Lyapunov methods - definition

Lapunov methods are a set of mathematical tools for analysing the stability of dynamic systems, developed by Alexander Lyapunov in the late 19th century. Their essence is to study the behaviour of a system near the point of equilibrium without finding its general solution. The core element of this method is the so-called Lyapunov function, an auxiliary function that acts as a measure of the energy of the system and allows to assess whether the trajectories of the system are approaching or moving away from the equilibrium point.

In the classical version, the method consists in finding a real function, positively determined in the surroundings of the equilibrium point, whose total derivative with respect to time (calculated along the solutions of the system) is negatively determined or not positive. Such a Lyapunov function proves asymptotic stability or stability in the Lyapunov sense, respectively. The advantage of this method is its generality - it can be applied to both linear systems, and non-linear, continuous and discrete systems.

In the context of engineering and Automation, Lyapunov methods provide a foundation for the design of stabilising, adaptive and non-linear control systems. They are used in the study of mechanical, electrical and biological systems, as well as in chaos theory and global dynamics analysis. Their practical implementation often requires the construction of appropriate Lyapunov functions on the basis of physical intuition or numerical procedures, including the use of symbolic optimisation, semi-definite programming and artificial intelligence tools.

Extensions of the classical method include time-dependent Lyapunov functions, methods with Lyapunov-Krasowski functions and generalisations to stochastic and distributed systems, among others. Due to their versatility and independence from the requirement to solve differential equations, Lyapunov methods are one of the fundamental tools for qualitative analysis in dynamical systems theory.

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