Advances in Study of Time-Delay Systems and Their Applications: A Second Edition

1. Introduction

The delay phenomenon is a common feature of industrial, communication, economic, biological, and similar systems and processes, and it significantly affects their stability and dynamics. It can markedly deteriorate the quality of control performance in feedback loops. Studying the influence of delays on system stability, dynamics, and control performance poses a challenging mathematical problem. System and control theories have addressed this issue for nearly a century, dating back to the seminal work by Volterra [1]. Modern control theory faces growing demands for enhanced quality and performance in both industrial applications and everyday life—demands that are difficult to meet using conventional methods. Achieving these goals requires a deeper understanding of controlled systems with delays. Despite significant advances in artificial intelligence techniques and strategies in recent years, distinguished scholars continue to propose innovative solutions to longstanding challenges and identify new open problems stemming from an increasingly profound understanding of this domain.

This collection focuses on recent developments in the analysis and control design of time-delay systems (TDSs). The goal was to attract quality and novel papers in the field of “Time-Delay Systems and Their Applications”. It is anticipated that these top-tier research papers will have a considerable impact on the international scientific community and will inspire further advancements in the field.

2. Overview of Special Issue Contributions

The call for this Special Issue attracted 28 leading researchers in the field of TDSs, who collectively submitted 10 manuscripts. After a rigorous review process, however, only eight high-quality contributions were accepted and published. Based on the affiliation of the corresponding authors, three of the accepted papers are from China [2,3,4], while the remaining research papers were authored by scholars from Bulgaria [5], Jordan [6], Russia [7], Saudi Arabia [8], and Spain [9].

Two of the papers primarily investigate the exponential stability of TDSs. Sedova and Druzhinina [7] addressed the problem of uniform exponential stability for nonlinear time-varying TDSs. They applied the Razumikhin method to establish sufficient conditions for exponential stability and proposed an extension of the approach to time-varying systems. Specifically, they allowed the time-derivative of the constructed Razumikhin function to be indefinite and derived a new upper estimate on its rate of decrease. The results obtained can be viewed as extensions of those presented in earlier works [10,11], as well as a representation of known sufficient conditions for the exponential stability of linear delay equations. Applying the exponential stability conditions derived in this study to specific examples revealed that previously related results were either inapplicable or yielded different exponential estimates for the solutions. It was noted that the Razumikhin method led to stability conditions that were sufficient but unnecessary, resulting in overly conservative outcomes. Various approaches were then proposed to reduce this conservatism, and numerous results were reported in this area. The paper included a comparison with other findings [10,12,13,14], and the authors concluded that no universally good stability criterion currently exists.

Another paper in the Special Issue, authored by Lang and Lu [4], evaluates the exponential incremental stability of TDSs and presents the relevant criteria for both continuous and discontinuous right-hand sides. The authors proposed and proved sufficient conditions for the exponential incremental stability of solutions in systems with continuous right-hand sides. Before addressing the incremental stability of systems with discontinuous right-hand sides, they established conditions for the existence and uniqueness of the Filippov solution. Then, by constructing a sequence of systems with continuous right-hand sides and applying an approximation method, they derived sufficient conditions for the exponential incremental stability of systems with discontinuous right-hand sides. Numerical experiments with a linear TDS and a Hopfield neural network with time delay were conducted to verify the theoretical results. For future research, the authors aim to develop methods for constructing continuous TDSs to approximate discontinuous systems more effectively.

The stability of TDSs is closely related to their asymptotic behavior. Two other papers are primarily focused on the asymptotic behavior of solutions in a particular nonlinear neutral TDS and on novel criteria for oscillation [8], as well as the asymptotic behavior of another type of neutral TDSs [6], respectively. The former contribution, authored by Mesmouli et al., investigates stability, asymptotic stability, and exponential stability using the Banach fixed-point theorem. Under certain conditions, the matrix measure serves as the key tool for tackling these three types of stability in a system that includes two Volterra terms and a nonlinear term. The results obtained generalize the findings of [15] and the authors’ recent research [16]. The work is another example of the advantage of using the fixed-point method over Lyapunov’s method.

Batiha et al. [6] studied the oscillatory behavior of solutions to second-order differential equations (DEs) with neutrality conditions. They developed analytical criteria that highlight the dynamics of these equations and explain the associated oscillation patterns. Notably, the paper contributes to a deeper understanding of the mathematical properties of this class of DEs, enhancing the ability of researchers to address similar problems in various contexts. In particular, it emphasizes the dual effect of neutrality conditions in shaping the behavior of solutions, underscoring the potential to extend existing models to more complex, real-world applications. This research also adds to the existing literature [17,18] by providing precise criteria for assessing the nature of oscillatory solutions, laying a strong foundation for future studies. Further exploration of oscillatory effects in the context of higher-order DEs may reveal new patterns and offer deeper insight into the mathematical structure of these systems, enabling broader applications and greater understanding of more general systems.

