This paper is concerned with the delay-dependent stability of systems with time-varying delays. The novelty relies on the use of the second-order Bessel-Legendre integral inequality which is less conservative than the Jensen and Wirtinger-based inequalities. Unlike similar contributions, the features of this inequality are fully integrated into the construction of augmented Lyapunov-Krasovskii functionals leading to novel stability criteria expressed in terms of linear matrix inequalities. The stability condition is tested on some classical numerical examples illustrating the efficiency of the proposed method.
A more general class of stochastic nonlinear systems with irreducible homogenous Markovian switching are considered in this paper. As preliminaries, the stability criteria and the existence theorem of strong solution are firstly presented by using the inequality of mathematic expectation of Lyapunov function. The state-feedback controller is designed by regarding Markovian switching as constant such that the closed-loop system has a unique solution, and the equilibrium is asymptotically stable in probability in the large. The output-feedback controller is designed based on a quadratic-plus-quartic-form Lyapunov function such that the closed-loop system has a unique solution with the equilibrium being asymptotically stable in probability in the large in unbiased case and has a unique bounded-in-probability solution in biased case.
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