(possibly nonlinear) continuous maps with $f_i(0)=0$, and $conv$ denotes the convex hull operation. The question of interest is (local or global) *asymptotic stability under arbitrary switching*. This means that we would like to know whether the origin is stable in the sense of Lyapunov (see [@Khalil:3rd.Ed] for a definition) and attracts all initial conditions (either in a neighborhood or globally) for *all* possible values that $\tilde{f}_k$ can take at each time step $k$.

The special case of this problem where the maps $f_1,\ldots,f_m$ are *linear* has been and continues to be the subject of intense study in the control community, as well as in the mathematics and computer science communities [@jsr-toolbox; @morris-ergodic; @cfbousch; @Shorten05stabilitycriteria; @jungers_lncis; @LeeD06; @bcv12; @liberzon-cdc10]. A switched linear system in this setting is given by $$\label{eq:switched.linear.system}
x_{k+1}\in conv\{A_ix_k\}, \quad i=1,\ldots,m,$$ where $A_1,\ldots,A_m$ are $m$ real $n\times n$ matrices. Local (or equivalently global) asymptotic stability under arbitrary switching of this system is equivalent to the *joint spectral radius* of these matrices being strictly less than one.

The [*joint spectral radius*]{} of a set of matrices ${{\cal{M}}}$ is defined as $$\label{eq:JSR}
\rho({{\cal{M}}})=\lim_{k\rightarrow \infty} \
\max_{A_1,\dots,A_k\in {{\cal{M}}}}||A_1\dots A_k||^{1/k},$$ where $\|\cdot\|$ is any matrix norm on $\mathbb{{{\mathbb R}}}^{n \times n}.$

Deciding 