Nonlinear Control Sy.. - Free

310 APPENDIX A. PROOFS
For the converse, assume that (A.3) is satisfied. Given el, let s C 1R+ be the set defined as
follows:
A = 161 E 1R+ : IIx(0)II < bl = 11x(t)II < ei}.
This set is bounded above and therefore has a supremum. Let 8' = sup(s). Thus,
IIx(0)II < a' = 11x(t)11 < el Vt > to.
The mapping 4): el -* 5*(el) satisfies the following: (i) 4i(O) = 0, (ii) P(e1) > 0 de > 0, and
(iii) it is nondecreasing, but it is not necessarily continuous. We can find a class 1C function
that satisfies fi(r) _< 4i(r), for each r E
K is also in the same class, we now define:
1R+. Since the inverse of a function in the class
and a(.) E 1C .
Given e, choose any x(0) such that a(IIx(0)II) = e. Thus IIx(0)II < b', and we have
11x(t)II < e = a(IIx(0)II) provided that IIx(0)II < P.
Lemma 3.3 The equilibrium xe = 0 of the system (3.1) is asymptotically stable if and only
if there exists a class 1CG function and a constant a such that
Proof: Suppose that A.4 is satisfied. Then
IIxoII < 8 = Ix(t)II 0. (A.4)
IIx(t)II < 3(Ilxo11, 0) whenever IIxoII < b
and the origin is stable by Lemma 3.2. Moreover, ,3 is in class /CC which implies that
,3(IIxoII,t) -* 0 as t --> oo. Thus, IIx(t)II -* 0 as t -4 oo and x = 0 is also convergent. It
then follows that x = 0 is asymptotically stable. The rest of the proof (i.e., the converse
argument) requires a tedious constructions of a 1CL function Q and is omitted. The reader
can consult Reference [41] or [88] for a complete proof.
Theorem 3.5 A function g(x) is the gradient of a scalar function V(x) if and only if the
matrix j19
xlxlxl
L9 0 09
8xz 8x, 8x2
19 L9
9X Rn e