Nonlinear Control Sy.. - Free

156 CHAPTER 6. INPUT-OUTPUT STABILITY
sufficiently rich to contain all input functions of interest as well as the corresponding outputs.
Mathematically this is a challenging problem since we would like to be able to consider
systems that are not well behaved, that is, where the output to an input in the space
X may not belong to X. The classical solution to this dilemma consists of making use of
the so-called extended spaces, which we now introduce.
6.1 Function Spaces
In Chapter 2 we introduced the notion of vector space, and so far our interest has been
limited to the nth-dimensional space ]R". In this chapter we need to consider "function
spaces," specifically, spaces where the "vectors," or "elements," of the space are functions
of time. By far, the most important spaces of this kind in control applications are the
so-called Lp spaces which we now introduce.
In the following definition, we consider a function u : pp+ --> R q, i.e., u is of the form:
ui (t)
u(t) = u2(t)
uq(t)
Definition 6.1 (The Space ,C2) The space .C2 consists of all piecewise continuous functions
u : 1R -> ]R9 satisfying
IIuIIc2 =f f -[Iu1I2 + Iu2I2 +... + Iu9I2] dt < oo. (6.1)
The norm IIuII1C2 defined in this equation is the so-called C2 norm of the function u.
Definition 6.2 (The Space Coo) The space Goo consists of all piecewise continuous functions
u : pt+ -1 1R4 satisfying
IIuIIc- dcf sup IIu(t)II. < oo. (6.2)
tEIR+
The reader should not confuse the two different norms used in equation (6.2). Indeed, the
norm IIuIIc_ is the ,Coonorm of the function u, whereas IIu(t)lloo represents the infinity norm
of the vector u(t) in 1R4, defined in section 2.3.1. In other words
IIuIIcm 4f sup (max Iuil) < oo 1 < i < q.
i
tE]R+