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Advances in Study of Time-Delay Systems and Their Applications, 2nd...
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Advances in Study of Time-Delay Systems and Their Applications, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C2: Dynamical Systems".

Deadline for manuscript submissions: 30 April 2025 | Viewed by 9918

Special Issue Editor

Dr. Libor Pekař

E-Mail Website
Guest Editor
Faculty of Applied Informatics, Tomas Bata University in Zlín, Nad Stráněmi 4511, 76005 Zlín, Czech Republic
Interests: analysis, modeling, identification, and control of time-delay systems; algebraic control methods; heat-exchanger processes; autotuning and optimization techniques
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In dynamical systems, the delay phenomenon is a generic part of industrial, communication, economical, biological, and similar processes that considerably affects their stability and dynamics. Moreover, it unambiguously deteriorates the quality of control performance in feedback loops. The study of the influence of delays on system stability, dynamics, and control performance poses a challenging mathematic exercise. System and control theories have worked to tackle this issue for almost a century, since the publication of the famous work by Volterra (1928). Modern theory is confronted with increasing requirements for the quality and performance of control systems in the industry, as well as in everyday reality, which can hardly be achieved using conventional methods. In order to meet these goals, more in-depth knowledge of the controlled delayed systems is a prerequisite.

This Special Issue of Mathematics is focused on recent developments in approaches to time-delay systems analysis and control design. We seek papers on system stability and dynamics analysis, including exponential, asymptotic, strong, delay-dependent, delay-independent, BIBO, H2, H∞, and other types of system stability. Hopf, fold, and pitchfork bifurcation; stability switching; and eigenvalue analyses are also welcome. Special consideration will be given to delays of neutral types and systems given by algebraic-differential equations; however, systems with retarded delays are also acceptable. In addition, the scope of this Special Issue includes modern control methods and their applications, such as switched systems, event-triggered control, Lyapunov–Razumikhin- and Krasovskii-type approaches, etc.

All submitted papers will be peer reviewed and selected on the basis of their quality and relevance to this Special Issue.

Dr. Libor Pekař
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • time-invariant and time-variant delayed systems
  • delay-varying models and their stability
  • linear and nonlinear delayed systems
  • dynamics and stability of systems with retarded and neutral delays
  • delayed systems described by algebraic-differential equations
  • exponential, asymptotic, strong, BIBO, H2, H∞ stability of time-delay systems
  • Hopf, fold, and pitchfork bifurcation, stability switching, and eigenvalue analysis
  • delay-dependent and delay-independent stability
  • finite dimension approximations
  • filtering and estimation of time-delay systems
  • switched systems with time delay and event-triggered control
  • Krasovskii-type and Lyapunov–Razumikhin-type stability and control approaches
  • new results in controllability and observability of time-delay systems
  • robust, algebraic, and adaptive control methods and their applications

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Published Papers (7 papers)

