  Sufficient conditions characterizing the asymptotic stability and the hybrid
$L_1/\ell_1$-gain of linear positive impulsive systems under minimum and range
dwell-time constraints are obtained. These conditions are stated as
infinite-dimensional linear programming problems that can be solved using sum
of squares programming, a relaxation that is known to be asymptotically exact
in the present case. These conditions are then adapted to formulate
constructive and convex sufficient conditions for the existence of
$L_1/\ell_1$-to-$L_1/\ell_1$ interval observers for linear impulsive and
switched systems. Suitable observer gains can be extracted from the
(suboptimal) solution of the infinite-dimensional optimization problem where
the $L_1/\ell_1$-gain of the system mapping the disturbances to the weighted
observation errors is minimized. Some examples on impulsive and switched
systems are given for illustration.
