Exponential stability and estimates for monotone and differential-difference systems
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
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.
- 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.
- Abdullin, R.Z., Anapolsky, L.Y., Kozlov, R.I., et al., Vector Lyapunov Functions in Stability Theory, Atlanta, 1996.
- Michel, A.N., Wang, K., and Hu, B., Qualitative Theory of Dynamical Systems. The Role of Stability-Preserving Mappings, New York, 2001.
- 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.
- 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.
- Bitsoris, G., Stability Analysis of Nonlinear Dynamical System, Internat. J. Control, 1983, vol. 38, no. 3, pp. 699–711. CrossRef
- 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.
- 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.
- Dem’yanov, V.F. and Rubinov, A.M., Osnovy negladkogo analiza i kvazidifferentsial’noe ischislenie (Foundations of Nonsmooth Analysis, and Quasidifferential Calculus), Moscow: Nauka, 1990.
- Gantmakher, F.R., Teoriya matrits (Theory of Matrices), Moscow: Nauka, 1988.
- Voevodin, V.V. and Kuznetsov, Yu.A., Matritsy i vychisleniya (Matrices and Calculations), Moscow: Nauka, 1984.
- 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.
- 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.
- Matrosov, V.M., Differential Equations and Inequalities with Discontinuous Right Hand Sides. I, II, Differ. Uravn., 1967, vol. 3, no. 3, pp. 395–409; no. 5, pp. 839–848.
- Kozlov, R.I., On the Theory of Differential Equations with Discontinuous Right-Hand Sides, Differ. Uravn., 1974, vol. 10, no. 7, pp. 1264–1275.
- Krasnosel’skii, M.A., Lifshits, E.A., and Sobolev, A.V., Pozitivnye lineinye sistemy (Positive Linear Systems), Moscow: Nauka, 1985.
- Filippov, A.V., Differentsial’nye uravneniya s razryvnoi pravoi chast’yu (Differential Equations with Discontinuous Right-Hand Sides), Moscow: Nauka, 1985.
- Exponential stability and estimates for monotone and differential-difference systems
Volume 48, Issue 9 , pp 1296-1307
- Cover Date
- Print ISSN
- Online ISSN
- SP MAIK Nauka/Interperiodica
- Additional Links