Editing Lyapunov stability (section)

{{Short description|Property of a dynamical system where solutions near an equilibrium point remain so}}
{{About|asymptotic stability of nonlinear systems|stability of linear systems|exponential stability}}
{{Lead rewrite|date=December 2021}}
{{Astrodynamics}}
Various types of [[Stability theory|stability]] may be discussed for the solutions of [[differential equation]]s or [[difference equation]]s describing [[dynamical system]]s. 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 <math>x_e</math> stay near <math>x_e</math> forever, then <math>x_e</math> is '''Lyapunov stable'''.  More strongly, if <math>x_e</math> is Lyapunov stable and all solutions that start out near <math>x_e</math> converge to <math>x_e</math>, then <math>x_e</math> 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.