Lagrange stability
Lagrange stability is a concept in the stability theory of dynamical systems, named after Joseph-Louis Lagrange. For any point in the state space, [math]\displaystyle{ x \in M }[/math] in a real continuous dynamical system [math]\displaystyle{ (T,M,\Phi) }[/math], where [math]\displaystyle{ T }[/math] is [math]\displaystyle{ \mathbb{R} }[/math], the motion [math]\displaystyle{ \Phi(t,x) }[/math] is said to be positively Lagrange stable if the positive semi-orbit [math]\displaystyle{ \gamma_x^+ }[/math] is compact. If the negative semi-orbit [math]\displaystyle{ \gamma_x^- }[/math] is compact, then the motion is said to be negatively Lagrange stable. The motion through [math]\displaystyle{ x }[/math] is said to be Lagrange stable if it is both positively and negatively Lagrange stable. If the state space [math]\displaystyle{ M }[/math] is the Euclidean space [math]\displaystyle{ \mathbb{R}^n }[/math], then the above definitions are equivalent to [math]\displaystyle{ \gamma_x^+, \gamma_x^- }[/math] and [math]\displaystyle{ \gamma_x }[/math] being bounded, respectively.
A dynamical system is said to be positively-/negatively-/Lagrange stable if for each [math]\displaystyle{ x \in M }[/math], the motion [math]\displaystyle{ \Phi(t,x) }[/math] is positively-/negatively-/Lagrange stable, respectively.
References
- Elias P. Gyftopoulos, Lagrange Stability and Liapunov's Direct Method. Proc. of Symposium on Reactor Kinetics and Control, 1963. (PDF)
- Bhatia, Nam Parshad; Szegő, Giorgio P. (2002). Stability theory of dynamical systems. Springer. ISBN 978-3-540-42748-3.
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