## Taiwanese Journal of Mathematics

### APPROXIMATE CONTROLLABILITY OF FRACTIONAL ORDER STOCHASTIC VARIATIONAL INEQUALITIES DRIVEN BY POISSON JUMPS

#### Abstract

This paper proposes the sufficient conditions of approximate controllability for a class of fractional order stochastic variational inequalities driven by Poisson jumps. The possibilities of finding the approximate controllability of a given problem of this type introduce the smoothing system corresponding to the fractional order stochastic variational inequalities driven by Poisson jumps. The results are achieved upon the Moreau-Yosida approximation of subdifferential operator. Sufficient conditions for the approximate controllability of smoothing system are discussed under the boundedness condition on control operator. The results are formulated and proved by using the fractional calculus, semigroup theory, stochastic analysis techniques. An example is provided to illustrate the obtained theory.

#### Article information

Source
Taiwanese J. Math., Volume 18, Number 6 (2014), 1721-1738.

Dates
First available in Project Euclid: 21 July 2017

https://projecteuclid.org/euclid.twjm/1500667493

Digital Object Identifier
doi:10.11650/tjm.18.2014.3885

Mathematical Reviews number (MathSciNet)
MR3284028

Zentralblatt MATH identifier
1357.34102

#### Citation

Muthukumar, P.; Rajivganthi, C. APPROXIMATE CONTROLLABILITY OF FRACTIONAL ORDER STOCHASTIC VARIATIONAL INEQUALITIES DRIVEN BY POISSON JUMPS. Taiwanese J. Math. 18 (2014), no. 6, 1721--1738. doi:10.11650/tjm.18.2014.3885. https://projecteuclid.org/euclid.twjm/1500667493

#### References

• H. M. Ahmed, Controllability of fractional stochastic delay equations, Internat. J. Nonlinear Sci., 8 (2009), 498-503.
• V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Space, Nordhoff Leiden, Netherlands, 1976.
• A. Bensoussan and A. Rascanu, Stochastic variational inequalities in infinite dimensional spaces, Numer. Funct. Anal. Optim., 18 (1977), 19-54.
• R. Cont and P. Tankov, Financial Modelling with Jump Processes, Financial Mathematics Series, Chapman and Hall/CRC, Boca Raton, 2004.
• J. Cui and L. Yan, Successive approximation of neutral stochastic evolution equations with infinite delay and Poisson jumps, Appl. Math. Comput., 218 (2012), 6776-6784.
• R. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, New York, Springer-Verlag, 1995.
• H. Huyen Dam, Variable fractional delay filter with sub-expression coefficients, Int. J. Innov. Comput., Inf. Control, 9 (2013), 2995-3003.
• J. Jeong and H. H. Roh, Approximate controllability for semilinear retarded systems, J. Math. Anal. Appl., 321 (2006), 961-975.
• J. Jeong, E. Y. Ju and K. Y. Lee, Controllability for variational inequalities of parabolic type with nonlinear perturbation, J. Inequali. Appl., 2010 (2010), Article ID: 768469.
• J. Jeong, E. Y. Ju and K. Y. Lee, Controllability for nonlinear variational inequalities of parabolic type, Taiwan. J. Math., 15 (2011), 857-873.
• A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
• C. Knoche, Mild Solutions of SPDEs Driven by Poisson Noise in Infinite Dimension and Their Dependence on Initial Conditions, Thesis, Bielefeld University, 2005.
• S. Kumar and N. Sukavanam, Approximate controllability of fractional order neutral control systems with delay, Int. J. Nonlinear Sci., 13 (2012), 454-462.
• V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 2677-2682.
• F. Li, X. Wang and P. Shi, Robust quantized $H_{\infty}$ control for network control systems with markovian jumps and time delays, Int. J. Innov. Comput., Inf. Control, 9 (2013), 4889-4902.
• H. Long, J. Hu and Y. Li, Approximate controllability of stochastic pde with infinite delays driven by Poisson jumps, IEEE International Conference on Information Science and Technology, (2012), 194-199.
• J. Luo and T. Taniguchi, The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps, Stochastic Dyn., 9 (2009), 135-152.
• C. M. Marle, Measures et Probabilites, Hermann, Paris, 1974.
• P. Muthukumar and C. Rajivganthi, Approximate controllability of fractional order neutral stochastic integro-differential system with nonlocal conditions and infinite delay, Taiwan. J. Math., 17 (2013), 1693-1713.
• A. Rascanu, Existence for a class of stochastic parabolic variational inequalities, Stochastics, 5 (1981), 201-239.
• Y. Ren, Q. Zhou and L. Chen, Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay, J. Optim. Theory Appl., 149 (2011), 315-331.
• R. Sakthivel and Y. Ren, Complete controllability of stochastic evolution equations with jumps, Reports Math. Phys., 68 (2011), 163-174.
• R. Sakthivel, S. Suganya and S. M. Anthoni, Approximate controllability of fractional stochastic evolution equations, Comput. Math. Appl., 63 (2012), 660-668.
• T. M. Steger, Stochastic growth under Wiener and Poisson uncertainty, Economics Letters, 86 (2005), 311-316.
• H. Tanabe, Equations of Evolutions, Pitman, London, 1979.
• R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 15 (1977), 407-411.
• K. Walde, Endogenous growth cycles, International Economic Review, 46 (2005), 867-894.
• H. Wang, B. Chen and C. Lin, Adaptive neural tracking control for a class of stochastic nonlinear systems with unknown dead-zone, Int. J. Innov. Comput., Inf. Control, 9 (2013), 3257-3269.
• H. Zhao, On existence and uniqueness of stochastic evolution equation with Poisson jumps, Statistics Probab. Lett., 79 (2009), 2367-2373.
• Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.