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This article considers the problems of achieving exponential stability and finite hybrid L2 × l2‐gain with respect to continuous and discrete disturbances for linear neutral time‐delay systems with impulsive effects. In the framework of descriptor system representation, a piecewise timer‐dependent Lyapunov functional is introduced to analyze the double impacts of state‐delay and impulse‐dwell‐time on stability of the underlying system. This functional depends on the state over an impulse interval and the state over a delay interval. Its construction is based on a delay‐fractioning scheme which ensures all delay subintervals contains at most one impulse instant. The relations among the augmented state variables are explored by using the impulse‐timer functions and the Newton–Leibniz formula. It is proved that the positive definiteness property of the functional at nonimpulse instants is not necessary for ensuring stability. The hybrid L2 × l2‐gain analysis is carried out by jointly using the introduced functional and a discrete‐time Lyapunov function. The obtain criteria for exponential stability and hybrid L2 × l2‐gain are expressed in terms of linear matrix inequalities. Their effectiveness is verified through two numerical examples. [ABSTRACT FROM AUTHOR]

This paper addresses the impulsive observers design problem of perturbed Lipschitz nonlinear systems subject to noisy delayed discrete measurements. The transmission delay, state perturbation, and measurement noise simultaneously hinder the convergence of impulsive observer. To deal with the difficulty, the observation error system with delayed impulses is represented as an augmented system with switching delay-free impulses. By taking into account the switching structure of the impulse dynamics, the impulse observation gain matrix is modelled as a function of the time interval between two consecutive sampling instant and update instant. Such a structure of the observation gain matrix is able to adapt to the variation of sampling period and transmission delay. To obtain a tractable design condition, the idea of piecewise linear interpolation is applied to represent the impulse observation gain matrix as a piecewise linear form, which is determined by finite static impulse gains. A piecewise Lyapunov function which matches the piecewise linear structure of the observation gain is constructed to analyze the exponential input-to-state stability (EISS) property of the observation error system. The EISS gain quantifies the robustness performance of the designed impulsive observer. Sufficient conditions for the existence of the proposed variable gain impulsive observers are presented. Two numerical examples are provided to illustrate the effectiveness of the developed impulsive observation strategy. • A methodology for designing impulsive observers with delayed measurements and disturbances is proposed. • Based on the augmented representation of the observation error system, a piecewise linear gain function is constructed. • The robustness performance of the proposed observers against external disturbance is characterized by the notion of EISS. [ABSTRACT FROM AUTHOR]

This paper is concerned with the positivity and impulsive stabilization of equilibrium points of delayed neural networks (DNNs) subject to bounded disturbances. With the aid of the continuous dependence theorem for impulsive delay differential equations, a relaxed positivity condition is derived, which allows the neuron interconnection matrix to be Metzler if the activation functions satisfy a certain condition. The notion of input-to-state stability (ISS) is introduced to characterize internal global stability and disturbance attenuation performance for impulsively controlled DNNs. The ISS property is analyzed by employing a time-dependent max-separable Lyapunov function which is able to capture the positivity characterization and hybrid structure of the considered DNNs. A ranged dwell-time-dependent ISS condition is obtained, which allows to design an impulsive control law via partial state variables. As a byproduct, an improved global exponential stability criterion for impulse-free positive DNNs is obtained. The applicability of the achieved results is illustrated through three numerical examples. • A relaxed positivity condition for impulsively controlled DNNs is proposed. • A novel time-dependent max-separable Lyapunov function is introduced for ISS analysis. • An improved criterion for positivity and global exponential stability of DNNs is obtained. [ABSTRACT FROM AUTHOR]