Approximate controllability of second-order non-autonomous stochastic impulsive differential systems

References

• Benchohra, M., Henderson, J., Ntouyas, S. (2006). Impulsive Differential Equations and Inclusions, Vol. 2. New York, NY: Hindawi Publishing Corporation.
   Google Scholar
• Yang, T. (2001). Impulsive Systems and Control: Theory and Applications. Berlin, Germany: Springer-Verlag.
   Google Scholar
• Arthi, G., Park, J. H., Jung, H. Y. (2014). Existence and controllability results for second-order impulsive stochastic evolution systems with state-dependent delay. Appl. Math. Comput. 248:328–341. DOI: 10.1016/j.amc.2014.09.084.
   Web of Science ®Google Scholar
• Chaudhary, R., Pandey, D. N. (2019). Existence results for a class of impulsive neutral fractional stochastic integro-differential systems with state dependent delay. Stochas. Anal. Appl. 37(5):865–892. DOI: 10.1080/07362994.2019.1621181.
   Web of Science ®Google Scholar
• Singh, V., Pandey, D. N. (2017). Weighted pseudo almost periodic solutions for fractional order stochastic impulsive differential equations. Cubo. 19(1):89–110. DOI: 10.4067/S0719-06462017000100006.
   Google Scholar
• Vijayakumar, V., Murugesu, R., Poongodi, R., Dhanalakshmi, S. (2017). Controllability of second-order impulsive nonlocal Cauchy problem via measure of noncompactness. Mediterr. J. Math. 14(1):1–14. DOI: 10.1007/s00009-016-0813-6.
   Web of Science ®Google Scholar
• Kalman, R. E. (1963). Controllability of linear systems. Contrib. Differ. Equat. 1:190–213.
   Google Scholar
• Singh, V. (2019). Controllability of hilfer fractional differential systems with non-dense domain. Numer. Funct. Anal. Optim. 40(13):1572–1592. DOI: 10.1080/01630563.2019.1615947.
   Web of Science ®Google Scholar
• Singh, V., Pandey, D. N. (2019). Controllability of second-order Sobolev-type impulsive delay differential systems. Math. Meth. Appl. Sci. 42(5):1377–1388. DOI: 10.1002/mma.5427.
   Web of Science ®Google Scholar
• Debbouche, A., Nieto, J. J. (2014). Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls. Appl. Math. Comput. 245:74–85. DOI: 10.1016/j.amc.2014.07.073.
   Web of Science ®Google Scholar
• Sakthivel, R., Ren, Y., Mahmudov, N. (2010). Approximate controllability of second-order stochastic differential equations with impulsive effects. Mod. Phys. Lett. B. 24(14):1559–1572. DOI: 10.1142/S0217984910023359.
   Web of Science ®Google Scholar
• Triggiani, R. (1977). A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control Optim. 15(3):407–411. DOI: 10.1137/0315028.
   Web of Science ®Google Scholar
• Naito, K. (1987). Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim. 25(3):715–722. DOI: 10.1137/0325040.
   Web of Science ®Google Scholar
• Bochenek, J. (1997). Existence of the fundamental solution of a second order evolution equation. Ann. Polon. Math. 66:15–35. DOI: 10.4064/ap-66-1-15-35.
   Google Scholar
• Kozak, M. (1995). A fundamental solution of a second-order differential equation in a Banach space. Univ. Iagel. Acta Math. 32:275–289.
   Google Scholar
• Mahmudov, N., Vijayakumar, V., Murugesu, R. (2016). Approximate controllability of second-order evolution differential inclusions in Hilbert spaces. Mediterr. J. Math. 13(5):3433–3454. DOI: 10.1007/s00009-016-0695-7.
   Web of Science ®Google Scholar
• Chaudhary, R., Malik, M., Pandey, D. N. (2020). Approximation of solution to second order impulsive differential equation with finite delay. Dyn. Contin. Discrete Impuls. Syst., Ser. B Appl. Algorithms. 27:223–243.
