Nonlinear Control Sy.. - Free

2.12. EXERCISES 59
Denoting t1 = to + p, we have that if u is such that
r
µ Lr+
then the solution x(t) is confined to B. This completes the proof.
It is important to notice that Theorem 2.13 provides a sufficient but not necessary
condition for the existence and uniqueness of the solution of the differential equation (2.30).
According to the theorem, the solution is guaranteed to exist only locally, i.e. in the interval
[to, to+s]. For completeness, below we state (but do not prove) a slightly modified version of
this theorem that provides a condition for the global existence and uniqueness of the solution
of the same differential equation. The price paid for this generalization is a stronger and
much more conservative condition imposed on the differential equation.
Theorem 2.14 (Global Existence and Uniqueness) Consider again the nonlinear differential
equation (2.30) and assume that f (x, t) is piecewise continuous in t and satisfies
Il f (x1, t) - f (x2, t) II