{"paper":{"title":"A Graph-Theoretic Approach to the $\\mathcal{H}_{\\infty}$ Performance of Dynamical Systems on Directed and Undirected Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Henrik Sandberg, Karl Henrik Johansson, Mohammad Pirani","submitted_at":"2018-04-27T13:09:12Z","abstract_excerpt":"We study a graph-theoretic approach to the $\\mathcal{H}_{\\infty}$ performance of leader following consensus dynamics on directed and undirected graphs. We first provide graph-theoretic bounds on the system $\\mathcal{H}_{\\infty}$ norm of the leader following dynamics and show the tightness of the proposed bounds. Then, we discuss the relation between the system $\\mathcal{H}_{\\infty}$ norm for directed and undirected networks for specific classes of graphs, i.e., balanced digraphs and directed trees. Moreover, we investigate the effects of adding directed edges to a directed tree on the resultin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10483","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}