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38 Introduction to Control Theory

• If x e is an equilibrium point at time t 0 of the non-autonomous system
(2.4.1), then x e is an equilibrium point of (2.4.1) for all t 1 ≥t 0 .

EXAMPLE 2.4–3: Equilibrium Point of Nonautonomous System

Consider the system

which is non-autonomous. It has NO EQUILIBRIUM POINTS. Although it
might seem that it has 2 equilibrium points x e1=-1 and x e2=1 at time t 0=1.
However, these are not equilibrium points for times since at times t≥1 the
conditions of equilibrium does not hold.

Some books also used the terms stationary or singular points to denote
equilibrium points.

EXAMPLE 2.4–4: Damped Pendulum

Recall the pendulum in Example 2.3.1 and let us try to find its equilibrium
points. Note first that the system is autonomous so that we do not need to
specify the particular time t 0 and then note that the pendulum is at equilibrium
if both

x 2=0 and sin(x 1)=0

Therefore the equilibrium points are at

It is obvious that the pendulum is at equilibrium when it is hanging straight
up or straight down with zero velocity.

DEFINITION 2.4–3 An equilibrium point x e at t 0 of (2.4.1) is said to be
isolated if there is a neighborhood N of x e which contains no other equilibrium
points besides x e.

Copyright © 2004 by Marcel Dekker, Inc.