The system is asymptotically stable in the large at the origin if :

Lyapunov's stability analysis is a technique for analyzing the stability of an equilibrium point in a continuous-time system. It's based on the Lyapunov stability criterion, which was developed by A. M. Lyapunov in 1892. The criterion is based on the concept of energy: if the total energy of a system is dissipated, then the system is always stable.  Lyapunov's stability theorem states that if a system has an equilibrium point at x = 0, and a positive scalar function V(x) is defined near the equilibrium point, then the system is stable if: For any ɛ > 0, there exists δ = δ(ɛ) > 0 to satisfy... Show more

Lyapunov's stability analysis is a technique for analyzing the stability of an equilibrium point in a continuous-time system. It's based on the Lyapunov stability criterion, which was developed by A. M. Lyapunov in 1892.

The criterion is based on the concept of energy: if the total energy of a system is dissipated, then the system is always stable. 

Lyapunov's stability theorem states that if a system has an equilibrium point at x = 0, and a positive scalar function V(x) is defined near the equilibrium point, then the system is stable if:
For any ɛ > 0, there exists δ = δ(ɛ) > 0 to satisfy ‖x(0)‖ < δ ⇒ ‖x(t)‖ < ɛ with ∀ t ≥ 0
There exists δ > 0 to satisfy ‖x(0)‖ < δ ⇒ lim ‖x(t)‖ = 0 

Lyapunov functions are scalar functions that can be used to verify the stability of equilibrium of an ordinary differential equation.

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