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On delay-dependent stability for a class of nonlinear stochastic delay-differential equations. (English) Zbl 1103.93043
Summary: Global asymptotic stability conditions for nonlinear stochastic systems with state delay are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov-Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of the system coefficients. Nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given.

MSC:
93E03 Stochastic systems in control theory (general)
93D20 Asymptotic stability in control theory
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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[1] Boukas E-K, Liu Z-K (2002) Deterministic and stochastic time-delay systems. Birkhäuser, Boston · Zbl 1056.93001
[2] De Souza CE, Li X (1997) Delay-dependent robust robust stability and stabilization of uncertain linear delay system: a linear matrix inequality approach. IEEE Trans. Automat Control 42:1144–1148 · Zbl 0889.93050
[3] De Souza C E, Li X (1999) Delay-dependent robust Hcontrol of uncertain linear state-delayed systems. Automatica 35:1313–1321 · Zbl 1041.93515
[4] Fridman L, Acosta P, Polyakov A (2001) Robust eigenvalue assignment for uncertain delay control systems. In: Proceedings of the 3rd IFAC workshop on time delay systems, Santa Fe, NM, December 8–10, 2001. Elsevier, London, pp 239–244
[5] Gu K, Han Q-L (2001) A revisit of some delay-dependent stability criteria for uncertain time-delay systems. In: Proceedings of the 3rd IFAC workshop on time delay systems, Santa Fe, NM, December 8–10, 2001. Elsevier, London, pp 153–158
[6] Gu K, Niculescu S-I (2003) Survey on recent results in the stability and control of time-delay systems. ASME Trans. J Dyn Syst Measur Control 125:158–165
[7] Hale JK, Verduyn-Lunel SM (1993) Introduction to functional differential equations. Springer, Berlin Heidelberg New York · Zbl 0787.34002
[8] Kharitonov V (1999) Robust stability analysis of time-delay systems: a survey. Annu Rev Control 23:185–196
[9] Kharitonov V, Melchor-Aguilar DA (2000) On delay-dependent stability conditions. Syst Control Lett 40:71–76 · Zbl 0977.93072
[10] Kolmanovskii VB, Myshkis AD (1992) Applied theory of functional differential equations. Kluwer, Dordrecht
[11] Kolmanovskii VB, Nosov VR (1986) Stability of functional differential equations. Academic, New York
[12] Kolmanovskii VB, Shaikhet LE (1996) Control of systems with aftereffect. American Mathematical Society, Providence
[13] Kolmanovskii VB, Niculescu S-I, Gu K (1999) Delay effects on stability: a survey. In: Proceedings of the 38th IEEE conference on decision and control, Phoenix, AZ, USA, December 7–10, 1999, pp 1993–1998
[14] Kolmanovskii VB, Niculescu S-I, Richard J-P (1999) On the Lyapunov–Krasovskii functionals for stability analysis of linear delay systems. Int J Control 72:374–382 · Zbl 0952.34057
[15] Liao X, Mao X (2000) Exponent stability of stochastic delay interval systems. Syst Control Letters 40:171–181 · Zbl 0949.60068
[16] Liptser RS, Shiryayev AN (1989) The martingale theory. Kluwer, Dordrecht
[17] Mao X, Koroleva N, Rodkina A (1998) Robust stability of uncertain stochastic differential delay equations. Syst Control Letters 35:325–336 · Zbl 0909.93054
[18] Melnikov AV, Rodkina AE (1993) Martingale approach to the procedures of stochastic approximation. In: Niemi H et al (eds) Frontiers in pure and applied probability, vol 1. TVP/VSP, Moscow, pp 165–182 · Zbl 0815.62054
[19] Niculescu S-I (2001) Delays effects on stability. a robust control approach. Springer, Berlin Heidelberg New York
[20] Nosov VR (2002) Stability of cubic equation with delay. In: Abstracts of international conference “Functional differential equations and applications”, June 9–13, 2002, Beer-Sheva, Israel, pp 48–49
[21] Orlov Y, Perruquetti W, Richard J-P (2003) Sliding mode control synthesis of uncertain time-delay systems. Asian J Control 5:568–577
[22] Pugachev VS, Sinitsyn IN (2001) Stochastic systems: theory and applications. World Scientific, Singapore
[23] Rodkina AE, Nosov VR (2003) On stability of some nonlinear scalar differential equations. Dyn Syst Appl 12:285–294 · Zbl 1070.34096
[24] Xie S, Xie L (2000) Stabilization of a class of uncertain large-scale stochastic systems with time delays. Automatica 36:161–167 · Zbl 0936.93050
[25] Xu S, Chen T (2002) Robust Hcontrol for uncertain stochastic systems with state delay. IEEE Trans Automat Control 47:2089–2094 · Zbl 1364.93755
[26] Yue D, Won S (2001) Delay-dependent robust stability of stochastic systems with time delay and nonlinear uncertainties. IEE Electron Letters 37:992–993 · Zbl 1190.93095
[27] Yue D, Fang J, Won S (2003) Delay-dependent robust stability of stochastic uncertain systems with time delay and Markovian jump parameters. Circ Syst Signal Process 22:351–365 · Zbl 1048.93095
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