Discontinuity, Nonlinearity, and Complexity
Controllability of Nonlinear Stochastic Fractional Systems with Lévy Noise
Discontinuity, Nonlinearity, and Complexity 6(3) (2017) 409420  DOI:10.5890/DNC.2017.09.009
R. Mabel Lizzy; K. Balachandran; M. Suvinthra
Department of Mathematics, Bharathiar University, Coimbatore 641046, India.
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Abstract
In this paper we study the controllability of linear and nonlinear stochastic fractional systems driven by Lévy noise. Here we use the LévyItô decomposition of an arbitrary Lévy process into Brownian and Poisson parts. The necessary and sufficient conditions for controllability of the linear system is obtained. Also, the nonlinear system is shown controllable under the assumption that the corresponding linear system is controllable and using the Banach contraction principle.
Acknowledgments
The work of the first author was supported by the University Grants Commission under grant number: MANF201517TAM50645 from the government of India.
References

[1]  Oksendal, B. (2014), Stochastic control of ItôLévy processes with applications to finance, Communications on Stochastic Analysis, 8, 115. 

[2]  Shlesinger, M.F., Zavslavsky, G.M., and Feisch, U. (1995), Lévy Flights and Related Topics in Physics, Springer Verlag: Berlin. 

[3]  Kilbas, A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier: New York. 

[4]  Balachandran, K. and Dauer, J.P. (2002), Controllability of nonlinear systems in Banach spaces: A survey, Journal of Optimization Theory and Applications, 115, 728. 

[5]  Balachandran, K. and Leelamani, A. (2006), Null controllability of neutral evolution integrodifferential systems with infinite delay, Mathematical Problems in Engineering, 2006, 45468:118. 

[6]  Bensoussan, A., Da Prato, G., Delfour, M.C., and Mitter, S.K. (1993), Representation and Control of Infinite Dimensional Systems, Birkh¨ auser: Boston. 

[7]  Balachandran, K. and Karthikeyan, S. (2008), Controllability of nonlinear Itô stochastic integrodifferential systems, Journal of the Franklin Institute, 345, 382391. 

[8]  Mahmudov, N.I. (2001), Controllability of linear stochastic systems in Hilbert spaces, Journal of Mathematical Analysis and Applications, 259, 6482. 

[9]  Applebaum, D. (2004), Lévy process and Stochastic Calculus, Cambridge University Press: Cambridge. 

[10]  Mandrekar, V. and Rüdiger, B. (2015), Stochastic Integration in Banach Spaces, Theory and Applications, Springer: Switzerland. 

[11]  Mohan, M.T. and Sritharan, S. (2016), Stochastic Euler equations of fluid dynamics with Lévy noise, Asymptotic Analysis, 99, 67103. 

[12]  Oksendal, B. and Sulem, A. (2007), Applied Stochastic Control of Jump Diffusions, Springer: Berlin. 

[13]  Xiao, H. (2013), Optimality conditions for optimal control of jumpdiffusion SDEs with correlated observations noises, Mathematical Problems in Engineering, 2013, 613159:17. 

[14]  Balachandran, K. and Kokila, J. (2012), On the controllability of fractional dynamical systems, International Journal of Applied Mathematics and Computer Science, 22, 523531. 

[15]  Balachandran, K., Govindaraj, V., RodríguezGerma, L., and Trujillo, J.J. (2013), Controllability results for nonlinear fractionalorder dynamical systems, Journal of Optimization Theory and Applications, 156, 3344. 

[16]  Balachandran, K. and Park, J.Y. (2009), Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear Analysis: Hybrid Systems, 3, 363367. 

[17]  Balachandran, K., Matar, M., and Trujillo, J.J. (2016), Note on controllability of linear fractional dynamical systems, Journal of Decision and Control, 3, 267279. 

[18]  Hanson, F.B. (2007), Applied Stochastic Processes and Control for JumpDiffusions: Modeling, Analysis and Computation, Society for Industrial and Applied Mathematics. 

[19]  Huynh, H.T., Lai, V.S., and Soumaré, I. (2009), Stochastic Simulation and Applications in Finance with MATLAB Programs, Wiley Publishing: England. 

[20]  Albeverio, S. and Rüdiger, B. (2005), Stochastic integrals and LévyItô decomposition on separable Banach spaces, Stochastic Analysis and Applications, 23, 217253. 

[21]  Da Prato, G. and Zabczyk, J. (1992), Stochastic Equations in Infinite Dimensions, Cambridge University Press: Cambridge. 

[22]  Jeanblanc, M., Yor,M., and Chesney,M. (2009),Mathematical Methods for Financial Markets, SpringerVerlag: New York. 

[23]  Mabel Lizzy, R. (2016), Controllability of nonlinear stochastic fractional integrodifferential systems in Hilbert spaces, Lecture Notes in Electrical Engineering, 407, 345356. 

[24]  Mabel Lizzy, R., Balachandran, K., and Suvinthra, M. (2017), Controllability of nonlinear stochastic fractional systems with distributed delay in control, Journal of control and decision, 4, 153167. 