The asymptotic behavior of solutions to nonlinear partial DEs (PDEs)—a key aspect in understanding their long-term dynamics—is the focus of another contribution to the Special Issue, authored by Ge and Du [3]. The paper addresses the global well-posedness and asymptotic behavior of solutions to non-autonomous Navier–Stokes equations (NANSEs) with infinite time delay and node determination. Using a mathematical framework based on a specialized function space, the authors demonstrated the well-posedness of the system, under the assumption of Lipschitz continuity with respect to time. In addition, they showed that the long-term behavior of strong solutions could be characterized by their values at a finite number of spatial nodes. The inclusion of an infinite delay term enabled the application of existing theoretical results to analyze the well-posedness and asymptotic behavior of the NANSEs under delay effects. This work provides theoretical support and specific examples for studying nonlinear PDEs with time-delay effects. It also validates the relevant conclusions obtained from time-delay DEs and offers insights into research methods applicable to nonlinear PDEs with other time-delay effects, further reinforcing findings from previous time-delay DE studies [19,20].

Nedzhibov [5] presented a detailed exposition of two spatio-temporal dynamic mode decomposition (STDMD) variants—the parallel and sequential STDMD methods. The author introduced the underlying matrix representations and highlighted their respective computational frameworks for analyzing spatio-temporal data. To address limitations inherited from the classic dynamic mode decomposition (DMD) algorithm, the study proposed extensions incorporating delay-embedding techniques [21]. In addition, numerical experiments were performed to validate the efficacy of the proposed extensions in overcoming the shortcomings of traditional DMD methods [22]. The results demonstrated the enhanced performance of delay-embedded STDMD, showcasing its utility in analyzing complex spatio-temporal datasets. Future research directions were also highlighted. This work advances spatio-temporal DMD methodologies by introducing extensions that enhance the robustness and accuracy of the analysis. The proposed approaches offer valuable tools for researchers and practitioners seeking more profound insights into the dynamics of complex spatio-temporal systems.

The problem of semi-global polynomial synchronization (SGPS) for high-order bidirectional associative memory neural networks (HOBAMNNs) with multiple proportional delays was addressed by Cong, Zhang, and Zhu in [2]. The authors initially derived delay-dependent SGPS criteria for the error dynamical system and then provided the corresponding controller gain. Illustrative examples were presented to demonstrate the applicability of their findings. The main contribution of this work lies in the first investigation of SGPS for HOBAMNNs with multiple proportional delays. By directly deriving easily implementable sufficient criteria based on the definition of SGPS, the proposed approach is applicable to a broader class of neural network models. Future research aims to explore general decay synchronization, which includes polynomial, exponential, logarithmic, and other forms of synchronization [23].

Finally, a comprehensive study on the evolution operator that generates the state trajectory of dynamical systems combining delay-free dynamics with several types of delays was conducted by De la Sen et al. [9]. In the first part of the study, explicit expressions were derived for the evolution associated with the state-trajectory solution of a class of linear time-varying differential delay-free systems. The impulsive-free part of the dynamics matrix function was assumed to be bounded, piecewise-continuous, and Lebesgue-integrable at all times. Both the absence and presence of impulsive actions in the system dynamics matrix were described. The obtained results were subsequently extended to systems with constant point delays, with the evolution operators that generate the trajectory solutions given explicitly. In the general case, these operators were non-unique in the impulsive scenario. The parameterization of impulsive actions at specific time instants occurred in the delay-free dynamics and in the various matrices of delayed dynamics, followed by an immediate return to the previous configuration. These impulsive actions were interpreted as instantaneous, abrupt switching changes in the parameterization, which might be non-unique, as the necessary impulsive gains to monitor the switched parameterizations can vary in achieving a suitable right limit of the solution trajectory. In addition, the boundedness of the solution trajectory of the impulsive TDS was investigated. It was found that an appropriate distribution of impulsive time instants can potentially stabilize a TDS, even if the delay-free dynamics is unstable.

3. Conclusions

All the above-mentioned authors deserve sincere gratitude for their valuable contributions to this Special Issue. We also extend our deep appreciation to the reviewers for their insightful comments and efforts in improving the quality of the submissions. We believe that these high-quality research papers will have a meaningful impact on the international scientific community and will further motivate research on the analysis and control of TDSs, both at the theoretical level and in real-world applications.