Behavior of Hilbert networks under perturbations of their elements (Q1901478)

One considers a nonlinear Hilbert network. This is an oriented, locally finite graph, whose countably infinite branches contain nonlinear elements (i.e. nonlinear relations in a Hilbert space between branch currents and voltages), such that Kirchhoff's laws are satisfied and the total energy is finite. Two hypotheses are required: each source distribution \(z\) determines uniquely a current distribution \(y\), and the mapping \(z @>h>>y\) is Lipschitz. Theorem 1 gives an estimate for the change of the current distribution caused by small perturbations of the elements, while Theorem 2 gives a bound for the change of current distribution due to a change of voltage sources. These estimates follow from the author's previous results on general input-output systems and, in particular, imply that under the above conditions the network is insensitive with respect to perturbations of its elements and exhibits robust stability.