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Research

32 pages, 401 KiB  
Article
On the Evolution Operators of a Class of Time-Delay Systems with Impulsive Parameterizations
by Manuel De la Sen, Asier Ibeas, Aitor J. Garrido and Izaskun Garrido
Mathematics 2025, 13(3), 365; https://doi.org/10.3390/math13030365 - 23 Jan 2025
Viewed by 598
Abstract
This paper formalizes the analytic expressions and some properties of the evolution operator that generates the state-trajectory of dynamical systems combining delay-free dynamics with a set of discrete, or point, constant (and not necessarily commensurate) delays, where the parameterizations of both the delay-free [...] Read more.
This paper formalizes the analytic expressions and some properties of the evolution operator that generates the state-trajectory of dynamical systems combining delay-free dynamics with a set of discrete, or point, constant (and not necessarily commensurate) delays, where the parameterizations of both the delay-free and the delayed parts can undergo impulsive changes. Also, particular evolution operators are defined explicitly for the non-impulsive and impulsive time-varying delay-free case, and also for the case of impulsive delayed time-varying systems. In the impulsive cases, in general, the evolution operators are non-unique. The delays are assumed to be a finite number of constant delays that are not necessarily commensurate, that is, all of them being integer multiples of a minimum delay. On the other hand, the impulsive actions through time are assumed to be state-dependent and to take place at certain isolated time instants on the matrix functions that define the delay-free and the delayed dynamics. Some variants are also proposed for the cases when the impulsive actions are state-independent or state- and dynamics-independent. The intervals in-between consecutive impulses can be, in general, time-varying while subject to a minimum threshold. The boundedness of the state-trajectory solutions, which imply the system’s global stability, is investigated in the most general case for any given piecewise-continuous bounded function of initial conditions defined on the initial maximum delay interval. Such a solution boundedness property can be achieved, even if the delay-free dynamics is unstable, by an appropriate distribution of the impulsive actions. An illustrative first-order example is developed in detail to illustrate the impulsive stabilization results. Full article
(This article belongs to the Special Issue Advances in Study of Time-Delay Systems and Their Applications, 2nd Edition)
19 pages, 308 KiB  
Article
Global Well-Posedness and Determining Nodes of Non-Autonomous Navier–Stokes Equations with Infinite Delay on Bounded Domains
by Huanzhi Ge and Feng Du
Mathematics 2025, 13(2), 222; https://doi.org/10.3390/math13020222 - 10 Jan 2025
Viewed by 554
Abstract
The asymptotic behavior of solutions to nonlinear partial differential equations is an important tool for studying their long-term behavior. However, when studying the asymptotic behavior of solutions to nonlinear partial differential equations with delay, the delay factor u(t+θ) [...] Read more.
The asymptotic behavior of solutions to nonlinear partial differential equations is an important tool for studying their long-term behavior. However, when studying the asymptotic behavior of solutions to nonlinear partial differential equations with delay, the delay factor u(t+θ) in the delay term may lead to oscillations, hysteresis effects, and other phenomena in the solution, which increases the difficulty of studying the well-posedness and asymptotic behavior of the solution. This study investigates the global well-posedness and asymptotic behavior of solutions to the non-autonomous Navier–Stokes equations incorporating infinite delays. To establish global well-posedness, we first construct several suitable function spaces and then prove them using the Galekin approximation method. Then, by accurately estimating the number of determining nodes, we reveal the asymptotic behavior of the solution. The results indicate that the long-term behavior of a strong solution can be determined by its values at a finite number of nodes. Full article
(This article belongs to the Special Issue Advances in Study of Time-Delay Systems and Their Applications, 2nd Edition)
17 pages, 289 KiB  
Article
Nonlinear Neutral Delay Differential Equations: Novel Criteria for Oscillation and Asymptotic Behavior
by Belal Batiha, Nawa Alshammari, Faten Aldosari, Fahd Masood and Omar Bazighifan
Mathematics 2025, 13(1), 147; https://doi.org/10.3390/math13010147 - 2 Jan 2025
Cited by 2 | Viewed by 828
Abstract
This research deals with the study of the oscillatory behavior of solutions of second-order differential equations containing neutral conditions, both in sublinear and superlinear terms, with a focus on the noncanonical case. The research provides a careful analysis of the monotonic properties of [...] Read more.
This research deals with the study of the oscillatory behavior of solutions of second-order differential equations containing neutral conditions, both in sublinear and superlinear terms, with a focus on the noncanonical case. The research provides a careful analysis of the monotonic properties of solutions and their derivatives, paving the way for a deeper understanding of this complex behavior. The research is particularly significant as it extends the scope of previous studies by addressing more complex forms of neutral differential equations. Using the linearization technique, strict conditions are developed that exclude the existence of positive solutions, which allows the formulation of innovative criteria for determining the oscillatory behavior of the studied equations. This research highlights the theoretical and applied aspects of this mathematical phenomenon, which contributes to enhancing the scientific understanding of differential equations with neutral conditions. To demonstrate the effectiveness of the results, the research includes two illustrative examples that prove the validity and importance of the proposed methodology. This work represents a qualitative addition to the mathematical literature, as it lays new foundations and opens horizons for future studies in this vital field. Full article
