Lyapunov, in his original 1892 work, proposed two methods for demonstrating stability. The first method developed the solution in a series which was then proved convergent within limits. The second method, which is almost universally used nowadays, makes use of a "Lyapunov function V(x)" which has an analogy to the potential function of classical dynamics. It is introduced as follows for a system having a point of equilibrium at x=0. Consider a function formula_34 such that
Then "V(x)" is called a Lyapunov function candidate and the system is stable in the sense of Lyapunov. (Note that formula_40 is required; otherwise for example formula_41 would "prove" that formula_42 is locally stable. An additional condition called "properness" or "radial unboundedness" is required in order to conclude global stability.) Furthermore, the system is asymptotically stable, in the sense of Lyapunov, if formula_43 with equality if and only if formula_36. Global asymptotic stability (GAS) follows similarly.

It is easier to visualize this method of analysis by thinking of a physical system (e.g. vibrating spring and mass) and considering the energy of such a system. If the system loses energy over time and the energy is never restored then eventually the system must grind to a stop and reach some final resting state. This final state is called the attractor. However, finding a function that gives the precise energy of a physical system can be difficult, and for abstract mathematical systems, economic systems or biological systems, the concept of energy may not be applicable.

Lyapunov's realization was that stability can be proven without requiring knowledge of the true physical energy, provided a Lyapunov function can be found to satisfy the above constraints.

Lyapunov's second method for stability
Stability for linear state space models
continuous-time ସିଷ୍ଟମ ଗୁଡ଼ିକ ପାଇଁ ସଂଜ୍ଞା
Barbalat's lemma and stability of time-varying systems