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In this paper we consider discrete-time positive switched systems, switching among autonomous
subsystems, characterized either by monomial matrices or by circulant matrices. For these two classes of
systems, some interesting necessary and sufficient conditions for (global uniform) asymptotic stability
and stabilizability are provided. Such conditions lead to simple algorithms that allow to easily detect,
under suitable conditions, whether a given positive switched system is not stabilizable.

In this paper exponential stabilizability
of continuous-time positive switched systems is in-
vestigated. For two-dimensional systems, exponential
stabilizability by means of a switching control law can
be achieved if and only if there exists a Hurwitz convex
combination of the (Metzler) system matrices. In the
higher dimensional case, it is shown by means of an
example that the existence of a Hurwitz convex combi-
nation is only sufficient for exponential stabilizability,
and that such a combination can be found if and only
if there exists a smooth, positively homogeneous and
co-positive control Lyapunov function for the system.
In the general case, exponential stabilizability ensures
the existence of a concave, positively homogeneous and
co-positive control Lyapunov function, but this is not
always smooth. The results obtained in the first part
of the paper are exploited to characterize exponential
stabilizability of positive switched systems with delays,
and to provide a description of all the “switched equi-
librium points” of an affine positive switched system.

In this paper we consider the class of discretetime
switched systems switching between p autonomous positive
subsystems. First, sufficient conditions for testing stability, based
on the existence of special classes of common Lyapunov functions,
are investigated, and these conditions are mutually related, thus
proving that if a linear copositive common Lyapunov function can
be found, then a quadratic positive definite common function can
be found, too, and this latter, in turn, ensures the existence of a
quadratic copositive common function. Secondly, stabilizability
is introduced and characterized. It is shown that if these
systems are stabilizable, they can be stabilized by means of a
periodic switching sequence, which asymptotically drives to zero
every positive initial state. Conditions for the existence of statedependent
stabilizing switching laws, based on the values of a
copositive (linear/quadratic) Lyapunov function, are investigated
and mutually related, too.
Finally, some properties of the patterns of the stabilizing
switching sequences are investigated, and the relationship between
a sufficient condition for stabilizability (the existence of
a Schur convex combination of the subsystem matrices) and an
equivalent condition for stabilizability (the existence of a Schur
matrix product of the subsystem matrices) is explored.