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  2. Lyapunov function - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_function

    A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).

  3. Lyapunov exponent - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_exponent

    Lyapunov proved that if the system of the first approximation is regular (e.g., all systems with constant and periodic coefficients are regular) and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically Lyapunov stable. Later, it was stated by O. Perron that the requirement of regularity of the ...

  4. Lyapunov stability - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_stability

    The definition for discrete-time systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts. Let (X, d) be a metric space and f : X → X a continuous function. A point x in X is said to be Lyapunov stable, if,

  5. Lyapunov equation - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_equation

    In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is A X A H − X + Q = 0 {\displaystyle AXA^{H}-X+Q=0} where Q {\displaystyle Q} is a Hermitian matrix and A H {\displaystyle A^{H}} is the conjugate transpose of A {\displaystyle A} , while the continuous-time Lyapunov equation is

  6. Floquet theory - Wikipedia

    en.wikipedia.org/wiki/Floquet_theory

    The real parts of the Floquet exponents are called Lyapunov exponents. The zero solution is asymptotically stable if all Lyapunov exponents are negative, Lyapunov stable if the Lyapunov exponents are nonpositive and unstable otherwise. Floquet theory is very important for the study of dynamical systems, such as the Mathieu equation.

  7. Lyapunov optimization - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_optimization

    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.

  8. Lyapunov fractal - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_fractal

    A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent) in the a−b plane for given periodic sequences of a and b. In the images, yellow corresponds to λ < 0 {\displaystyle \lambda <0} (stability), and blue corresponds to λ > 0 {\displaystyle \lambda >0} (chaos).

  9. Control-Lyapunov function - Wikipedia

    en.wikipedia.org/wiki/Control-Lyapunov_function

    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.

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