Nonlinear Control Sy.. - Free

116 CHAPTER 4. LYAPUNOV STABILITY II: NONAUTONOMOUS SYSTEMS
Proof of theorem 4.1: Choose R > 0 such that the closed ball
BR = {xER':IIxII to
al(IIxII) < W(x(t),t) < W(xo,to).
Also W is continuous with respect to x and satisfies W(0, to) = 0. Thus, given to, we can
find 6 > 0 such that
11x011 < 6 = W(xo, to) < a(R)
which means that if 1xoHI < 6, then
ai(I1xII) < a(R) => Ix(t)II < R Vt > to.
This proves stability. If in addition W (x, t) is decreasing, then there exists a positive
function V2(x) such that
I W(x, t)1 < V2(x)
and then E3a2 in the class K such that
ai(IIxH) < V1(x) < W(x, t) < V2(x) < a2(11xI1) Vx E BR, Vt > to.
By the properties of the function in the class K , for any R > 0, 3b = f (R) such that
a2(6) < a, (R)
=> ai(R) > a2(a) > W(xo,to) > W(x,t) > a(1Ix(t)II)
which implies that
11x(t)II < R `dt > to.
However, this 6 is a function of R alone and not of to as before. Thus, we conclude that the
stability of the equilibrium is uniform. This completes the proof.
Proof of theorem 4.2: Choose R > 0 such that the closed ball
BR={XE]R":11x11