Nonlinear Control Sy.. - Free

A.1. CHAPTER 3 309
Figure A.2: Asymptotically stable equilibrium point.
This proves that there exists E X such that
The existence of E 1C such that
can be proved similarly.
al(jlxlj) < V(x) for IxMI < r
V(x) < a2(lIXIl) for IxII < r
To simplify the proof of the following lemma, we assume that the equilibrium point
is the origin, xe = 0.
Lemma 3.2: The equilibrium xe of the system (3.1) is stable if and only if there exists a
class 1C function a constant a such that
IIx(0)II < 8 = IIx(t)II < a(IIx(0)II) Vt > 0. (A.2)
Proof: Suppose first that (A.2) is satisfied. We must show that this condition implies that
for each e1i361 = al(es) > 0:
Ix(0)II < 81 = IIx(t)II < el Vt > to. (A.3)
Given e1, choose 81 = min{8, a-'(fl) }. Thus, for 11x(0)I1 < 61i we have
Ilx(t)II C a(Ilx(O)II) < a(a-1(el)) = el.