Locally asymptotically stable definition
system is locally asymptotically stable (L. A. S. ) near or at xe if there is an R 0 s. t. kx(0)x e k R x(t) x e as t Basic Lyapunov theory 122dulum is damped, the stable equilibrium point is locally asymptotically stable. By shifting the origin of the system, we may assume that the equilibrium point of interest occurs at x 0. If multiple equilibrium points exist, we will need to study the stability of each by appropriately shifting the origin. 43 locally asymptotically stable definition
What would be a physical analogy where given system is asymptotically stable . A pendulum without friction is stable at its lowest point. A pendulum with friction is asymptotically stable at its lowest point.
Locally (uniformly) stable: if V(y, t) is lpdf and V'(y, t)0 locally in x and for all t. Locally (uniformly) asymptotically stable: if V(y, t) is lpdf and decrescent and V'(y, t) is lpdf. The definitions of lpdf and decresent are available in the notes and involve the identification of suitable alpha functions (see here for a related faq). is asymptotically stable (in fact, exponentially stable) if the joint spectral radius of the set, , is smaller than one. Stability for systems with inputs [ edit A system with inputs (or controls) has the formlocally asymptotically stable definition Definition [Ref. 1 [Asymptotic Stability and Uniform Asymptotic Stability The equilibrium state 0 of (1) is (locally) asymptotically stable if 1. It is stable in the sense of Lyapunov and 2. There exists a (to) such that, if xt xt t (), , ()o then as0.