{"paper":{"title":"On Riccati equations in Banach algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.RA"],"primary_cat":"math.OC","authors_text":"Amol J. Sasane, Ruth F. Curtain","submitted_at":"2010-08-20T09:08:04Z","abstract_excerpt":"Let $R$ be a commutative complex Banach algebra with the involution $\\cdot ^\\star$ and suppose that $A\\in R^{n\\times n}$, $B\\in R^{n\\times m}$, $C\\in R^{p\\times n}$. The question of when the Riccati equation $$ PBB^\\star P-PA-A^\\star P-C^\\star C=0 $$ has a solution $P\\in R^{n\\times n}$ is investigated. A counterexample to a previous result in the literature on this subject is given, followed by sufficient conditions on the data guaranteeing the existence of such a $P$. Finally, applications to spatially distributed systems are discussed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3462","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"}