The most important type is that concerning the stability of solutions near to a point of equilibrium.
In simple terms, if the solutions that start out near an equilibrium point
The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge.
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.
Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.
Lyapunov stability is named after Aleksandr Mikhailovich Lyapunov, a Russian mathematician who defended the thesis The General Problem of Stability of Motion at Kharkov University in 1892.
[1] A. M. Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium.
His work, initially published in Russian and then translated to French, received little attention for many years.
The mathematical theory of stability of motion, founded by A. M. Lyapunov, considerably anticipated the time for its implementation in science and technology.
He did not have doctoral students who followed the research in the field of stability and his own destiny was terribly tragic because of his suicide in 1918 [citation needed].
For several decades the theory of stability sank into complete oblivion.
The Russian-Soviet mathematician and mechanician Nikolay Gur'yevich Chetaev working at the Kazan Aviation Institute in the 1930s was the first who realized the incredible magnitude of the discovery made by A. M. Lyapunov.
The contribution to the theory made by N. G. Chetaev[2] was so significant that many mathematicians, physicists and engineers consider him Lyapunov's direct successor and the next-in-line scientific descendant in the creation and development of the mathematical theory of stability.
The interest in it suddenly skyrocketed during the Cold War period when the so-called "Second Method of Lyapunov" (see below) was found to be applicable to the stability of aerospace guidance systems which typically contain strong nonlinearities not treatable by other methods.
A large number of publications appeared then and since in the control and systems literature.
[3][4][5][6][7] More recently the concept of the Lyapunov exponent (related to Lyapunov's First Method of discussing stability) has received wide interest in connection with chaos theory.
Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems.
(There are examples showing that attractivity does not imply asymptotic stability.
), one can formulate similar definitions of stability near an arbitrary solution
However, one can reduce the more general case to that of an equilibrium by a change of variables called a "system of deviations".
, obeying the differential equation: This is no longer an autonomous system, but it has a guaranteed equilibrium point at
Lyapunov, in his original 1892 work, proposed two methods for demonstrating stability.
[1] The first method developed the solution in a series which was then proved convergent within limits.
The second method, which is now referred to as the Lyapunov stability criterion or the Direct Method, makes use of a Lyapunov function V(x) which has an analogy to the potential function of classical dynamics.
An additional condition called "properness" or "radial unboundedness" is required in order to conclude global stability.
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.
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.
The definition below provides this, using an alternate language commonly used in more mathematical texts.
[14] Instead, Barbalat's lemma allows for Lyapunov-like analysis of these non-autonomous systems.
Assuming f is a function of time only: Barbalat's Lemma says: An alternative version is as follows: In the following form the Lemma is true also in the vector valued case: The following example is taken from page 125 of Slotine and Li's book Applied Nonlinear Control.