×
Dzhalladova, Irada A.; Baštinec, Jaromír; Diblík, Josef; Khusainov, Denys Y.

Estimates of exponential stability for solutions of stochastic control systems with delay. (English) Zbl 1217.93150

Abstr. Appl. Anal. 2011, Article ID 920412, 14 p. (2011).
Summary: A nonlinear stochastic differential-difference control system with delay of neutral type is considered. Sufficient conditions for the exponential stability are derived by using Lyapunov-Krasovskii functionals of quadratic form with exponential factors. Upper bound estimates for the exponential rate of decay are derived.

Cited in 12 Documents

MSC:

93D20 Asymptotic stability in control theory
93E15 Stochastic stability in control theory
34K50 Stochastic functional-differential equations
93C10 Nonlinear systems in control theory

Keywords:

nonlinear stochastic differential-difference control system; exponential stability; Lyapunov-Krasovskii functionals
PDFBibTeX XMLCite
\textit{I. A. Dzhalladova} et al., Abstr. Appl. Anal. 2011, Article ID 920412, 14 p. (2011; Zbl 1217.93150)
Full Text: DOI
OA License

References:

[1] L. Glass and M. C. Mackey, From Clocks to Chaos, The Rhythms of Life, Princeton University Press, Princeton, NJ, USA, 1988. · Zbl 0705.92004
[2] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional-Differential Equations, vol. 463 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netheralnds, 1999. · Zbl 0917.34001
[3] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. · Zbl 0777.34002
[4] I. I. Gikhman and A. V. Skorokhod, Stochastic Differential Equations, Naukova Dumka, Kiev, Ukraine, 1968. · Zbl 0169.48702
[5] R. Z. Khasminskii, Stability of Systems of Differential Equations under Stochastic Perturbations of Parameters, Science Publishing House, Moscow, Russia, 1969. · Zbl 0214.15903
[6] A. A. Martynuk and A. A. Martynjuk, Stability of Motion of Complex Systems, Naukova Dumka, Kiev, Ukraine, 1975.
[7] E. A. Barbashin, Lyapunov’s Functions, Nauka, Moskow, Russia, 1970.
[8] E. A. Barbashin, Introduction to the Theory of Stability, Wolters-Noordhoff, Groningen, The Netherlands, 1970. · Zbl 0198.19703
[9] D. Ya. Khusainov and A. V. Shatyrko, Method of Lyapunov Functions in the Study of Stability Functional Differential Systems, Publishing Company of Kiev National University, Kiev, Ukraine, 1997. · Zbl 0911.34066
[10] B. S. Razumikhin, Stability in Hereditary Systems, Science Publishing House, Moscow, Russia, 1988.
[11] D. G. Korenevskii, Stability of Solutions in Deterministic and Stochastic Differential-Difference Equations, Algebraic Criteria, Naukova Dumka, Kiev, Ukraine, 1992. · Zbl 0864.34067
[12] N. N. Krasovskii, Some Problems in the Stability Theory of Motion, Fiziko-Tekhnicheskikh, Moscow, Russia, 1959.
[13] A. S. Bychkov and D. Khusainov, “An exponential decay of solutions of neutral type stochastic equations,” Random Operators and Stochastic Equations, vol. 3, no. 3, pp. 245-256, 1995. · Zbl 0834.60065 · doi:10.1515/rose.1995.3.3.245
[14] I. A. Dzhalladova, “Investigation of the stabilization of a mathematical model of a dynamical system with random action in the case of resonance,” Ukrainian Mathematical Journal, vol. 49, no. 9, pp. 1324-1329, 1997, translated from Ukrans ki Matematichni Zhurnal, vol. 49, no. 9, pp. 1177-1181. · Zbl 0933.93073 · doi:10.1007/BF02487339
[15] I. A. Dzhalladova, “Investigation of the stabilization of a mathematical model of a dynamical system with random influence in the resonance case,” Ukrainian Mathematical Journal, vol. 50, no. 8, pp. 1299-1307, 1998, translated from Ukrans ki Matematichni Zhurnal, vol. 50, no. 8, pp. 1137-1143. · Zbl 0940.34066 · doi:10.1007/BF02513085
[16] D. Ya. Khusainov and A. S. Bychkov, “Stability of stochastic systems of neutral type with small deviations of the argument,” Differential Equations, vol. 28, no. 12, pp. 2060-2069, 1992. · Zbl 0798.60061
[17] F. L. Lewis, Optimal Estimation, With an Introduction to Stochastic Control Theory, Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1986. · Zbl 0665.93065
[18] D. G. Korenevskiĭ, Stability of Dynamical Systems under Random Perturbations of Parameters, Algebraic Criteria, Naukova Dumka, Kiev, Ukraine, 1989. · Zbl 0769.93081
[19] I. G. Nechaeva and D. Y. Khusainov, “Obtaining estimates for the stability of solutions of stochastic functional-differential equations,” Differential Equations, vol. 28, no. 3, p. 405-414, 547, 1992. · Zbl 0839.60055
[20] J.-H. Park and S. Won, “A note on stability of neutral delay-differential systems,” Journal of the Franklin Institute, vol. 336, no. 3, pp. 543-548, 1999. · Zbl 0969.34066 · doi:10.1016/S0016-0032(98)00040-4
[21] V. Rezvan, Absolute Stability Automatic System with Delay, Nauka, Moscow, Russia, 1983. · Zbl 0990.93519
[22] M. L. Sverdan, E. F. Tsarkov, and V. K. Yasinskii, Stability in Stochastic Modeling of Complex Dynamical Systems, Svyatyn, ”Nad Prutom”, 1996.
[23] E. F. Tsarkov, Random Perturbations of Functional Difefrential Equations, Zinatne, Riga, Latvia, 1989. · Zbl 0725.34092
[24] E. F. Tsarkov and V. K. Yasinskii, Quasi-Linear Stochastic Functional Differential Equations, Orientir, Riga, Latvia, 1992.
[25] X.-x. Liu and B. Xu, “A further note on stability criterion of linear neutral delay-differential systems,” Journal of the Franklin Institute, vol. 343, no. 6, pp. 630-634, 2006. · Zbl 1114.34335 · doi:10.1016/j.jfranklin.2006.02.024
[26] M. A. Ajzerman and F. R. Gantmacher, Absolute Stability Regulations System, Science Publishing House, Moscow, Russia, 1963. · Zbl 0111.08702
[27] A. H. Gelig, G. A. Leonov, and V. A. Jakubovi\vc, The Stability of Nonlinear Systems with a Nonunique Equilibrium State, Nauka, Moscow, Russia, 1978. · Zbl 0544.93051
[28] A. K. Gelig and G. A. Leonov, Nonlinear Systems, Nauka, Moscow, Russia, 1978.
[29] A. D. Myshkis, Linear Differential Equations with Delayed Argument, Nauka, Moscow, Russia, 2nd edition, 1972. · Zbl 0261.34040
[30] E. Boukas, Deterministic and Stochastic Time Delay Systems, Birkhäuser, Boston, Mass, USA, 2002. · Zbl 1056.93001
[31] X. Liao and P. Yu, Absolute Stability of Nonlinear Control Systems, vol. 25 of Mathematical Modelling: Theory and Applications, Springer, New York, NY, USA, 2nd edition, 2008. · Zbl 1151.93002 · doi:10.1007/978-1-4020-8482-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.