Linear systems subject to input saturation and time delay: Global asymptotic and Lp-stabilization

This Note deals with two problems on stabilization of linear systems by static feedbacks which are bounded and time-delayed, namely global asymptotic stabilization and finite gain $L^p$-stabilization, $p\in [1,\infty ]$. Regarding the first issue, we provide, under standard necessary conditions, two types of solutions for arbitrary small bound on the control and large (constant) delay. The first solution is based on the knowledge of a static stabilizing feedback in the zero-delay case and the second solution is of nested saturation type, which extends results of Mazenc et al. [IEEE Trans. Automat. Contr. 48 (1) (2003) 57–63]. For the finite-gain $L^p$-stabilization issue, we assume that the system is neutrally stable. We show the existence of a linear feedback such that, for arbitrary small bound on the control and large (constant) delay, finite gain $L^p$-stability holds with respect to every $L^p$-norm, $p\in [1,\infty ]$. Moreover, the corresponding $L^p$-gain is delay-independent. To cite this article: K. Yakoubi, Y. Chitour, C. R. Acad. Sci. Paris, Ser. I 340 2005).