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Sufficient conditions characterizing the asymptotic stability and the hybrid$L_1/\ell_1$-gain of linear positive impulsive systems under minimum and rangedwell-time constraints are obtained. These conditions are stated asinfinite-dimensional linear programming problems that can be solved using sumof squares programming, a relaxation that is known to be asymptotically exactin the present case. These conditions are then adapted to formulateconstructive and convex sufficient conditions for the existence of$L_1/\ell_1$-to-$L_1/\ell_1$ interval observers for linear impulsive andswitched systems. Suitable observer gains can be extracted from the(suboptimal) solution of the infinite-dimensional optimization problem wherethe $L_1/\ell_1$-gain of the system mapping the disturbances to the weightedobservation errors is minimized. Some examples on impulsive and switchedsystems are given for illustration.

$L_1/\ell_1$-to-$L_1/\ell_1$ analysis of linear positive impulsive  systems with application to the $L_1/\ell_1$-to-$L_1/\ell_1$ interval  observation of linear impulsive and switched systems