(This article belongs to the Special Issue Advances in Study of Time-Delay Systems and Their Applications, 2nd Edition)
18 pages, 1022 KiB  
Article
Delay-Embedding Spatio-Temporal Dynamic Mode Decomposition
by Gyurhan Nedzhibov
Mathematics 2024, 12(5), 762; https://doi.org/10.3390/math12050762 - 4 Mar 2024
Cited by 3 | Viewed by 2170
Abstract
Spatio-temporal dynamic mode decomposition (STDMD) is an extension of dynamic mode decomposition (DMD) designed to handle spatio-temporal datasets. It extends the framework so that it can analyze data that have both spatial and temporal variations. This facilitates the extraction of spatial structures along [...] Read more.
Spatio-temporal dynamic mode decomposition (STDMD) is an extension of dynamic mode decomposition (DMD) designed to handle spatio-temporal datasets. It extends the framework so that it can analyze data that have both spatial and temporal variations. This facilitates the extraction of spatial structures along with their temporal evolution. The STDMD method extracts temporal and spatial development information simultaneously, including wavenumber, frequencies, and growth rates, which are essential in complex dynamic systems. We provide a comprehensive mathematical framework for sequential and parallel STDMD approaches. To increase the range of applications of the presented techniques, we also introduce a generalization of delay coordinates. The extension, labeled delay-embedding STDMD allows the use of delayed data, which can be both time-delayed and space-delayed. An explicit expression of the presented algorithms in matrix form is also provided, making theoretical analysis easier and providing a solid foundation for further research and development. The novel approach is demonstrated using some illustrative model dynamics. Full article
(This article belongs to the Special Issue Advances in Study of Time-Delay Systems and Their Applications, 2nd Edition)
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9 pages, 262 KiB  
Article
Asymptotic Behavior of Solutions in Nonlinear Neutral System with Two Volterra Terms
by Mouataz Billah Mesmouli, Abdelouaheb Ardjouni and Hicham Saber
Mathematics 2023, 11(12), 2676; https://doi.org/10.3390/math11122676 - 13 Jun 2023
Cited by 1 | Viewed by 1143
Abstract
In this manuscript, we generalise previous results in the literature by providing sufficient conditions for the matrix measure to guarantee the stability, asymptotic stability and exponential stability of a neutral system of differential equations. This is achieved by constructing a suitable operator from [...] Read more.
In this manuscript, we generalise previous results in the literature by providing sufficient conditions for the matrix measure to guarantee the stability, asymptotic stability and exponential stability of a neutral system of differential equations. This is achieved by constructing a suitable operator from our system and applying the Banach fixed point theorem. Full article
(This article belongs to the Special Issue Advances in Study of Time-Delay Systems and Their Applications, 2nd Edition)
26 pages, 1361 KiB  
Article
Criteria on Exponential Incremental Stability of Dynamical Systems with Time Delay
by Yingying Lang and Wenlian Lu
Mathematics 2023, 11(10), 2242; https://doi.org/10.3390/math11102242 - 10 May 2023
Viewed by 1267
Abstract
Incremental stability analysis for time-delay systems has attracted more and more attention for its contemporary applications in transportation processes, population dynamics, economics, satellite positions, etc. This paper researches the criteria for exponential incremental stability for time-delay systems with continuous or discontinuous right-hand sides. [...] Read more.
Incremental stability analysis for time-delay systems has attracted more and more attention for its contemporary applications in transportation processes, population dynamics, economics, satellite positions, etc. This paper researches the criteria for exponential incremental stability for time-delay systems with continuous or discontinuous right-hand sides. Firstly, the sufficient conditions for exponential incremental stability for time-delay systems with continuous right-hand sides are studied, and several corollaries for specific cases are provided. As for time-delay systems with discontinuous right-hand sides, after expounding the relevant conditions for the existence and uniqueness of the Filippov solution, by using approximation methods, sufficient conditions for exponential incremental stability are obtained. The conclusions are applied to linear switched time-delay systems and Hopfield neural network systems with composite right-hand sides. Full article
(This article belongs to the Special Issue Advances in Study of Time-Delay Systems and Their Applications, 2nd Edition)
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15 pages, 404 KiB  
Article
Exponential Stability of Nonlinear Time-Varying Delay Differential Equations via Lyapunov–Razumikhin Technique
by Natalya O. Sedova and Olga V. Druzhinina
Mathematics 2023, 11(4), 896; https://doi.org/10.3390/math11040896 - 10 Feb 2023
Cited by 1 | Viewed by 2207
Abstract
In this article, some new sufficient conditions for the exponential stability of nonlinear time-varying delay differential equations are given. An extension of the classical asymptotical stability theorem in terms of a Lyapunov–Razumikhin function is obtained. The condition of non-positivity of the time derivative [...] Read more.
In this article, some new sufficient conditions for the exponential stability of nonlinear time-varying delay differential equations are given. An extension of the classical asymptotical stability theorem in terms of a Lyapunov–Razumikhin function is obtained. The condition of non-positivity of the time derivative of a Razumikhin function is weakened. Additionally, the resulting sufficient asymptotic stability conditions allow us to guarantee uniform exponential stability and evaluate the exponential convergence rate of the system solutions. The effectiveness of the results is demonstrated by some examples. Full article
(This article belongs to the Special Issue Advances in Study of Time-Delay Systems and Their Applications, 2nd Edition)
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