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2.6 Stability Theory 53

x e is convergent, if it is convergent for any t 0 . See Figure 2.6.2 for
illustration.

Figure 2.6.2: Convergence of x e at t 0

4. Asymptotic Stability: x e is asymptotically stable (AS) at t 0 if states
starting sufficiently close to x e will stay close and will eventually
converge to it. More precisely, x e is AS at t 0 if it is both convergent
and stable at t 0 . x e is AS if it is AS for any t 0 . An illustration of an AS
equilibrium point is shown in Figure 2.6.3.

Figure 2.6.3: Asymptotic stability of x e at t 0

5. Global Asymptotic Stability: x e is globally asymptotically stable
(GAS) at t 0 if any initial state will stay close to x e and will eventually
converge to it. In other words, x e is GAS if it is stable at t 0, and if
every x(t) converges to x e as time goes to infinity. x e is GAS if it is

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