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The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [ 1 ] [ 2 ] In particular, the discrete-time Lyapunov equation (also known as Stein equation ) for X {\displaystyle X} is
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory .
This example shows a system where a Lyapunov function can be used to prove Lyapunov stability but cannot show asymptotic stability. Consider the following equation, based on the Van der Pol oscillator equation with the friction term changed:
[1] [2] [3] Unlike the classical CLT, which requires that the random variables in question have finite variance and be both independent and identically distributed, Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg. [4]
Lyapunov functions are used extensively in control theory to ensure different forms of system stability. The state of a system at a particular time is often described by a multi-dimensional vector. A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state.
Note that in the formula above is assumed to be only positive definite. It can be easily proved, [ 13 ] that if V {\displaystyle V} is an iISS-Lyapunov function with α ∈ K ∞ {\displaystyle \alpha \in {\mathcal {K}}_{\infty }} , then V {\displaystyle V} is actually an ISS-Lyapunov function for a system ( 1 ).
The exact limit values of finite-time Lyapunov exponents, if they exist and are the same for all , are called the absolute ones [3] {+ (,)} = {()} {} and used in the Kaplan–Yorke formula. Examples of the rigorous use of the ergodic theory for the computation of the Lyapunov exponents and dimension can be found in. [11] [12] [13]
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.