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

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