  This paper presents novel stabilizability conditions for switched linear
systems with arbitrary and uncontrollable underlying switching signals. We
distinguish and study two particular settings: i) the \emph{robust} case, in
which the active mode is completely unknown and unobservable, and ii) the
\emph{mode-dependent} case, in which the controller depends on the current
active switching mode. The technical developments are based on graph-theory
tools, relying in particular on the path-complete Lyapunov functions framework.
The main idea is to use directed and labeled graphs to encode Lyapunov
inequalities to design robust and mode-dependent piecewise linear
state-feedback controllers. This results in novel and flexible conditions, with
the particular feature of being in the form of linear matrix inequalities
(LMIs). Our technique thus provides a first controller-design strategy allowing
piecewise linear feedback maps and piecewise quadratic (control) Lyapunov
functions by means of semidefinite programming. Numerical examples illustrate
the application of the proposed techniques, the relations between the graph
order, the robustness, and the performance of the closed loop.
