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, $x\in M$ in a real continuous dynamical system $(T,M,\mathrm{\Phi })$, where $T$ is $\mathbb{ℝ}$, the motion $\mathrm{\Phi }(t,x)$ is said to be positively Lagrange stable if the positive semi-orbit ${\gamma }_x^{+}$ is compact. If the negative semi-orbit ${\gamma }_x^{-}$ is compact, then the motion is said to be negatively Lagrange stable. The motion through $x$ is said to be Lagrange stable if it is both positively and negatively Lagrange stable. If the state space $M$ is the Euclidean space ${\mathbb{ℝ}}^n$, then the above definitions are equivalent to ${\gamma }_x^{+},{\gamma }_x^{-}$ and ${\gamma }_x$ being bounded, respectively.

A dynamical system is said to be positively-/negatively-/Lagrange stable if for each $x\in M$, the motion $\mathrm{\Phi }(t,x)$ 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.