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Lyapunov stability of 2D finite-dimensional behaviours

2011; Taylor & Francis; Volume: 84; Issue: 4 Linguagem: Inglês

10.1080/00207179.2011.575800

1366-5820

Diego Napp, Paolo Rapisarda, Paula Rocha,

Gene Regulatory Network Analysis

Abstract In this article we investigate a Lyapunov approach to the stability of finite-dimensional 2D systems. We use the behavioural framework and consider a notion of stability following the ideas in Pillai and Shankar [H. Pillai and S. Shankar (1998 Pillai, H and Shankar, S. 1998. A Behavioral Approach to Control of Distributed Systems. SIAM Journal of Control and Optimization, 37: 388–408. [Crossref], [Web of Science ®] , [Google Scholar]). A Behavioral Approach to Control of Distributed Systems, SIAM Journal of Control and Optimization, 37, 388–408], Rocha [P. Rocha (2008 Rocha, P. 2008. Stabilization of Multidimensional Behaviors. Multidimensional Systems and Signal Processing, 19: 273–286. [Crossref] , [Google Scholar]). Stabilization of Multidimensional Behaviors, Multidimensional Systems and Signal Processing, 19, 273–286], Valcher [M. Valcher (2000 Valcher, M. 2000. Characteristic Cones and Stability Properties of Two-dimensional Autonomous Behaviors. IEEE Transactions on Circuits and Systems, Part I, CAS-47: 290–302. [Crossref] , [Google Scholar]). Characteristic Cones and Stability Properties of Two-dimensional Autonomous Behaviors, IEEE Transactions on Circuits and Systems, Part I, CAS-47, 290–302]. We characterise stability in terms of the existence of a (quadratic) Lyapunov function and provide a constructive algorithm for the computation of all such Lyapunov functions. Keywords: 2-D systemLyapunov functionquadratic difference formbehavioural approach Acknowledgements P. Rapisarda gratefully acknowledges the financial support of The Royal Society and of the Engineering and Physical Sciences Research Council for financially supporting the visit to the Faculty of Electrical Engineering and Computer Science of the University of Oporto (Portugal) during which the results presented in this article were obtained. The research of the other two authors has been financed by the Fundação para a Ciência e a Tecnologia, through the R&D Unit Centro de Investigação e Desenvolvimento em Matématica e Aplicações. The authors also gratefully acknowledge the British Council and the Council of Rectors of the Portuguese Universities for a Treaty of Windsor award.