  Starting from a finite family of continuously differentiable positive
definite functions, we study conditions under which a function obtained by
max-min combinations is a Lyapunov function, establishing stability for two
kinds of nonlinear dynamical systems: a) Differential inclusions where the
set-valued right-hand-side comprises the convex hull of a finite number of
vector fields, and b) Autonomous switched systems with a state-dependent
switching signal. We investigate generalized notions of directional derivatives
for these max-min functions, and use them in deriving stability conditions with
various degrees of conservatism, where more conservative conditions are
numerically more tractable. The proposed constructions also provide nonconvex
Lyapunov functions, which are shown to be useful for systems with
state-dependent switching that do not admit a convex Lyapunov function. Several
examples are included to illustrate the results.
