Lyapunov stability of 2D finite-dimensional behaviours

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.