Nonlinear Control Sy.. - Free

58
It follows that
CHAPTER 2. MATHEMATICAL PRELIMINARIES
IFx - xollc= max IFx - xoll < 6(Lr+
tE[TT,t0+6]
and then, choosing 6 < means that F maps S into S.
Step (2):To show that F is a contraction on S, we consider x1, x2 E S and proceed as
follows:
II (Fx1)(t) - (Fx2)(t)II = t t
o,[
f(x1(T), T) - f(x2(T), T) ] dr
J t 11
to
t
f (x1(T ), 1) - f (x2 (T ), T) 11 dT
< L Ilxl(T) - x2(T)II d7
to
t
< LIlx1-x2IIc ft dT
IIFxi - Fx211c < L6II xl - x211, < PIIx1 - x211c for 6 < L
Choosing p < 1 and 6 < i , we conclude that F is a construction. This implies that there
is a unique solution of the nonlinear equation (2.30) in S.
Step (3): Given that S C X, to complete the proof, we must show that any solution of
(2.30) in X must lie in S. Starting at xo at to, a solution can leave S if and only if at some
t = t1, x(t) crosses the border B for the first time. For this to be the case we must have
Thus, for all t < tl we have that
It follows that
Ilx(tl)-xoll=r.
IIx(t) -x011 < ft[ IIf(x(r),r) -f(xo,T)II + Ilf(xo,T)II J dT
o
< j[L(Ilx(r)-xoII)+]dr
t
< j[Lr+]dr.
o
r = 11x(ti) - xo II < (Lr + .) (tl - to).
o