51
(2) f^(t, x) satisfies the criterion
sup Ilf. (t,x)n < LlixII L > 0 (4-10)
t>t
o
Then if we have L suitably small so that
(mL a) < 0, (4-11)
the system (4-6) is exponentially asymptotically stable for the
equilibrium point x = 0.
Proof: Let $(t,tQ) be the state transition matrix for the system
(4-8). Consequently, we can write the solution to (4-6) as
t
x(t) = *(t,t )x + / *(t, t)f,(t, x(t))dx (4-12)
*o
By taking the norm of both sides of (4-12) and using (4-9) it easily
follows that
IIx(t) II < me"a^"^o^ llxQII + / mea^"x^L IIx(x) IIdr (4-13)
Multiplying through by e gives
. at t
eaT* iix(t) il < me 0 llxQ ll + mL / eaT IIx(x) lldx
t_
(4-14)
We may now apply the Bellman-Gronwall inequality (see [17]) to obtain