   Google Scholar
• Henríquez, H. R. (2011). Existence of solutions of non-autonomous second order functional differential equations with infinite delay. Nonlinear Anal. Theory Methods Appl. 74(10):3333–3352. DOI: 10.1016/j.na.2011.02.010.
   Web of Science ®Google Scholar
• Henríquez, H. R., Poblete, V., Pozo, J. C. (2014). Mild solutions of non-autonomous second order problems with nonlocal initial conditions. J. Math. Anal. Appl. 412(2):1064–1083. DOI: 10.1016/j.jmaa.2013.10.086.
   Web of Science ®Google Scholar
• Henríquez, H. R., Pozo, J. C. (2016). Existence of solutions of abstract non-autonomous second order integro-differential equations. Bound. Value Probl. 2016(1):1–24. DOI: 10.1186/s13661-016-0675-7.
   Google Scholar
• Ren, Y., Hou, T., Sakthivel, R., Cheng, X. (2014). A note on the second-order non-autonomous neutral stochastic evolution equations with infinite delay under Carathéodory conditions. Appl. Math. Comput. 232:658–665. DOI: 10.1016/j.amc.2014.01.091.
   Web of Science ®Google Scholar
• Nirmalkumar, R., Murugesu, R. (2018). Approximate controllability of second-order stochastic non-autonomous integrodifferential inclusions by resolvent operators. Communications. 68(1):929–943. DOI: 10.31801/cfsuasmas.489730.
   Google Scholar
• Vijayakumar, V., Murugesu, R. (2019). Controllability for a class of second-order evolution differential inclusions without compactness. Applic. Anal. 98(7):1367–1385. DOI: 10.1080/00036811.2017.1422727.
   Web of Science ®Google Scholar
• Vijayakumar, V., Udhayakumar, R., Dineshkumar, C. (2020). Approximate controllability of second order nonlocal neutral differential evolution inclusions. IMA J Math Control Inform. DOI: 10.1093/imamci/dnaa001.
   Google Scholar
• Nagaraj, M., Kavitha, V., Baleanu, D., Arjunan, M. M. (2019). Approximate controllability of second-order nonlocal impulsive partial functional integro-differential evolution systems in Banach spaces. Filomat. 33(18):5887–5912. DOI: 10.2298/FIL1918887N.
   Web of Science ®Google Scholar
• Faraci, F., Iannizzotto, A. (2006). A multiplicity theorem for a perturbed second-order non-autonomous system. Proc. Edinburgh Math. Soc. 49(2):267–275. DOI: 10.1017/S001309150400149X.
   Google Scholar
• Travis, C. C., Webb, G. F. (1977). Compactness, regularity, and uniform continuity properties of strongly continuous cosine families. Houston J. Math. 3(4):555–567.
   Google Scholar
• Fattorini, H. O. (1985). Second Order Linear Differential Equations in Banach Spaces, Vol. 99. Amsterdam, the Netherlands: North-Holland Publishing Co.
   Google Scholar
• Serizawa, H., Watanabe, M. (1986). Perturbation for cosine families in Banach spaces. Houston J. Math. 12(1):117–124.
   Web of Science ®Google Scholar
• Mahmudov, N. I. (2003). Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim. 42(5):1604–1622. DOI: 10.1137/S0363012901391688.
   Web of Science ®Google Scholar
• Liu, K. (2006). Stability of infinite dimensional stochastic differential equations with applications. Pitman Monographs and Surveys in Pure and Applied Mathematics, Volume 135. Boca Raton, FL: Chapman & Hall/CRC Press.
   Google Scholar
• Mahmudov, N. I. (2001). Controllability of linear stochastic systems in Hilbert spaces. J. Math. Anal. Appl. 259(1):64–82. DOI: 10.1006/jmaa.2000.7386.
   Web of Science ®Google Scholar
• Mahmudov, N., Denker, A. (2000). On controllability of linear stochastic systems. Int. J. Control. 73(2):144–151. DOI: 10.1080/002071700219849.
   Web of Science ®Google Scholar