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  2. Lyapunov stability - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_stability

    More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis). The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge.

  3. Stability theory - Wikipedia

    en.wikipedia.org/wiki/Stability_theory

    The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand ...

  4. Exponential stability - Wikipedia

    en.wikipedia.org/wiki/Exponential_stability

    An exponentially stable LTI system is one that will not "blow up" (i.e., give an unbounded output) when given a finite input or non-zero initial condition. Moreover, if the system is given a fixed, finite input (i.e., a step ), then any resulting oscillations in the output will decay at an exponential rate , and the output will tend ...

  5. 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).

  6. Marginal stability - Wikipedia

    en.wikipedia.org/wiki/Marginal_stability

    In the theory of dynamical systems and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable.Roughly speaking, a system is stable if it always returns to and stays near a particular state (called the steady state), and is unstable if it goes further and further away from any state, without being bounded.

  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. Input-to-state stability - Wikipedia

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

    System is called globally asymptotically stable at zero (0-GAS) if the corresponding system with zero input ...

  9. Asymptotic analysis - Wikipedia

    en.wikipedia.org/wiki/Asymptotic_analysis

    The function f(n) is said to be "asymptotically equivalent to n 2, as n → ∞". This is often written symbolically as f (n) ~ n 2, which is read as "f(n) is asymptotic to n 2". An example of an important asymptotic result is the prime number theorem.