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 $(T,M,\Phi )$, where $T$ is $ℝ$, the motion $\Phi (t,x)$ 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 ${ℝ}^n$, 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 $\Phi (t,x)$ 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)