(2) fl(t, x) satisfies the criterion
sup lif1(t,x)ii < Ltixii , L > 0 tt
0
(4-10)
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,t ) be the state transition matrix for the system (4-8). Consequently, we can write the solution to (4-6) as
x(t) = t(t,t0)X0 + f t(t, T)f1(T, x(T))dT
(4-12)
By taking the norm of both sides of (4-12) and using (4-9) it easily follows that
Iix(t) ii < me-a(t-to)x11 + f mea(t)L Ix(T) ItdT
(4-13)
Multiplying through by eat gives
ato t
eatilx(t)ii 4 me tox0 + mL f eaT ilx(T)IdT to
(4-14)
We may now apply the Bellman-Gronwall inequality (see [17]) to obtain