Lyapunov stability

Various types of <a href="/facts/Stability_theory/Su5sHnxs">stability</a> may be discussed for the solutions of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> or <a href="/facts/Difference_equation/JolXC71M">difference equations</a> describing <a href="/facts/Dynamical_system/782gREl5">dynamical systems</a>. 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 <a href="/facts/Aleksandr_Lyapunov/1b0TlYBz">Aleksandr Lyapunov</a>. In simple terms, if the solutions that start out near an equilibrium point 
 
 
 
 
 x
 
 e
 
 
 
 
 {\displaystyle x_{e}}
 
 stay near 
 
 
 
 
 x
 
 e
 
 
 
 
 {\displaystyle x_{e}}
 
 forever, then 
 
 
 
 
 x
 
 e
 
 
 
 
 {\displaystyle x_{e}}
 
 is Lyapunov stable. More strongly, if 
 
 
 
 
 x
 
 e
 
 
 
 
 {\displaystyle x_{e}}
 
 is Lyapunov stable and all solutions that start out near 
 
 
 
 
 x
 
 e
 
 
 
 
 {\displaystyle x_{e}}
 
 converge to 
 
 
 
 
 x
 
 e
 
 
 
 
 {\displaystyle x_{e}}
 
, then 
 
 
 
 
 x
 
 e
 
 
 
 
 {\displaystyle x_{e}}
 
 is said to be asymptotically stable (see <a href="/facts/Asymptotic_analysis/kt87qLpu">asymptotic analysis</a>). The notion of <a href="/facts/Exponential_stability/mXW6BYgG">exponential stability</a> 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 <a href="/facts/Structural_stability/aoUYsDub">structural stability</a>, which concerns the behavior of different but "nearby" solutions to differential equations. <a href="/facts/Input-to-state_stability/Sr1tSfgA">Input-to-state stability</a> (ISS) applies Lyapunov notions to systems with inputs.

Lyapunov stability open-in-new

Lyapunov stability