  In this paper, we consider the robustness of a basic model of a dynamical
distribution network. In the first problem, i.e., optimal weight allocation, we
minimize the H-inf- norm of the dynamical distribution network subject to
allocation of the weights on the edges. It is shown that this optimization
problem can be formulated as a semi-definite program. Next we consider the
semi-definiteness of the weighted graph Laplacian matrix with negative weights
on the edges. A necessary and sufficient condition, using the effective
resistance matrix, is established to guarantee the positive semi-definiteness
of the Laplacian matrix. Furthermore, the bounded real lemma is derived for
state-space symmetric systems.
