of nonlinear fractional impulisve stochastic differential equations with nonlocal conditions and infinite delay, Nonlinear Dyn. Syst. Theory, Volume 16 (2016), pp. 35-48

[10] Byszewski, L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., Volume 162 (1991), pp. 494-505

[11] 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., Volume 40 (1991), pp. 11-19

[12] Chen, M. Approximate controllability of stochastic equations in a Hilbert space with fractional Brownian motions, Stoch. Dyn., Volume 15 (2015), pp. 1-16

[13] Curtain, R. F.; Zwart, H. An Introduction to Infinite Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995

[14] Desch, W.; Grimmer, R.; Schappacher, W. Some considerations for linear integrodifferential equations, J. Math. Anal. Appl., Volume 104 (1984), pp. 219-234

[15] Dieye, M.; Diop, M. A.; Ezzinbi, K. On exponential stability of mild solutions for some stochastics partial integrodifferential equations, Statist. Probab. Lett., Volume 123 (2017), pp. 61-76

[16] Dieye, M.; Diop, M. A.; Ezzinbi, K. Optimal feedback control law for some stochastic integrodifferential equations on Hilbert spaces, Eur. J. Control, Volume 37 (2017), pp. 54-62

[17] Dineshkumar, C.; Udhayakumar, R. Results on approximate controllability of fractional stochastic Sobolev-type Volterra-Fredholm integro-differential equation of order $1<r<2$, Math. Methods Appl. Sci., Volume 45 (2022), pp. 6691-6704

[18] Dineshkumar, C.; Udhayakumar, R.; Vijayakumar, V.; Nisar, K. S.; Shukla, A. A note concerning to approximate controllability of Atangana-Baleanu fractional neutral stochastic systems with infinite delay, Chaos Solitons Fractals, Volume 157 (2022), pp. 1-17

[19] Diop, M. A.; Guindo, P. A. D.; Fall, M.; Diakhaby, A. Optimal controls for stochastic functional integrodifferential equations, Electron. J. Math. Anal. Appl., Volume 9 (2021), pp. 241-261

[20] Ezzinbi, K.; Degla, G.; Ndambomve, P. Controllability for some Partial Functional Integrodifferential Equations with Nonlocal Conditions in Banach Spaces, Discuss. Math. Differ. Incl. Control Optim., Volume 35 (2015), pp. 25-46

[21] Ezzinbi, K.; Toure, H.; Zabsone, I. Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces, Nonlinear Anal., Volume 70 (2009), pp. 2761-2771

[22] Fu, X. L.; Liu, X. B. Existence of periodic solutions for abstract neutral non-autonomous equations with infinite delay, J. Math. Anal. Appl., Volume 325 (2007), pp. 249-267 | DOI

[23] Grimmer, R. C. Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., Volume 273 (1982), pp. 333-349 | DOI

[24] Haddad, W. M.; Chellaboina, V.; Nersesov, S. G. Stability, Dissipativity, and Control, Princeton University Press, Princeton, 2006

[25] Hale, J. K.; Lunel, S. M. Verduyn Introduction to functional differential equations, Springer-Verlag, New York, 1993 | DOI

[26] Hernandez, E.; Prokopczyk, A.; Ladeira, L. A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. Real World Appl., Volume 7 (2006), pp. 510-519

[27] Hernandez, E.; Sakthivel, R.; Aki, S. Tanaka Existence results for impulsive evolution differential equations with statedependent delay, Electron. J. Differential Equations, Volume 2008 (2008), pp. 1-11 | EuDML

[28] Hino, Y.; Murakami, S. Stability properties of linear Volterra integrodifferential equations in a Banach space, Funkcial. Ekvac., Volume 48 (2005), pp. 367-392

[29] Hino, Y.; Murakami, S.; Naito, T. Functional differential equations with infinite delay, Springer-Verlag, Berlin, 1991

[30] Huang, H.; Fu, X. Approximate controllability of semi-linear stochastic integrodifferential equations with infinite delay, IMA J. Math. Control Infor., Volume 37 (2020), pp. 1133-1167

