Lagrange stability explained

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)
• Book: Bhatia . Nam Parshad . Szegő . Giorgio P. . Stability theory of dynamical systems . Springer . 2002. 978-3-540-42748-3 .