Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-18T05:20:00.798Z Has data issue: false hasContentIssue false
  • English
  • Français

Controllability of stochastic nonlinear oscillating delay systems driven by the Rosenblatt distribution

Part of: Control systems Functional-differential and differential-difference equations

Published online by Cambridge University Press:  03 March 2020

T. Sathiyaraj ,
JinRong Wang  and
D. O'Regan
T. Sathiyaraj
Affiliation:
Department of Mathematics, Guizhou University, Guiyang, Guizhou550025, P.R. China
JinRong Wang*
Affiliation:
Department of Mathematics, Guizhou University, Guiyang, Guizhou550025, P.R. China and School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong273165, P.R. China (jrwang@gzu.edu.cn)
D. O'Regan
Affiliation:
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Republic of Ireland
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the controllability of second-order nonlinear stochastic delay systems driven by the Rosenblatt distributions in finite dimensional spaces. A set of sufficient conditions are established for controllability of nonlinear stochastic delay systems using fixed point theory, delayed sine and cosine matrices and delayed Grammian matrices. Furthermore, controllability results for second-order stochastic delay systems driven by Rosenblatt distributions via the representation of solution by delayed sine and cosine functions are presented. Finally, our theoretical results are illustrated through numerical simulation.

Keywords

Stochastic delay systemsDelayed perturbations matrixControllability

MSC classification

Primary: 34K50: Stochastic functional-differential equations
Secondary: 93C05: Linear systems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 217 - 239
Check for updates
Copyright
Copyright © The Author(s), 2020. The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Diblík, J., Khusainov, D. Y. and Růžičková, M.. Controllability of linear discrete systems with constant coefficients and pure delay. SIAM J. Control Optim. 47 (2008), 1140–1149.CrossRefGoogle Scholar
2Diblík, J., Fečkan, M. and Pospíšil, M.. Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices. Ukr. Math. J. 65 (2013), 64–76.CrossRefGoogle Scholar
3Diblík, J., Fečkan, M. and Pospíšil, M.. On the new control functions for linear discrete delay systems. SIAM J. Control Optim. 52 (2014), 1745–1760.CrossRefGoogle Scholar
4Diblík, J., Khusainov, D. Y., Lukáčová, J. and Růžičková, M.. Control of oscillating systems with a single delay. Adv. Differ. Equ. 2010 (2010), 108218, 1–15 .CrossRefGoogle Scholar
5Gihman, I. I. and Skorokhod, A. Y.. Stochastic differential equations (New York: Springer-Verlag, 1972).CrossRefGoogle Scholar
6Hughes, P. C. and Skelton, R. E.. Controllability and observability of linear matrix second order systems. ASME J. Appl. Mech. 47 (1980), 415–420.CrossRefGoogle Scholar
7Itô, K.. Stochastic differential equations. Mem. Am. Math. Soc. 4 (1951), 1–51.Google Scholar
8Klamka, J.. Stochastic controllability of linear systems with state delays. Int. J. Appl. Math. Comput. 17 (2007), 5–13.CrossRefGoogle Scholar
9Klamka, J.. Stochastic controllability of systems with variable delay in control. Bull. Pol. Ac.: Tech. 56 (2008), 279–284.Google Scholar
10Krasnoselskii, M.. Topological methods in the theory of nonlinear integral equations (New York: Pergamon Press, 1964).Google Scholar
11Khusainov, D. Y. and Shuklin, G. V.. Linear autonomous time-delay system with permutation matrices solving. Stud. Univ. Žilina 17 (2003), 101–108.Google Scholar
12Khusainov, D. Y. and Shuklin, G. V.. Relative controllability in systems with pure delay. Int. Appl. Mech. 41 (2005), 210–221.CrossRefGoogle Scholar
13Khusainov, D. Y., Diblík, J., Růžičková, M. and Lukáčová, J.. Representation of a solution of the Cauchy problem for an oscillating system with pure delay. Nonlinear Oscillat. 11 (2008), 276–285.CrossRefGoogle Scholar
14Laub, A. J. and Arnold, W. F.. Controllability and observability criteria for multivariable linear second-order model. IEEE Trans. Autom. Control 29 (1984), 163–165.CrossRefGoogle Scholar
15Liang, C., Wang, J. R. and O'Regan, D.. Controllability of nonlinear delay oscillating systems. Electron. J. Qual. Theo. 47 (2017), 1–18.Google Scholar
16Liang, C., Wei, W. and Wang, J.. Stability of delay differential equations via delayed matrix sine and cosine of polynomial degrees. Adv. Differ. Equ. 131 (2017), 1–17.Google Scholar
17Mahmudov, N. I. and Zorlu, S.. Controllability of non-linear stochastic systems. Int. J. Control 76 (2003), 95–104.CrossRefGoogle Scholar
18Maejima, M. and Tudor, C. A.. Selfsimilar processes with stationary increments in the second Wiener chaos. Probab. Math. Stat. 32 (2012), 167–186.Google Scholar
19Maejima, M. and Tudor, C. A.. On the distribution of the Rosenblatt process. Stat. Probab. Lett. 83 (2013), 1490–1495.CrossRefGoogle Scholar
20Rosenblatt, M., Independence and dependence. In Proc. 4th Berkeley Symp. Math. Statist. Probab., Vol. 2, pp. 431–443 (Berkeley, California: University of California Press, 1961).Google Scholar
21Sakthivel, R. and Kim, J. H.. On controllability of nonlinear stochastic systems. Rep. Math. Phys. 58 (2006), 433–443.CrossRefGoogle Scholar
22Sharma, J. P. and George, R. K.. Controllability of matrix second order systems: a trigonometric matrix approach. Electron. J. Differ. Equ. 80 (2007), 1–14.Google Scholar
23Shen, G. J. and Ren, Y.. Neutral stochastic partial differential equations with delay driven by Rosenblatt process in a Hilbert space. J. Korean Stat. Soc. 44 (2015), 123–133.CrossRefGoogle Scholar
24Shen, G., Sakthivel, R., Ren, Y. and Li, M.. Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process. Collect. Math. 71 (2020), 63–82.CrossRefGoogle Scholar
25Tudor, C. A.. Analysis of the Rosenblatt process. ESAIM Probab. Stat. 12 (2008), 230–257.CrossRefGoogle Scholar
26Wang, J., Luo, Z. and Fečkan, M.. Relative controllability of semilinear delay differential systems with linear parts defined by permutable matrices. Eur. J. Control 38 (2017), 39–46.CrossRefGoogle Scholar