Nonlinear Control Sy.. - Free

4.4. PROOF OF THE STABILITY THEOREMS 117
is contained in D. By the assumptions of the theorem there exist class IC functions al, a2,
and a3 satisfying
a1(IIxII) < W(x,t) < a2(IIXII) Vt,Vx E BR (4.19)
a3(114) < -W(x,t) Vt,Vx E BR. (4.20)
Given that the at's are strictly increasing and satisfy a,(0) = 0, i = 1, 2, 3, given e > 0, we
can find 5i, b2 > 0 such that
x2(81) < al(R)
a2(52) < min[a1(c),a2(b1)]
(4.21)
(4.22)
where we notice that b1 and b2 are functions of e but are independent of to. Notice also
that inequality (4.22) implies that 61 > 62. Now define
Conjecture: We claim that
T = C" (R)
a3(a2)
11x011 < 61 = lx(t')II < b2
for some t = t` in the interval to < t' < to + T.
(4.23)
To see this, we reason by contradiction and assume that 1xo11 < 61 but llx(t`)f1 > b2
for all t in the interval to < t' < to + T. We have that
0 < a,(52) al is class IC
al(b2) b2. Also, since W is a
decreasing function of t this implies that
But
0< al(a2) < W(x(to +T), to +T)
to+T
< W(x(to), to) + J W(x(t), t) dt
to
< W(x(to),to) -Ta3(d2) from (4.20) and (4.23).
W(x(to),to) < a2(11x(to)11) a2(5i), since 11xoll < 61, by assumption.
by (4.19)