Nonlinear Control Sy.. - Free

2.8. DIFFERENTIABILITY 51
Theorem 2.6 Suppose that f maps an open set D C IR' into IRm. Then f is continuously
differentiable in D if and only if the partial derivatives , 1 < i < m, 1 < j < n exist and
are continuous on D.
Because it is relatively easier to work with the partial derivatives of a function, Definition
2.23 is often restated by saying that
A function f : 1R' --> Rm is said to be continuously differentiable at a point
xo if the partial derivatives ', 1 < i < m, 1 < j < n exist and are continuous
at xo. A function f : IR" -- 1W is said to be continuously differentiable on a set
D C IR'`, if it is continuously differentiable at every point of D.
Summary: Given a function f : R" -i 1R' with continuous partial derivatives, abusing the
notation slightly, we shall use f'(x) or D f (x) to represent both the Jacobian matrix and
the total derivative of f at x. If f : IR' -+ R, then the Jacobian matrix is the row vector
of of of
f1(X) axl' ax2' ... axn
This vector is called the gradient of f because it identifies the direction of steepest ascent
of f. We will frequently denote this vector by either
ax
or V f (x)
of of
of
vf(x) = of = .
Ox axl'ax2' axn
It is easy to show that the set of functions f : IR'2 -+ IR' with continuous partial
derivatives, together with the operations of addition and scalar multiplication, form a vector
space. This vector space is denoted by Cl. In general, if a function f : R -> lR' has
continuous partial derivatives up to order k, the function is said to be in Ck, and we write
f E Ck. If a function f has continuous partial derivatives of any order, then it is said to be
smooth, and we write f E C. Abusing this terminology and notation slightly, f is said to
be sufficiently smooth where f has continuous partial derivatives of any required order.
2.8.1 Some Useful Theorems
We collect a number of well known results that are often useful.