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This article is about asymptotic stability of nonlinear systems. For stability of linear systems, see exponential stability. Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov. 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 is Lyapunov stable and 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 behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs. Contents

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•1 History

•2 Definition for continuous-time systems

o2.1 Lyapunov's second method for stability

•3 Definition for discrete-time systems

•4 Stability for linear state space models

•5 Stability for systems with inputs

•6 Example

•7 Barbalat's lemma and stability of time-varying systems •8 References

•9 Further reading

•10 External links

[edit] History

Lyapunov stability is named after Aleksandr Lyapunov, a Russian mathematician who published his book "The General Problem of Stability of Motion" in 1892.[1] Lyapunov was the first to consider the modifications necessary in nonlinear systems to the linear theory of stability based on linearizing near a point of equilibrium. His work, initially published in Russian and then translated to French, received little attention for many years. Interest in it started suddenly during the Cold War (1953-1962) period when the so-called "Second Method of Lyapunov" was found to be applicable to the stability of aerospace guidance systems which typically contain strong nonlinearities not treatable by other methods. A large number of publications appeared then and since in the control and systems literature.[2][3][4][5][6] More recently the concept of the Lyapunov exponent (related to Lyapunov's First Method of discussing stability) has received wide interest in connection with chaos theory. Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems.[7] [edit] 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 . Suppose has an equilibrium . 1.The equilibrium of the above system is said to be Lyapunov stable, if, for every , there exists a such that, if , then , for every . 2.The equilibrium of the above system is said to be asymptotically stable if it is Lyapunov stable and if there exists such that if , then . 3.The equilibrium 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: 1.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. 2.Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium. 3.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)...