[31] Kalamani, P.; Baleanu, D.; Selvarasu, S.; Arjunan, M. M. On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions, Adv. Difference Equ., Volume 2016 (2016), pp. 1-36

[32] Klamka, J. Constrained controllability of semilinear systems with delays, Nonlinear Dynam., Volume 56 (2009), pp. 169-177

[33] Krasnosel’skii, M. A. Positive Solutions of Operator Equations, Noordhoff Ltd., Groningen, 1964

[34] Lakshmikanthan, V.; Bainov, D. D.; Simeonov, P. S. Theory of Impulsive Differential Equations, World Scientific Publishers Co., Teaneck, 1989

[35] Lunardi, A. On the linear heat equation with fading memory, SIAM J. Math. Anal., Volume 21 (1990), pp. 1213-1224

[36] Mahaffy, J. M.; Belair, J.; Mackey, M. C. Hematopoietic model with moving boundary condition and state-dependent delay: Applications in erythropoiesis, J. Theoret. Biol., Volume 190 (1998), pp. 135-146

[37] Mahmudov, N. I. Controllability of linear stochastic systems in Hilbert spaces, J. Math. Anal. Appl., Volume 259 (2001), pp. 64-82

[38] Mahmudov, N. I. Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., Volume 42 (2003), pp. 1604-1622

[39] Mahmudov, N. I.; Zorlu, S. On the approximate controllability of fractional evolution equations with compact analytic semigroup, J. Comput. Appl. Math., Volume 259 (2014), pp. 194-204

[40] Mokkedem, F. Z.; Fu, X. L. Approximate controllability of semilinear neutral integrodifferential systems with finite delay, Appl. Math. Comput., Volume 242 (2014), pp. 202-215

[41] Muthukumar, P.; Rajivganthi, C. Approximate controllability of impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces, J. Control Theory Appl., Volume 13 (2013), pp. 351-358

[42] Nunziato, J. W. On heat conduction in materials with memory, Quart. Appl. Math., Volume 29 (1971), pp. 187-204

[43] Samoilenko, A. M.; Perestyuk, N. A. Impulsive Differential Equations, World Scientific Publishing Co., River Edge, 1995

[44] Sivasankar, S.; Udhayakumar, R. A note on approximate controllability of second-order neutral stochastic delay integrodifferential evolution inclusions with impulses, Math. Methods Appl. Sci., Volume 45 (2022), pp. 6650-6676

[45] Smart, D. R. Fixed Point Theorems, Cambridge University Press, London-New York, 1974

[46] Triggiani, R. A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., Volume 15 (1977), pp. 407-411

[47] Vijayakumar, V.; Udhayakumar, R.; Panda, S. K.; Nisar, K. S. Results on approximate controllability of Sobolev type fractional stochastic evolution hemivariational inequalities, Numer. Methods Partial Differ. Equ., Volume 2020 (2020), pp. 1-20

[48] Vijayakumar, V.; Udhayakumar, R.; Zhou, Y.; Sakthivel, N. Approximate controllability results for Sobolev-type delay differential system of fractional order without uniqueness, Numer. Methods Partial Differ. Equ., Volume 2020 (2020), pp. 1-20

[49] Vrabie, I. I. $C_0$-Semigroups and Applications, North-Holland Publishing Co., Amsterdam, 2003

[50] Wang, J. R.; Wei, W. A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces, Results Math., Volume 58 (2010), pp. 379-397

[51] Yan, Z. M. Nonlocal problems for delay integrodifferential equations in Banach spaces, Differ. Equ. Appl., Volume 2 (2010), pp. 15-25

[52] Yan, Z. M.; Yan, X. X. Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay, Collect. Math., Volume 64 (2013), pp. 235-250

[53] Zuazua, E. Controllability and observability of partial differential equations, some results and open problems in Handbook of Differential Equations, Handbook of Differential Equations: Evolutionary Equations, Volume 3 (2007), pp. 527-621

Cité par Sources :