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Various types of stability may be discussed for the solutions of differential equations or difference 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 Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point ${\displaystyle x_{e}}$ stay near ${\displaystyle x_{e}}$ forever, then ${\displaystyle x_{e}}$ is Lyapunov stable. More strongly, if ${\displaystyle x_{e}}$ is Lyapunov stable and all solutions that start out near ${\displaystyle x_{e}}$ converge to ${\displaystyle x_{e}}$, then ${\displaystyle x_{e}}$ is said to be asymptotically stable (see asymptotic analysis). 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.

History

Lyapunov stability is named after Aleksandr Mikhailovich Lyapunov, a Russian mathematician who defended the thesis The General Problem of Stability of Motion at Kharkov University in 1892.^[1] A. M. Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium. His work, initially published in Russian and then translated to French, received little attention for many years. The mathematical theory of stability of motion, founded by A. M. Lyapunov, considerably anticipated the time for its implementation in science and technology. Moreover Lyapunov did not himself make application in this field, his own interest being in the stability of rotating fluid masses with astronomical application. He did not have doctoral students who followed the research in the field of stability and his own destiny was terribly tragic because of his suicide in 1918 ^{[citation needed]}. For several decades the theory of stability sank into complete oblivion. The Russian-Soviet mathematician and mechanician Nikolay Gur'yevich Chetaev working at the Kazan Aviation Institute in the 1930s was the first who realized the incredible magnitude of the discovery made by A. M. Lyapunov. The contribution to the theory made by N. G. Chetaev^[2] was so significant that many mathematicians, physicists and engineers consider him Lyapunov's direct successor and the next-in-line scientific descendant in the creation and development of the mathematical theory of stability.

The interest in it suddenly skyrocketed during the Cold War period when the so-called "Second Method of Lyapunov" (see below) 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.^{[3][4][5][6][7]} 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.^[8]

Definition for continuous-time systems

Consider an autonomous nonlinear dynamical system

${\displaystyle {\dot {x}}=f(x(t)),\;\;\;\;x(0)=x_{0}}$,

where ${\displaystyle x(t)\in {\mathcal {D}}\subseteq \mathbb {R} ^{n}}$ denotes the system state vector, ${\displaystyle {\mathcal {D}}}$ an open set containing the origin, and ${\displaystyle f:{\mathcal {D}}\rightarrow \mathbb {R} ^{n}}$ is a continuous vector field on ${\displaystyle {\mathcal {D}}}$. Suppose ${\displaystyle f}$ has an equilibrium at ${\displaystyle x_{e}}$ so that ${\displaystyle f(x_{e})=0}$ then

1. This equilibrium is said to be Lyapunov stable if for every ${\displaystyle \epsilon >0}$ there exists a ${\displaystyle \delta >0}$ such that if ${\displaystyle \|x(0)-x_{e}\|<\delta }$ then for every ${\displaystyle t\geq 0}$ we have ${\displaystyle \|x(t)-x_{e}\|<\epsilon }$.
2. The equilibrium of the above system is said to be asymptotically stable if it is Lyapunov stable and there exists ${\displaystyle \delta >0}$ such that if ${\displaystyle \|x(0)-x_{e}\|<\delta }$ then ${\displaystyle \lim _{t\rightarrow \infty }\|x(t)-x_{e}\|=0}$.
3. The equilibrium of the above system is said to be exponentially stable if it is asymptotically stable and there exist ${\displaystyle \alpha >0,~\beta >0,~\delta >0}$ such that if ${\displaystyle \|x(0)-x_{e}\|<\delta }$ then ${\displaystyle \|x(t)-x_{e}\|\leq \alpha \|x(0)-x_{e}\|e^{-\beta t}}$ for all ${\displaystyle t\geq 0}$.

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 ${\displaystyle \delta }$ from it) remain "close enough" forever (within a distance ${\displaystyle \epsilon }$ from it). Note that this must be true for any ${\displaystyle \epsilon }$ 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 ${\displaystyle \alpha \|x(0)-x_{e}\|e^{-\beta t}}$.

The trajectory ${\displaystyle x(t)=\phi (t)}$ is (locally) attractive if

${\displaystyle \|x(t)-\phi (t)\|\rightarrow 0}$ as ${\displaystyle t\rightarrow \infty }$

for all trajectories ${\displaystyle x(t)}$ that start close enough to ${\displaystyle \phi (t)}$, 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 examples showing that attractivity does not imply asymptotic stability.^[9][10][11] Such examples are easy to create using homoclinic connections.)

If the Jacobian of the dynamical system at an equilibrium happens to be a stability matrix (i.e., if the real part of each eigenvalue is strictly negative), then the equilibrium is asymptotically stable.

System of deviations

Instead of considering stability only near an equilibrium point (a constant solution ${\displaystyle x}$Zdroj:https://en.wikipedia.org?pojem=Lyapunov_stability
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