Approximate controllability of second-order stochastic non-autonomous integrodifferential inclusions by resolvent operators

Year 2019, Volume: 68 Issue: 1, 929 - 943, 01.02.2019

R. Nirmalkumar R. Murugesu

https://doi.org/10.31801/cfsuasmas.489730

Cited By: 2

Abstract

In this paper, we formulate a set of sufficient conditions for the approximate controllability for a class of second-order stochastic non-autonomous integrodifferential inclusions in Hilbert space. We establish the results with the help of resolvent operators and Bohnenblust-Karlin's fixed point theorem is to prove the main result. An application is given to illustrate the main result.

Keywords

Approximate controllability, stochastic integrodifferential inclusions, resolvent operators, evolution equations, mild solutions

References

• Balasubramaniam, P., Ntouyas, S.K., Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space, Journal of Mathematical Analysis and Applications, 324, (2006),161 - 176.
• Balasubramaniam, P.,Vinayagam, D., Existence of solutions of nonlinear neutral stochastic differential inclusions in a Hilbert space, Stoch. Anal. Appl, 23, (2005), 137-151.
• Balasubramaniam, P. and Tamilalagan, P., Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi's function, Applied Mathematics and Computation, 256, (2015),232-246.
• Batty, C.J.K., Chill, R. and Srivastava, S., Maximal regularity for second order non-autonomous cauchy problems, studia Mathematica, 189, (2008), 205-223.
• Bohenblust, H.F. and Karlin, S., On a Theorem of Ville, in Contributions to the Theory of Games, Princeton University Press, Princeton,NJ, 155-160, 1950.
• Caraballo, T., Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics, 33, (1990), 27-47.
• Chalishajar, D.N., Controllability of second order impulsive neutral functional differential inclusions with infinite delay, Journal of Optimization Theory and Applications, 154 (2), (2012), 672-684.
• Chang, Y.K. and Li, W.T., Controllability of Sobolev type semilinear functional differential and integrodifferential inclusions with an unbounded delay, Georgian Mathematical Journal, 13 (1), (2006), 11-24.
• Chang, Y.K., Controllability of impulsive functional differential systems with infinite delay in Banach space, Chaos Solitons Fractals, 33, (2007), 1601-1609.
• Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge. 1992.
• Deimling, K., Mutivalued Differential Equations, De Gruyter, Berlin. 1992.
• Faraci, F and Iannizzotto, A., A multiplicity theorem for a pertubed second order non-autonmous system. Proceedings of Edinburg Mathematical society, 49, (2006), 267-275.
• Fattorini HO., In Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, vol 108, North-Holland, Amsterdam. 1985.
• Grimmer, R., Resolvent operators for integral equations in Banach space, Transactions of American Mathematical Society, 273, (1982), 333-349.
• Grimmer, R. and Pritchard, A.J., Analytic Resolvent operators for integral equations in Banach space, Journal of Differential equations, 50, (1983), 234-259.
• Henríquez, H.R., Existence of solutions of non-autonomous second order functional differential equations with infinite delay, Nonlinear Analysis: Theory, Method and Applications, 74, (2011), 3333-3352.
• Henríquez, H.R., Existence of solutions of the nonautonomous abstract cauchy problem of second order. Semigroup fourm, (2013), Doi:10.1007/s00233-013-9458-8.
• Henríquez, H.R. and Hernández, E., Existence of solutions of a second order abstract functional cauchy problem with nonlocal conditions, Annales Polonici Mathematici, 88(2), (2006), 141-159.
• Henríquez, H.R. and Hernández, E., Approximate controllability of second order distributed implicit functional system, Nonlinear Analysis, 70, (2009), 1023-1039.
• Henríquez, H.R. and Pozo, J.C., Existence of solutions of abstract non-autonomous second order integrodifferential equtaions, Boundary Value Problems, 168, (2016), 1-24.
• Hernández, E and Henríquez, H.R. and dos Santos, J.P.C., Existence results for abstract partial neutral integro-differential equation with unbounded delay, Electronic Journal of Qualitative Theory of Differential equations, 29, (2009), 1-23.
• Hu, S and Papageorgiou, N.S., Handbook of Multivalued Analysis(Theory), Kluwer Academic Publishers, Dordrecht Boston, London. 1997.
• Mahmudov, N.I., Controllability of linear stochastic systems in Hilbert spaces, Journal of Mathematical Analysis Applications, 259, (2001), 64-82.
• Mahmudov, N.I. and Denker, A., On controllability of linear stochastic systems, Int. J. Control, 73, (2000), 144-151.
• Mahmudov, N.I., Vijayakumar, V. and Murugesu, R., Approximate controllability of second-order evolution differential inclusions in Hilbert spaces, Mediterranean Journal of Mathematics, 13(5), (2016),3433-3454.
• Mao, X., Stochastic Differential equations and Applications, Horwood, Chichester. 1997.
• Oksendal, B., Stochastic Differential Equations, An Introduction with Applications, Springer-Verlag. 2000.
• Ren, Y. and Sun, D.D., Second-order neutral stochastic evolution equations with infinite delay under Caratheodory conditions, J. Optim. Theory Appl, 147, 569-582.
• Revathi, P., Sakthivel, R. and Ren, Y., Stochastic functional differential equations of Sobolev-type with infinite delay, Statistics and Probability Letters, 109, (2016), 68-77.
• Sakthivel, R, Mahmudov, N.I. and Lee, S.G., Controllability of non-linear impulsive stochastic systems, International Journal of Control, 82, (2009), 801-807.
• Shen, L.J. and Sun, J.T., Approximate controllability of stochastic impulsive functional systems with infinite delay, Automatica, 48, (2012), 2705-2709.
• Sobczyk, K. Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic, London. 1991.
• Travis, C.C. and Webb, G.F., Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hung , 32, (1978), 76-96.
• Vijayakumar, V., Ravichandran, C. and Murugesu, R., Approximate controllability for a class of fractional neutral integro-differential inclusions with state-dependent delay, Nonlinear Studies, 20 (4), (2013), 511-530.
• Vijayakumar, V., Selvakumar, A. and Murugesu, R., Controllability for a class of fractional neutral integro-differential equations with unbounded delay, Applied Mathematics and Computation, 232, (2014), 303-312.
• Vijayakumar, V. (2016), Approximate controllability results for analytic resolvent integrodifferential inclusions in Hilbert spaces, International Journal of Control, doi: 10.1080/00207179.2016.1276633.
• Vijayakumar, V., Murugesu, R., Poongodi, R. and Dhanalakshmi, S. Controllability of second order impulsive nonlocal Cauchy problem via measure of noncompactness, Mediterranean Journal of Mathematics, 14 (1), (2017), 29-51.
• Zhou, Y., Vijayakumar, V. and Murugesu, R., Controllability for fractional evolution inclusions without compactness, Evolution Equations and Control Theory, 4 (4), (2015), 507-524.