- Lyapunov stability
mathematics, the notion of Lyapunov stability occurs in the study of dynamical systems. In simple terms, if all solutions of the dynamical system that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if all solutions that start out near converge to , then is asymptotically stable. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behaviour of different but "nearby" solutions to differential equations.
Definition for continuous-time systems
Consider an autonomous nonlinear dynamical system
where denotes the system state vector, an open set containing the origin, and continuous on . Without loss of generality, we may assume that the origin is an equilibrium.
# The origin of the above system is said to be Lyapunov stable, if, for every , there exists a such that, if , then , for every .
# The origin of the above system is said to be asymptotically stable if it is Lyapunov stable and if there exists such that if , then .
# The origin of the above system is said to be exponentially stable if it is asymptotically stable and if there exist such that if , then , for .
Conceptually, the meanings of the above terms are the following:
# Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a distance from it) remain "close enough" forever (within a distance from it). Note that this must be true for "any" that one may want to choose.
# Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium.
# Exponential stability means that solutions not only converge, but in fact converge faster than or at least as fast as a particular known rate .
The trajectory "x" is (locally) "attractive" if
for for all trajectories that start close enough, and "globally attractive" if this property holds for all trajectories.
That is, if "x" belongs to the interior of its
stable manifold. It is "asymptotically stable" if it is both attractive and stable. (There are counterexamples showing that attractivity does not imply asymptotic stability. Such examples are easy to create using homoclinic connections.)
Definition for iterated systems
The definition for discrete-time systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts.
Let be a
metric spaceand a continuous function. A point is said to be Lyapunov stable, if, for each , there is a such that for all , if
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