Nonlinear Control Sy.. - Free

158 CHAPTER 6. INPUT-OUTPUT STABILITY
Notice that according to definition 6.3, PT satisfies
(1) [PT(u + v)J(t) = UT(t) + VT(t) Vu, V E Xe .
(ii) [PT(au)](t) = auT(t) Vu E Xe , a E R.
Thus, the truncation operator is a linear operator.
Definition 6.4 The extension of the space X, denoted Xe is defined as follows:
Xe = {u : ]R+ - IR9, such that XT E X VT E R+}. (6.7)
In other words, X. is the space consisting of all functions whose truncation belongs to X,
regardless of whether u itself belongs to X. In the sequel, the space X is referred to as the
"parent" space of Xe. We will assume that the space X satisfy the following properties:
(i) X is a normed linear space of piecewise continuous functions of the form u : IR+ -> IR9.
The norm of functions in the space X will be denoted II - IIx
(ii) X is such that if u E X, then UT E X VT E IR+, and moreover, X is such that
u = limTc,, UT. Equivalently, X is closed under the family of projections {PT}.
(iii) If u E X and T E R+, then IIUTIIX 5 IIxIIx ; that is, IIxTIIx
function of T E IR+.
(iv) If u E Xe, then u E X if and only if limT-+,, IIxTIIx < oo.
is a nondecreasing
It can be easily seen that all the Lp spaces satisfy these properties. Notice that although
X is a normed space, Xe is a linear (not normed) space. It is not normed because in
general, the norm of a function u E Xe is not defined. Given a function u E Xe , however,
using property (iv) above, it is possible to check whether u E X by studying the limit
limT,,,. IIUTII
Example 6.2 Let the space of functions X be defined by
X = {x x : 1R -+ 1R x(t) integrable and Jx x(t) dt < o0
In other words, X is the space of real-valued function in Li. Consider the function x(t) = t.
We have
XT(t) =
IIXTII = J
f t, 0 < t < T
1f0, t>T
T
I XT(t) I dt = J T t dt = 22