Nonlinear Control Sy.. - Free

52 CHAPTER 2. MATHEMATICAL PRELIMINARIES
Property 2.1 (Chain Rule) Let fl : D1 C R" -+ R", and f2 : D2 C R" -+ IR". If f2 is
differentiable at a E D2 and fl is differentiable at f2(a), then fl o f2 is differentiable at a,
and
D(fl o f2)(a) = DF1(f2(a))Df2(a).
Theorem 2.7 (Mean-Value Theorem) Let f : [a, b] -> R be continuous on the closed
interval [a, b] and differentiable in the open interval (a, b). Then there exists a point c in
(a, b) such that
f(b) - f(a) = f'(c) (b - a).
A useful extension of this result to functions f : R" -+ R' is given below. In the following
theorem S2 represents an open subset of lR'.
Theorem 2.8 Consider the function f : 0 C R" --> R'" and suppose that the open set w
contains the points a, and b and the line segment S joining these points, and assume that
f is differentiable at every point of S. The there exists a point c on S such that
If (b) - f (a)lb = Ilf'(c)(b -
Theorem 2.9 (Inverse Function Theorem) Let f : Rn -> R" be continuously differentiable
in an open set D containing the point xo E R", and let f'(xo) # 0. Then there exist an open
set Uo containing xo and an open set Wo containing f(xo) such that f : Uo -+ Wo has a
continuous inverse f -1 : WO - Uo that is differentiable and for all y = f (x) E WO satisfies
2.9 Lipschitz Continuity
a) 11.
Df-1(y) = [Df(x)] 1 = [Df-1(y)]-1.
We defined continuous functions earlier. We now introduce a stronger form of continuity,
known as Lipschitz continuity. As will be seen in Section 2.10, this property will play a
major role in the study of the solution of differential equations.
Definition 2.24 A function f (x) : Rn - lRm is said to be locally Lipschitz on D if every
point of D has a neighborhood Do C D over which the restriction of f with domain D1
satisfies
If(XI) - f (X2)11 < LJJx1 - x2j1.
(2.19)
It is said to be Lipschitz on an open set D C R" if it satisfies (2.19) for all x1i x2 E D with
the same Lipschitz constant. Finally, f is said to be globally Lipschitz if it satisfies (2.19)
with D = Rn.