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

Optimal Weight Allocation of Dynamic Distribution Networks and Positive Semi-definiteness of Signed Laplacians