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
in a real continuous dynamical system
(
T
,
M
,
Φ
)
, where
T
is
R
, the motion
Φ
(
t
,
x
)
is said to be positively Lagrange stable if the positive semi-orbit
γ
x
+
is compact. If the negative semi-orbit
γ
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
γ
x
+
,
γ
x
−
and
γ
x
being bounded, respectively.
A dynamical system is said to be positively-/negatively-/Lagrange stable if for each
x
∈
M
, the motion
Φ
(
t
,
x
)
is positively-/negativey-/Lagrange stable, respectively.
