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If ˙ is negative definite, then the global asymptotic stability of the origin is a consequence of Lyapunov's second theorem. The invariance principle gives a criterion for asymptotic stability in the case when V ˙ ( x ) {\displaystyle {\dot {V}}(\mathbf {x} )} is only negative semidefinite.
An additional condition called "properness" or "radial unboundedness" is required in order to conclude global stability. Global asymptotic stability (GAS) follows similarly. It is easier to visualize this method of analysis by thinking of a physical system (e.g. vibrating spring and mass) and considering the energy of such a system. If the ...
A discrete-time input-to-output LTI system is exponentially stable if and only if the poles of its transfer function lie strictly within the unit circle centered on the origin of the complex plane. Systems that are not LTI are exponentially stable if their convergence is bounded by exponential decay .
A telecommunications device for the deaf (TDD) is a teleprinter, an electronic device for text communication over a telephone line, that is designed for use by persons with hearing or speech difficulties. Other names for the device include teletypewriter (TTY), textphone (common in Europe), and minicom (United Kingdom).
In physics and other fields of science, one frequently comes across problems of an asymptotic nature, such as damping, orbiting, stabilization of a perturbed motion, etc. Their solutions lend themselves to asymptotic analysis ( perturbation theory ), which is widely used in modern applied mathematics , mechanics and physics .
A hearing loop consists of one or more physical loops of cable which are placed around a designated area, usually a room or a building. The cable generates an electromagnetic field throughout the looped space which can be picked up by a telecoil-equipped hearing aid, a cochlear implant (CI) processor, or a specialized hand-held hearing loop receiver for individuals without telecoil-compatible ...
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation , for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature ...
To prove asymptotic stability, either a new Lyapunov candidate function needs to be considered, or LaSalle's invariance principal needs to be applied. I agree. To use a Lyapunov function itself to prove asymptotic stability (without resorting to the invariance principle or Barbalat's Lemma), you have to choose a different function.