Geodesic
Global asymptotic stability of generalized homogeneous dynamical systems
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2023), pp. 52-82.

Voir la notice de l'article provenant de la source Math-Net.Ru

The goal of the paper is to study the relationship between asymptotic stability and exponential stability of the solutions of generalized homogeneous nonautonomous dynamical systems. This problem is studied and solved within the framework of general non-autonomous (cocycle) dynamical system. The application of our general results for differential and difference equations is given. 
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David Cheban. Global asymptotic stability of generalized homogeneous dynamical systems. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2023), pp. 52-82. https://geodesic-test.mathdoc.fr/item/BASM_2023_2_a5/

[1] Artstein Z., “Uniform Asymptotic Stability via the Limiting Equations”, Journal of Differential Equations, 27:2 (1978), 172–189 | DOI | MR | Zbl

[2] A. Bacciotti, L. Rosier, Liapunov Functions and Stability in Control Theory, Springer-Verlag, Berlin–Heidelberg, 2005, xiii+236 pp. | MR | Zbl

[3] Bebutov V. M., “On the shift dynamical systems on the space of continuous functions”, Bull. of Inst. of Math. of Moscow University, 2:5 (1940), 1–65 (in Russian)

[4] Extensions of Minimal Transformation Group, Sijthoff Noordhoff, Alphen aan den Rijn, The Netherlands Germantown, Maryland USA, 1979 | MR

[5] Mathematical Notes, 63:1 (1998), 115–126 | DOI | MR | Zbl

[6] Cheban D. N., Lyapunov Stability of Non-Autonomous Dynamical Systems, Nova Science Publishers Inc, New York, 2013, xii+275 pp.

[7] Cheban D. N., Global Attractors of Nonautonomous Dynamical and Control Systems, Interdisciplinary Mathematical Sciences, 18, 2nd Edition, World Scientific, River Edge, NJ, 2015, xxv+589 pp. | DOI | MR

[8] D. N. Cheban, Nonautonomous Dynamics: Nonlinear oscillations and Global attractors, Springer Nature Switzerland AG, 2020, xxii+434 pp. | MR | Zbl

[9] M'Closkey R. T., Murray R. M., “Extending Exponential Stabilizers for nonholonomic systems from Kinematic Controllers to Dynamic Controllers”, Preprints of the Fourth IFAC Symposium on Robot Control, 1994, 243–248

[10] Efimov D., Perruquetti M., Richard J.-P., “Development of Homogeneity Concept for Time-Delay Systems”, SIAM J. Control Optim., 52:3 (2014), 1547–1566 | DOI | MR | Zbl

[11] M. Gil, Difference Equations in Normed Spaces. Stability and Oscilations, North-Holland, Elsevier, 2007, xxii+362 pp. | MR

[12] W. Hahn, Stability of Motion, Springer-Verlag, Berlin–Heidelberg–New York, 1967, xi+446 pp. | MR | Zbl

[13] Kawski M., “Geometric Homogeneity and Stabilization”, IFAC Proceedings Volumes, 28:14 (1995), 147–152 | DOI

[14] Lakshmikanthan V., Donato Trigiante, Theory of Difference Equations: Numerical Methods and Applications, Marcel Dekker, Inc., New York–Basel, 2002, x+299 pp. | MR

[15] Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, 1982, xi+211 pp. | MR | MR | Zbl | Zbl

[16] A. Polyakov, Generalized Homogeneity in Systems and Control, Springer Nature Switzerland AG, 2020, xviii+447 pp. | MR | Zbl

[17] J.-B. Pomet, C. Samson, Time-varying exponential stabilization of nonholonomic systems in power form, Research Report RR-2126, ffinria-00074546v2ff, INRIA, 1993, 27 pp.

[18] Rosier L., Etude de quelques problemes de stabilisation, PhD Thesis, Ecole Normale Supérieure de Cachan (France), 1993

[19] Sell G. R., “Non-Autonomous Differential Equations and Topological Dynamics. II. Limiting equations”, Trans. Amer. Math. Soc., 127 (1967), 263–283 | DOI | MR | Zbl

[20] Sell G. R., Lectures on Topological Dynamics and Differential Equations, Van Nostrand Reinhold math. studies, 2, Van Nostrand-Reinbold, London, 1971 | MR

[21] Shcherbakov B. A., Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Shtiintsa, Kishinev, 1972, 231 pp. (in Russian) | MR

[22] Shcherbakov B. A., Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Shtiintsa, Kishinev, 1985, 147 pp. (in Russian) | MR | Zbl

[23] Introduction to Topological Dynamics, Noordhoff, Leiden, 1975, ix+163 pp. | MR | MR | Zbl

[24] Methods of A. M. Lyapunov and Their Applications, United States, Atomic Energy Commission, Groningen, 1964 | MR | MR