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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.
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.
The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution.
The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z ...
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
Stability by Liapunov's direct method. Academic Press. Haddad, W.M.; Chellaboina, VS (2008). Nonlinear Dynamical Systems and Control, a Lyapunov-based approach. Princeton University Press. ISBN 9780691133294. Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system ... which is based on the direct Lyapunov method with special Lyapunov ...
The ordinary Lyapunov function is used to test whether a dynamical system is (Lyapunov) stable or (more restrictively) asymptotically stable. Lyapunov stability means that if the system starts in a state x ≠ 0 {\displaystyle x\neq 0} in some domain D , then the state will remain in D for all time.