Nonlinear Control Sy.. - Free

6.1. FUNCTION SPACES 157
Both L2 and L,,. are special cases of the so-called LP spaces. Given p : 1 < p < oo, the
space LP consists of all piecewise continuous functions u : IIt+ -4 Rq satisfying
00 1/P
IIUIIcp f (f (f[Iu1I'
+ IU2IP + ... + IugIP] dt) < oo. (6.3)
Another useful space is the so-called L1. From (6.3), L1 is the space of all piecewise
continuous functions u : pg+ -4 Rq satisfying:
IIUIIc1 f (jiuii + IU2I + + jugI] dt) < oo. (6.4)
Property: (Holder's inequality in Lp spaces). If p and q are such that 1 + 1 = 1 with
1 < p < oo and if f E LP and g E Lq, then f g E L1, and
I(f9)TII1=
T
f f(t)9(t)dt < (fT f(t)Idt) (fT I9(t)I
o
dt) . (6.5)
For the most part, we will focus our attention on the space L2, with occasional
reference to the space L. However, most of the stability theorems that we will encounter
in the sequel, as well as all the stability definitions, are valid in a much more general setting.
To add generality to our presentation, we will state all of our definitions and most of the
main theorems referring to a generic space of functions, denoted by X.
6.1.1 Extended Spaces
We are now in a position to introduce the notion of extended spaces.
Definition 6.3 Let u E X. We define the truncation operator PT : X -> X by
def
(PTU)(t) = UT(t) =
u(t), t < T
0, t > T
t, T E R+ (6.6)
Example 6.1 Consider the function u : [0, oo) -+ [0, oo) defined by u(t) = t2. The truncation
of u(t) is the following function:
_ t2, 0 < t < T
uT(t) 0, t > T