A remark on Lagrange stability of nonlinear systems stabilization. (English) Zbl 0773.93058

The problem of stability (in the sense of Lagrange) for nonlinear control systems is considered. For simplificity only the single input-single output systems are treated. It is proved under suitable assumptions, that if there is a bounded input \(u\) such that the corresponding output is bounded for any initial condition, then the system is \(u\)-Lagrange stable. Furthermore, some assumptions are proposed under which the Lagrange stability with respect to any bounded input implies existence of a stability feedback law.


93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C10 Nonlinear systems in control theory
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[1] Chanane, B.; Banks, P., Nonlinear input-output maps for bilinear systems and stability, Journal of Mathematical Control & Information, 6, 309-315 (1989) · Zbl 0696.93012
[2] Gauthier, J. P.; Hammouri, H.; Othman, S., A simple observer for nonlinear systems. Applications to bioreactors, IEEE Trans. Automat. Control (April, 1992), (to appear).
[3] Gauthier, J. P.; Bornard, G., Observability for any \(u(t)\) of a class of bilinear systems, IEEE Trans. Automt. Control, 26, 922-926 (1981) · Zbl 0553.93014
[4] Jurdjevic, V.; Quinn, H., Controllability and observability, Journal of Differential Equations, 12 (1978) · Zbl 0417.93012
[5] Gauthier, J. P.; Bornard, G., Outils et modèles mathématiques pour l’automatique. L’analyse de systèmes et le trairtement du signal, CNRS Edit., 307-324 (1981)
[6] LaSalle, J. P., Stability theory for ordinary differential equations, Journal of Differential Equations (1968) · Zbl 0159.12002
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