Abstract
This article considers the control Lyapunov inequality with a priori prescribed decrease over the space of Minkowski functions generated by compact convex sets containing the origin in the interior within the setting of unconstrained and constrained linear discrete time systems. Within the unconstrained setting, the article derives a novel necessary and sufficient condition, which characterizes all Minkowski functions that satisfy the considered control Lyapunov inequality. This novel necessary and sufficient condition is provided for Minkowski functions of general compact convex sets containing the origin in the interior, and it is also refined for Minkowski functions of polytopic sets containing the origin as an interior point. In addition, the article characterizes, and establishes topological properties of, the related stabilizing set-valued control map. The novel results derived for the unconstrained setting are also extended to the constrained setting.
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