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Recent works have shown that the $L_1$ and $L_\infty$-gains are naturalperformance criteria for linear positive systems as they can be characterizedusing linear programs. Those performance measures have also been extended tolinear positive impulsive and switched systems through the concept of hybrid$L_1\times\ell_1$-gain. For LTI positive systems, the $L_\infty$-gain is knownto coincide with the $L_1$-gain of the transposed system and, as a consequence,one can use linear copositive Lyapunov functions for characterizing the$L_\infty$-gain of LTI positive systems. Unfortunately, this does not hold inthe time-varying setting and one cannot characterize the hybrid$L_\infty\times\ell_\infty$-gain of a linear positive impulsive system in termsof the hybrid $L_1\times\ell_1$-gain of the transposed system. An alternativeapproach based on the use of linear copositive max-separable Lyapunov functionsis proposed. We first prove very general necessary and sufficient conditionscharacterizing the exponential stability and the $L_\infty\times\ell_\infty$-and $L_1\times\ell_1$-gains using linear max-separable copositive and linearsum-separable copositive Lyapunov functions. Results characterizing thestability and the hybrid $L_\infty\times\ell_\infty$-gain of linear positiveimpulsive systems under arbitrary, constant, minimum, and range dwell-timeconstraints are then derived from the previously obtained general results.These conditions are then exploited to yield constructive convex stabilizationconditions via state-feedback. By reformulating linear positive switchedsystems as impulsive systems with multiple jump maps, stability andstabilization conditions are also obtained for linear positive switchedsystems. It is notably proven that the obtained conditions generalize existingones of the literature.

Hybrid $L_\infty\times\ell_\infty$-Performance Analysis and Control of Linear Time-Varying Impulsive and Switched Positive Systems