Nonlinear Control Sy.. - Free

54 CHAPTER 2. MATHEMATICAL PRELIMINARIES
Definition 2.25 A function f (x, t) : lR' x R -> IRn is said to be locally Lipschitz in x on an
open set D x [to, T] C In x IR if every point of D has a neighborhood D1 C D over which the
restriction off with domain D1 x [to, T] satisfies (2.20). It is said to be locally Lipschitz on
D C 1Rn x (to, oo) if it is locally Lipschitz in x on every D1 x [to, T] C D x [to, oo). It is said
to be Lipschitz in x on D x [to, T] if it satisfies (2.20) for all x1, x2 E D and all t E [to, T.
Theorem 2.11 Let f : 1R' x IR -* IR' be continuously differentiable on D x [to, Tj and
assume that the derivative of f satisfies
of (x, t) < L (2.21)
on D x [to, T]. The f is Lipschitz continuous on D with constant L:
11f(x,t)_f(y,t)1I S has one and only one x E S
such that f (x) = x.
Proof: A point xo E X satisfying f (xo) = xo is called a fixed point. Thus, the contraction
mapping principle is sometimes called the fixed-point theorem. Let xo be an arbitrary point
in S, and denote x1 = f (xo), x2 = A X0, , xn+1 = f (xn). This construction defines a
sequence {xn} = { f (xn_ 1) }. Since f maps S into itself, xk E S Vk. We have
d(xn+1, xn) = d(f (xn), f (xn-1) d(xn, xn-1) (2.23)