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