Lyapunov stability

The primary result of the theory is the concept of Lyapunov stability.

Researchers applying mathematical models from system dynamics usually use Lyapunov stability.

Lyapunov stability is typically used to derive control adaptation laws and show convergence.

Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems.

They are related to some concept of stability in the dynamical systems sense, often Lyapunov stability.

See Lyapunov stability, which gives a definition of asymptotic stability for more general dynamical systems.

The stability of a general dynamical system with no input can be described with Lyapunov stability criteria.

An alternative approach to determining stability, which does not involve calculating eigenvalues, is to analyze the system's Lyapunov stability.

To address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability.

Lyapunov stability (i.e., asymptotic stability)

Unlike Lyapunov stability, which considers perturbations of initial conditions for a fixed system, structural stability deals with perturbations of the system itself.

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.

Lyapunov stability is named after Aleksandr Lyapunov, a Russian mathematician who published his book "The General Problem of Stability of Motion" in 1892.

Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a distance from it) remain "close enough" forever (within a distance from it).

It uses the fact that positive recurrent Markov chains exhibit a notion of "Lyapunov stability" in terms of returning to any state while starting from it within a finite time interval.

Stability for nonlinear systems that take an input is input-to-state stability (ISS), which combines Lyapunov stability and a notion similar to BIBO stability.

This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

More precisely, if all eigenvalues are negative real numbers or complex numbers with negative real parts then the point is a stable attracting fixed point, and the nearby points converge to it at an exponential rate, cf Lyapunov stability and exponential stability.

Topics include Lyapunov stability theory, partial stability, Lagrange stability, boundedness, ultimate boundedness, input-to-state stability, input-output stability, finite-time stability, semistability, stability of sets, stability of periodic orbits, and stability theorems via vector Lyapunov functions.