Exponential stability and estimates for monotone and differential-difference systems
- First Online:
We present necessary and sufficient conditions for the exponential stability in the nonnegative cone and refine exponential estimates for solutions of systems of autonomous difference equations with monotone nondecreasing right-hand sides, including discontinuous ones, as well as for solutions of some class of systems of differential-difference equations with monotonicity. Unlike well-known criteria, the new ones are free of some additional assumptions on the right-hand sides of the considered models other than the original monotonicity conditions.
We show that, in the nonsmooth and discontinuous cases, the traditional exponential stability conditions based on “linearization” can lead to negative or very coarse results.
Unable to display preview. Download preview PDF.
- 1.Vasil’ev, S.N., A Reduction Method and Qualitative Analysis of Dynamical Systems: I, II, Izv. Ross. Akad. Nauk Teor. Sist. Upravl., 2006, no. 1, pp. 21–29; no. 2, pp. 5–17.Google Scholar
- 2.Abdullin, R.Z., Anapolsky, L.Y., Kozlov, R.I., et al., Vector Lyapunov Functions in Stability Theory, Atlanta, 1996.Google Scholar
- 3.Michel, A.N., Wang, K., and Hu, B., Qualitative Theory of Dynamical Systems. The Role of Stability-Preserving Mappings, New York, 2001.Google Scholar
- 4.Kozlov, R.I. and Burnosov, S.V., Asymptotic Behavior of and Estimates for the Solutions of Monotone Difference Equations, in Metod funktsii Lyapunova v analize dinamiki sistem (The Method of Lyapunov Functions in the Analysis of the Dynamics of Systems), Novosibirsk: Nauka, 1987, pp. 85–93.Google Scholar
- 5.Kozlov, R.I., Teoriya sistem sravneniya v metode vektornykh funktsii Lyapunova (Theory of Comparison Systems in the Method of Vector Lyapunov Functions), Novosibirsk: Nauka, 2001.Google Scholar
- 7.Anapol’skii, L.Yu., The Comparison Method in the Dynamics of Discrete Systems, in Vektor-funktsii Lyapunova i ikh postroenie (Vector-Valued Lyapunov Functions and Their Construction), Novosibirsk: Nauka, 1980, pp. 92–128.Google Scholar
- 8.Kozlov, R.I. and Kozlova, O.R., Investigation of the Stability of Nonlinear Continuous-Discrete Models of Economic Dynamics by the Method of Vector Lyapunov Functions. I, II, Izv. Ross. Akad. Nauk Teor. Sist. Upravl., 2009, no. 2, pp. 104–113; no. 3, pp. 41–50.Google Scholar
- 12.Vassilyev, S.N., Kozlov, R.I., and Sivasundaram, S., Toward a Qualitative Theory of Systems with Discrete-Continuous Time and Impulsive Effects, Proc. of ICNPAA-2000, vol. 2, Cambridge, 2001, pp. 667–680.Google Scholar
- 13.Vassilyev, S., Kozlov, R., Lakeyev, A., and Zherlov, A., Control Methods for Some Classes of Logical-Dynamic Systems under Uncertainties and Perturbations, J. Hybrid Systems, 2002, vol. 2, no. 1, pp. 87–97.Google Scholar
- 16.Krasnosel’skii, M.A., Lifshits, E.A., and Sobolev, A.V., Pozitivnye lineinye sistemy (Positive Linear Systems), Moscow: Nauka, 1985.Google Scholar
- 17.Filippov, A.V., Differentsial’nye uravneniya s razryvnoi pravoi chast’yu (Differential Equations with Discontinuous Right-Hand Sides), Moscow: Nauka, 1985.Google Scholar