Nonlinear Control Sy.. - Free

112 CHAPTER 4. LYAPUNOV STABILITY II: NONAUTONOMOUS SYSTEMS
(ii) Wl(x,t)>O Vx540, VtE1R.
However, limt,, Wl (x, t) = 0 Vx. Thus, Wl is positive semi definite, but not positive
definite.
Example 4.2 Let
W(xt) =
(x1 + x2)(12 + 1)
V2(x)(t2 + 1),
V2(x) = ((x1
+2))
Thus, V2(x) > 0 Vx E 1R2 and moreover W2(x,t) > V2(x) Vx E ]R2, which implies that
is positive definite. Also
lim W2(x,t) = oo Vx E 1R2.
taco
Thus it is not possible to find a positive definite function V() such that IW2(x, t)I < V(x) Vx,
and thus W2(x,t) is not decrescent. IW2(x,t)l is not radially unbounded since it does not
tend to infinity along the xl axis.
Example 4.3 Let
W3(x, t) _ (x1+x2)(12 + 1)
V3(x)(12+1)
V3(x)deJ(xl+x2)
Following a procedure identical to that in Example 4.2 we have that W3(x) is positive definite,
radially unbounded and not decrescent.
Example 4.4 Let
2 2
W4(x t) _ (x 2+X2 ) (xl + 1) .
Thus, W4(., ) > OVx E 1R2 and is positive definite. It is not time-dependent, and so it is
decrescent. It is not radially unbounded since it does not tend to infinity along the xl axis.
Example 4.5 Let
W5(x t)
_ (x2 +x2)(12 + 1)
(t2 + 2)
= V5(x)(t + 2),
V5 (x) def (xl + x2)
2