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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.
This solution is asymptotically stable as t → ∞ ("in the future") if and only if for all eigenvalues λ of A, Re(λ) < 0. Similarly, it is asymptotically stable as t → −∞ ("in the past") if and only if for all eigenvalues λ of A, Re(λ) > 0. If there exists an eigenvalue λ of A with Re(λ) > 0 then the solution is unstable for t → ...
The darker more stable isotope region departs from the line of protons (Z) = neutrons (N), as the element number Z becomes larger. This is a list of chemical elements by the stability of their isotopes. Of the first 82 elements in the periodic table, 80 have isotopes considered to be stable. [1] Overall, there are 251 known stable isotopes in ...
System is called globally asymptotically stable at zero (0-GAS) if the corresponding system with zero input ...
The 251 known stable nuclides include tantalum-180m, since even though its decay is automatically implied by its being "metastable", this has not been observed. All "stable" isotopes (stable by observation, not theory) are the ground states of nuclei, except for tantalum-180m, which is a nuclear isomer or excited state. The ground state ...
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).
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.
If >, when ˙ hold only for in some neighborhood of the origin, and the set {˙ =}does not contain any trajectories of the system besides the trajectory () =,, then the local version of the invariance principle states that the origin is locally asymptotically stable.