Nonlinear Control Sy.. - Free

108 CHAPTER 4. LYAPUNOV STABILITY H. NONAUTONOMOUS SYSTEMS
xe = 0 can be a translation of a nonzero trajectory. Indeed, consider the nonautonomous
system
th = f(x,t) (4.2)
and assume that x(t) is a trajectory, or a solution of the differential equation (4.2) for t > 0.
Consider the change of variable
y = X(t) - x(t).
We have
But i = f (x(t), t). Thus
f (x, t) - x(t)
def
= g(y, t)
g(y,t) = f(y + x(t),t) - f(x(t),t)
0 if y=0
that is, the origin y = 0 is an equilibrium point of the new system y = g(y, t) at t = 0.
Definition 4.1 The equilibrium point x = 0 of the system (4.1) is said to be
Stable at to if given c > 0, 36 = b(e, to) > 0 :
11x(0)11 < S = 11x(t)II < c Vt > to > 0
Convergent at to if there exists 51 = 51(to) > 0 :
11x(0)11 < Si = slim x(t) = 0.
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Equivalently (and more precisely), xo is convergent at to if for any given cl > 0, 3T =
T(elito) such that
1x(0)11 < Si = Ix(t)11 < El dt > to +T (4.5)
Asymptotically stable at to if it is both stable and convergent.
Unstable if it is not stable.