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This paper presents an alternative approach for obtaining a converse Lyapunov theorem for discrete-time systems. The proposed approach is constructive, as it provides an explicit Lyapunov function. The developed converse theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and semi-global practical Lyapunov functions for globally asymptotically stable systems. Furthermore, for specific classes of systems, the developed converse theorem can be used to establish non-conservatism of a particular type of Lyapunov functions. Most notably, a proof that conewise linear Lyapunov functions are non-conservative for GES conewise linear systems is given and, as a by-product, tractable construction of polyhedral Lyapunov functions for linear systems is attained.

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DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:

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a b s t r a c tPolytopic delay difference inclusions (DDIs) have received increasing attention recently, mostly due to their ability to model a wide variety of relevant processes, including networked control systems. One of the fundamental problems for DDIs that poses a non-trivial challenge is stabilization. This paper embraces the Razumikhin approach and provides several solutions to the stabilization problem as follows. Firstly, a method to synthesize a control Lyapunov-Razumikhin function (cLRF) is presented for unconstrained DDIs. Secondly, for constrained DDIs, a receding horizon controller based on the cLRF for the unconstrained system is proposed, along with a closed-loop stability analysis. Thirdly, it is shown that a tractable implementation of the developed control algorithm can be attained even for large delays, by means of an on-line Minkowski set addition. An advantageous feature of the developed methodology is that all the synthesis algorithms can be formulated as a low complexity semi-definite programming problem for quadratic cLRF candidates. A comparison with alternative synthesis methods demonstrates the advances provided by the developed theory.

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