Nonlinear Control Sy.. - Free

2.11. SOLUTION OF DIFFERENTIAL EQUATIONS 57
existence of a fixed point of this map, in turn, can be verified by the contraction mapping
theorem. To start with, we define the sets X and the S C X as follows:
X = C[to, to + 8]
thus X is the set of all continuous functions defined on the interval [to,to+a]. Given x E X,
we denote
Ilxlic = 1X(t)11 (2.33)
tE m, x
and finally
S={xEX:IIx-xoMIc S. In step (2) we show that F is a contraction from S into S.
This, in turn implies that there exists one and only one fixed point x = Fx E S, and thus a
unique solution in S. Because we are interested in solutions of (2.30) in X (not only in S).
The final step, step (3), consists of showing that any possible solution of (2.30) in X must
be in S.
Step (1): From (2.31), we obtain
t
(Fx)(t) - xo = f (r, x(r)) d7-
fo
The function f is bounded on [to, tl] (since it is piecewise continuous). It follows that we
can find such that
tmaxjf(xo,t)jI
IIFx-xoll < j[Mf(x(r)r)_f(xor)+ t
If(xo,r)I] dr
o
f[L(Ix(r)
o
and since for each x E S, IIx - xo II < r we have
-xoll)+C]dr
t
IFx - xoll < f[Lr+]dr
< (t - to) [Lr + ].