  We provide novel dissipativity conditions for bounding the incremental L-1
gain of systems. Moreover, we adapt existing results on the L-infinity gain to
the incremental setting and relate the incremental L-1 and L-infinity gain
bounds through system adjoints. Building on work on optimization based
approaches to constructing polyhedral Lyapunov functions, we make use of these
conditions to obtain a Linear Programming based algorithm that can provide
increasingly sharp bounds on the gains as a function of a given candidate
polyhedral storage function or polyhedral set. The algorithm is also extended
to allow for the design of linear feedback controllers for performance, as
measured by the bounds on the incremental gains. We apply the algorithm to a
couple of numerical examples to illustrate the power, as well as some
limitations, of this approach.
