Dissipativity-Based Scalable $\mathcal{L}_{2}$-Gain Analysis for Nonlinear Networked Systems

By making use of dissipativity theory we can provide an easily verifiable condition which ensures a scalable
$\mathcal{L}_{2^{-}}$
gain in a network of nonlinear systems, i.e. a gain independent on the number of systems. However, ensuring bounded
$\mathcal{L}_{2}$
-gains may become insufficient, since the energy of the input may grow unbounded. Therefore, we can also give a proof that the same condition may be used to bound the energy of the local systems. Such a bound ensures that the effects of increasing network size do not accumulate in any of the systems. We end the paper with an example in which we demonstrate that scaling the network size does not lead to an accumulation in any part of the network.