Necessary and sufficient conditions for bounded global stability of certain nonlinear systems
Author:
R. K. Brayton
Journal:
Quart. Appl. Math. 29 (1971), 237-244
MSC:
Primary 34.51; Secondary 94.00
DOI:
https://doi.org/10.1090/qam/288366
MathSciNet review:
288366
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Abstract: Nonlinear systems having the form \[ \dot x = - Ax + By\] \[ \dot y = Cx - f\left ( y \right )\], where $\partial f/\partial y$ is a symmetric matrix, are considered. Such systems include the class of nonlinear reciprocal networks where the nonlinearity is voltage (or current) controlled. Also included, provided ${c^T}b \ne 0$, are the equations of nonlinear feedback systems, \[ \dot x = Ax + bf\left ( {{c^T}x} \right )\], considered by Aizerman [1], A type of stability called bounded global stability is considered which requires that all bounded solutions decay as $t \to \infty$ to the set of equilibrium points. A necessary and sufficient condition on the linear parts of these systems for their bounded global stability is given. It is also shown that this condition insures the existence of at least one stable equilibrium point.
M. A. Aǐzerman, On a problem concerning the stability “in the large” of dynamical systems, Uspehi Mat. Nauk 4, no. 4(32); 187–188 (1949)
V. A. Pliss, Certain problems in the theory of stability in the whole, Lenigrad Univ. Press, Leningrad, 1958
J. C. Willems, Perturbation theory for the analysis of instability in nonlinear feedback systems, Fourth Annual Allerton Conference on Circuit and System Theory, 1966, pp. 836–848
- Jürgen Moser, On nonoscillating networks, Quart. Appl. Math. 25 (1967), 1–9. MR 209567, DOI https://doi.org/10.1090/S0033-569X-1967-0209567-X
- R. K. Brayton and J. K. Moser, A theory of nonlinear networks. I, Quart. Appl. Math. 22 (1964), 1–33. MR 169746, DOI https://doi.org/10.1090/S0033-569X-1964-0169746-7
J. P. LaSalle, An invariance principle in the theory of stability, Division of Appl. Math., TR 66-1, Brown University, Providence, R. I.
M. A. Aǐzerman, On a problem concerning the stability “in the large” of dynamical systems, Uspehi Mat. Nauk 4, no. 4(32); 187–188 (1949)
V. A. Pliss, Certain problems in the theory of stability in the whole, Lenigrad Univ. Press, Leningrad, 1958
J. C. Willems, Perturbation theory for the analysis of instability in nonlinear feedback systems, Fourth Annual Allerton Conference on Circuit and System Theory, 1966, pp. 836–848
J. K. Moser, On nonoscillating networks, Quart. Appl. Math. 25, 1–9 (1967)
R. K. Brayton and J. K. Moser, A theory of nonlinear networks I, Quart. Appl. Math. 22, 1–33 (1964)
J. P. LaSalle, An invariance principle in the theory of stability, Division of Appl. Math., TR 66-1, Brown University, Providence, R. I.
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© Copyright 1971
American Mathematical Society