Generalized Sarymsakov Matrices

• Published in: IEEE transactions on automatic control Vol. 64; no. 8; pp. 3085 - 3100
• Main Authors: Xia, Weiguo, Liu, Ji, Cao, Ming, Johansson, Karl Henrik, Basar, Tamer
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.08.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: 1974; 1984; 2005; 2005 Fourth International Symposium on Information Processing in Sensor Networks4th International Symposium on Information Processing in Sensor Networks; 2011; 2011 50TH IEEE CONFERENCE ON DECISION AND CONTROL AND EUROPEAN CONTROL CONFERENCE (CDC-ECC)50th IEEE Conference of Decision and Control (CDC)/European Control Conference (ECC); 2012; 2012 IEEE 51ST ANNUAL CONFERENCE ON DECISION AND CONTROL (CDC)51st IEEE Annual Conference on Decision and Control (CDC); 2014; 2014 IEEE 53RD ANNUAL CONFERENCE ON DECISION AND CONTROL (CDC)53rd IEEE Annual Conference on Decision and Control (CDC); 2015; 2015 54TH IEEE CONFERENCE ON DECISION AND CONTROL (CDC)54th IEEE Conference on Decision and Control (CDC); APR 25-27; Combinatorial analysis; Convergence; Cooperative control; DEC 10-13; DEC 12-15; DEC 15-17; DEC 15-18; doubly stochastic matrices; GROOT MH; IEEE TRANSACTIONS ON AUTOMATIC CONTROL; Indexes; JAPAN; JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION; Linear matrix inequalities; Los Angeles; multi-agent systems; Multiplication; ndrickx Julien M; Orlando; Osaka; P118 uri Behrouz; P19 ao L; P2118 u Ji; P2718; P2835 vaei Javad; P3991 itsiklis J. N; P437 ndrickx Julien M; P5070 u Ji; P63 uri Behrouz; Problems in decentralized decision making and computation; products of stochastic matrices; Robot sensing systems; Sarymsakov matrices; Stochastic processes; Symmetric matrices; V57; V59; V69
• ISSN: 0018-9286; 1558-2523; 1558-2523
• DOI: 10.1109/TAC.2018.2878476

Within the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class of Sarymsakov matrices is the largest known subset that is closed under matrix multiplication, and more critically whose compact subsets are all consensus sets. This paper shows that a larger subset with these two properties can be obtained by generalizing the standard definition for Sarymsakov matrices. The generalization is achieved by introducing the notion of the SIA index of a stochastic matrix, whose value is 1 for Sarymsakov matrices, and then exploring those stochastic matrices with larger SIA indices. In addition to constructing the larger set, this paper introduces another class of generalized Sarymsakov matrices, which contains matrices that are not SIA, and studies their products. Sufficient conditions are provided for an infinite product of matrices from this class, converging to a rank-one matrix. Finally, as an application of the results just described and to confirm their usefulness, a necessary and sufficient combinatorial condition, the "avoiding set condition," for deciding whether or not a compact set of stochastic matrices is a consensus set is revisited. In addition, a necessary and sufficient combinatorial condition is established for deciding whether or not a compact set of doubly stochastic matrices is a consensus set.