Nonlinear Control Sy.. - Free

Chapter 4
Lyapunov Stability II:
Nonautonomous Systems
In this chapter we extend the results of Chapter 3 to nonautonomous system. We start by
reviewing the several notions of stability. In this case, the initial time instant to warrants
special attention. This issue will originate several technicalities as well as the notion of
uniform stability, to be defined.
4.1 Definitions
We now extend the several notions of stability from Chapter 3, to nonautonomous systems.
For simplicity, in this chapter we state all our definitions and theorems assuming that the
equilibrium point of interest is the origin, xe = 0.
Consider the nonautonomous systems
x=f(x,t) f:DxIIF+->1R' (4.1)
where f : D x [0, oo) -a IR is locally Lipschitz in x and piecewise continuous in t on
D x [0, oc). We will say that the origin x = 0 E D is an equilibrium point of (4.1) at t = to
if
f(0,t) = 0 dt > to.
For autonomous systems equilibrium points are the real roots of the equation f (xe) = 0.
Visualizing equilibrium points for nonautonomous systems is not as simple. In general,
'Notice that, as in Chapter 3, (4.1) represents an unforced system.
107