{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T16:25:58Z","timestamp":1762100758504},"reference-count":18,"publisher":"University of Zielona G\u00f3ra, Poland","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2007,12,1]]},"abstract":"<jats:title>The Choice of the Forms of Lyapunov Functions for a Positive 2D Roesser Model<\/jats:title><jats:p>The appropriate choice of the forms of Lyapunov functions for a positive 2D Roesser model is addressed. It is shown that for the positive 2D Roesser model: (i) a linear form of the state vector can be chosen as a Lyapunov function, (ii) there exists a strictly positive diagonal matrix<jats:italic>P<\/jats:italic>such that the matrix<jats:italic>A<jats:sup>T<\/jats:sup>PA - P<\/jats:italic>is negative definite. The theoretical deliberations will be illustrated by numerical examples.<\/jats:p>","DOI":"10.2478\/v10006-007-0039-7","type":"journal-article","created":{"date-parts":[[2008,1,9]],"date-time":"2008-01-09T14:16:23Z","timestamp":1199888183000},"page":"471-475","source":"Crossref","is-referenced-by-count":68,"title":["The Choice of the Forms of Lyapunov Functions for a Positive 2D Roesser Model"],"prefix":"10.61822","volume":"17","author":[{"given":"Tadeusz","family":"Kaczorek","sequence":"first","affiliation":[]}],"member":"37438","reference":[{"key":"1","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1109\/TAC.2004.826715","article-title":"A tutorial on the positive realization problem","volume":"5","author":"L. Benvenuti","year":"2004","journal-title":"IEEE Transactions on Automatic Control"},{"key":"2","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-5225-6","author":"N. Bose","year":"1985","journal-title":"Multidimensional Systems Theory Progress, Directions and Open Problems"},{"key":"3","doi-asserted-by":"crossref","DOI":"10.1002\/9781118033029","author":"L. Farina","year":"2000","journal-title":"Positive Linear Systems. Theory and Applications"},{"key":"4","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/BF01776566","article-title":"Double indexed dynamical systems","volume":"12","author":"E. Fornasini","year":"1978","journal-title":"Mathematical Systems Theory"},{"key":"5","first-page":"484","volume":"AC-21","author":"E. Fornasini","year":"1976","journal-title":"State-space realization theory of two- dimensional filters"},{"key":"6","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/0024-3795(94)00242-8","article-title":"On the spectral and combinatorial structure of 2D positive systems","volume":"245","author":"E. Fornasini","year":"1996","journal-title":"Linear Algebra and Its Applications"},{"key":"7","first-page":"101","article-title":"Recent developments in 2D positive systems theory","volume":"4","author":"E. Fornasini","year":"1997","journal-title":"International Journal of Applied Mathematics and Computer Science"},{"key":"8","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1109\/81.554329","article-title":"Elementary operation approach to state space realization of 2D systems","volume":"2","author":"K. Ga\u0142kowski","year":"1997","journal-title":"IEEE Transaction on Circuits and Systems"},{"key":"9","first-page":"227","article-title":"Externally positive 2D linear systems","volume":"3","author":"T. Kaczorek","year":"1999","journal-title":"Bulletin of the Polish Academy of Sciences: Technical Sciences"},{"key":"10","first-page":"405","article-title":"Reachability and controllability of non-negative 2D Roesser type models","volume":"4","author":"T. Kaczorek","year":"1996","journal-title":"Bulletin of the Polish Academy of Sciences: Technical Sciences"},{"key":"11","author":"T. Kaczorek","year":"2000","journal-title":"Positive 1D and 2D Systems"},{"key":"12","first-page":"221","article-title":"When the equilibrium of positive 2D Roesser model are strictly positive","volume":"3","author":"T. Kaczorek","year":"2002","journal-title":"Bulletin of the Polish Academy of Sciences: Technical Sciences"},{"key":"13","author":"T. Kaczorek","year":"1985","journal-title":"Two-Dimensional Linear Systems"},{"key":"14","first-page":"319","author":"J. Klamka","year":"1999","journal-title":"Controllability of 2D linear systems"},{"key":"15","author":"J. Klamka","year":"1991","journal-title":"Controllability of dynamical systems"},{"key":"16","first-page":"600","volume":"2","author":"J. Kurek","year":"1985","journal-title":"The general state-space model for a two-dimensional linear digital systems"},{"key":"17","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1109\/81.995672","article-title":"Stability of positive 2D systems described by the Roesser model","volume":"4","author":"J. Kurek","year":"2002","journal-title":"IEEE Transactions on Circuits and Systems I"},{"key":"18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TAC.1975.1100844","article-title":"A discrete state-space model for linear image processing","volume":"1","author":"R. Roesser","year":"1975","journal-title":"IEEE Transactions on Automatic Control"}],"container-title":["International Journal of Applied Mathematics and Computer Science"],"original-title":[],"link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/amcs\/17\/4\/article-p471.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/view\/j\/amcs.2007.17.issue-4\/v10006-007-0039-7\/v10006-007-0039-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T10:27:37Z","timestamp":1709202457000},"score":1,"resource":{"primary":{"URL":"https:\/\/content.sciendo.com\/doi\/10.2478\/v10006-007-0039-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,12,1]]},"references-count":18,"journal-issue":{"issue":"4"},"URL":"https:\/\/doi.org\/10.2478\/v10006-007-0039-7","relation":{},"ISSN":["1641-876X"],"issn-type":[{"value":"1641-876X","type":"print"}],"subject":[],"published":{"date-parts":[[2007,12,1]]}}}