Consider a nonlinear system of the following form
where 
, and
and 
are continuous functions with f(0,0)=0 and h(0)=0.
Since we are interested in whether a system can be stabilized by
switching between different controllers we introduce a set of so
called basic controllers of the form
where 
is
continuous, 
, and 
is closed.
To determine when to use a basic controller we also introduce a
switching function I(x) which maps the set of state
measurements into the set of integers 
. Hence, the
switching function induces a partition of the state space with
different enumerated components. We use the following notation
for the control law obtained by choosing basic controller with respect to the switching function I. In the sequel we assume that (1) is controlled by the basic controllers (2) with respect to the switching function I. Such systems may arise in cases when the controller consists of a number of relay nonlinearities which can be used to change the structure of the system (1).
Notice that the right hand side of the closed loop system
is bounded as long as the state belongs to some compact subset
of 
. Hence, 
is bounded
on .
Sometimes it can be desired to implement a switching controller using some hysteresis elements, for example to avoid sliding modes. Observe that this kind of controllers cannot be described by a switching function since the hysteresis element introduce a discrete state. We will use the term switching rule to distinguish these cases.
Here we state a number of definitions and results that we need for investigating stability and stabilizability of switched systems, see for example [8, 9].
If the origin is asymptotically stable it has a region of
attraction, i.e., a set of initial states 
such that
for all 
.
Note that sometimes the term positively invariant is used to describe the above property. However, we do not adopt that terminology here.
The main problem we consider in this report is:
What are the conditions for existence and how is it possible to design a (globally) stabilizing switching controller (3)?Much of the stability analysis of smooth systems can be generalized to nonsmooth systems.
