Lagrange stability

Lagrange stability is a concept in the stability theory of dynamical systems, named after Joseph-Louis Lagrange. 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 xin M} in a real continuous dynamical system ( T , M , Φ ) {displaystyle (T,M,Phi )} , where T {displaystyle T} is R {displaystyle mathbb {R} } , the motion Φ ( t , x ) {displaystyle Phi (t,x)} 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 xin M} , the motion Φ ( t , x ) {displaystyle Phi (t,x)} is positively-/negativey-/Lagrange stable, respectively.