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  2. Lyapunov equation - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_equation

    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

  3. Lyapunov stability - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_stability

    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:

  4. 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).

  5. Conley's fundamental theorem of dynamical systems - Wikipedia

    en.wikipedia.org/wiki/Conley's_fundamental...

    Conley's decomposition is characterized by a function known as complete Lyapunov function. Unlike traditional Lyapunov functions that are used to assert the stability of an equilibrium point (or a fixed point) and can be defined only on the basin of attraction of the corresponding attractor, complete Lyapunov functions must be defined on the whole phase-portrait.

  6. 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.

  7. 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.

  8. Lyapunov exponent - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_exponent

    In 1930 O. Perron constructed an example of a second-order system, where the first approximation has negative Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of the original nonlinear system is Lyapunov unstable. Furthermore, in a certain neighborhood of this zero solution almost all ...

  9. Input-to-state stability - Wikipedia

    en.wikipedia.org/wiki/Input-to-state_stability

    ISS unified the Lyapunov and input-output stability theories and revolutionized our view on stabilization of nonlinear systems, design of robust nonlinear observers, stability of nonlinear interconnected control systems, nonlinear detectability theory, and supervisory adaptive control. This made ISS the dominating stability paradigm in ...