Nonlinear Control Sy.. - Free

4.2. POSITIVE DEFINITE FUNCTIONS 111
uniformly on t. Equivalently, W(., ) is radially unbounded if given M, AN > 0 such that
for all t, provided that Ixjj > N.
W (x, t) > M
Remarks: Consider now function W (x, t). By Definition 4.5, W (, ) is positive definite in
D if and only if 3V1 (x) such that
Vi (x) < W (x, t) , Vx E D (4.10)
and by Lemma 3.1 this implies the existence of such that
al(IjxII) < Vl(x) < W(x,t) , Vx E Br C D. (4.11)
If in addition is decrescent, then, according to Definition 4.6 there exists V2:
W (x, t) < V2(x) , Vx E D (4.12)
and by Lemma 3.1 this implies the existence of such that
W(x,t) < V2(x) < 012(IIxII) , Vx E Br C D. (4.13)
It follows that is positive definite and decrescent if and only if there exist a (timeinvariant)
positive definite functions and V2(.), such that
Vi(x) < W(x,t) < V2 (x) , Vx E D (4.14)
which in turn implies the existence of function and E IC such that
al(IIxII) < W(x,t) < a2(1IxII) , Vx E Br C D. (4.15)
Finally, is positive definite and decrescent and radially unbounded if and only if
al(.) and a2() can be chosen in the class ]Cc,,,.
4.2.1 Examples
In the following examples we assume that x = [x1, x2]T and study several functions W (x, t).
Example 4.1 Let Wl (x, t) _ (x1 + x2)e ,t a > 0. This function satisfies
(i) Wl (0, t) = 0 e"t = 0