On minimal rank solutions to symmetric Lyapunov equations in Euclidean Jordan algebra. (1st February 2016)

• Record Type: Journal Article
• Authors: Luo, Ziyan; Xiu, Naihua
• Is Part Of: Optimization. Volume 65:Number 2(2016)
• Page Start: 433
• Page End: 442
• Publication Date: 2016-02-01
• Subjects: minimal rank solution -- symmetric Lyapunov equation -- exact relaxation -- Schatten -norm -- Euclidean Jordan algebra; 90C26 -- 90C25 -- 90C59 -- 15A18 -- 15A24; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2015.1053882
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract : The minimal rank solutions of continuous-time and discrete-time symmetric Lyapunov equations, which have important applications in dynamical systems, are generally difficult to achieve due to the involved rank minimization. By employing the decomposition techniques of Euclidean Jordan algebra and the symmetric Lyapunov operators, we show that in the setting of Euclidean Jordan algebra, these minimal rank solutions of both symmetric continuous-time and discrete-time Lyapunov equations are unique and can be exactly solved by the corresponding Schatten -norm ( ) relaxation problems under some easy-to-check conditions. Moreover, both the upper and lower bounds for the minimal ranks are proposed.