Lyapunovstöðugt

Lyapunovstöðugt is an Icelandic term that translates to "Lyapunov stable" in English, referring to a concept in the theory of dynamical systems. A dynamical system is considered Lyapunov stable if, for any initial condition close to an equilibrium point, the system's trajectory remains close to that equilibrium point over time. More formally, if an equilibrium point is $x_e$, the system is Lyapunov stable if for every $\epsilon > 0$, there exists a $\delta > 0$ such that if the initial state $x(0)$ is within $\delta$ of $x_e$, then the state $x(t)$ remains within $\epsilon$ of $x_e$ for all $t \geq 0$. This definition does not require the trajectories to converge to the equilibrium point, only to stay arbitrarily close to it if they start sufficiently close.

Lyapunov stability is a fundamental concept in control theory and the analysis of differential equations. It is often a prerequisite for achieving asymptotic stability, where trajectories not only stay close but also converge to the equilibrium point. While Lyapunov stability guarantees that small perturbations do not lead to large deviations from the equilibrium, it does not imply robustness against sustained disturbances. The term is named after the Russian mathematician Aleksandr Lyapunov, who pioneered the study of stability of motion.