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Mathematics
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Impulsive Control Systems and Complexity
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Impulsive Control Systems and Complexity

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

Deadline for manuscript submissions: closed (30 June 2020) | Viewed by 40503

Special Issue Editors

Prof. Dr. Xiaodi Li

grade E-Mail Website
Guest Editor
School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250014, China
Interests: impulsive control theory; hybrid systems; time-delay systems; neural networks and applied mathematics
Special Issues, Collections and Topics in MDPI journals
Dr. Ivanka Stamova

E-Mail Website
Guest Editor
Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA
Interests: nonlinear analysis; mathematical modeling; fractional order systems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Gani Stamov

E-Mail Website
Guest Editor
1. Department of Mathematical Physics, Technical University of Sofia, Sliven, Bulgaria
2. Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA
Interests: nonlinear analysis; control theory; mathematical modeling; differential equations; fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Many complex real word phenomena exist under the conditions of disorder, chaos, randomness, uncertainty, or in general, under the conditions of entropy. The design of efficient impulsive controllers for such chaotic systems is the main objective of numerous researchers. The impulsive control of complex phenomena arises naturally in a wide variety of applications. Indeed, impulsive control dynamical systems are used for the mathematical simulation of processes which are subject to impulses during their evolution. Such types of processes are observed in numerous fields of science and technology: Control theory, population dynamics, biotechnologies, industrial robotics, etc.

The dynamic of impulsive control dynamical systems has long been and will continue to be one of the dominant themes in mathematics and mathematics applications due to its theoretical and practical significance. During the last couple of decades, the analysis of impulsive control complex systems and related models has attracted the attention of a wide audience of professionals, such as mathematicians, applied researchers, and practitioners. For example, impulsive control and synchronization are the most interesting and important collective behaviors of complex networking systems and have aroused great interest in researchers in fields such as secure communication and information processing. There are many cases where impulsive control can give a better performance than continuous control. Sometimes even only impulsive control can be used for control purposes. Impulsive control methodology is very effective and allows synchronization of a complex system using only small control impulses, even though the system’s behavior may follow unpredictable patterns.

In spite of the amount of published results recently focused on impulsive control complex systems, there remain many challenging open questions. The theory and applications of these systems are still very active areas of research.

In this Special Issue, we provide an international forum for researchers to contribute with original research as well as review papers focusing on the latest achievements in the theory and applications of impulsive control complex dynamical systems.

Prof. Dr. Xiaodi Li
Prof. Dr. Ivanka Stamova
Prof. Dr. Gani Stamov
Guest Editors

Manuscript Submission Information

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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

  • complex dynamical systems
  • impulsive control
  • synchronization
  • stability
  • impulsive state feedback control
  • chaos control
  • complex networks
  • cluster synchronization
  • hybrid control
  • time-varying delays
  • finite-time synchronization

Published Papers (15 papers)

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Research

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20 pages, 7291 KiB  
Article
Neural-Impulsive Pinning Control for Complex Networks Based on V-Stability
by Daniel Ríos-Rivera, Alma Y. Alanis and Edgar N. Sanchez
Mathematics 2020, 8(9), 1388; https://doi.org/10.3390/math8091388 - 19 Aug 2020
Cited by 4 | Viewed by 2001
Abstract
In this work, a neural impulsive pinning controller for a twenty-node dynamical discrete complex network is presented. The node dynamics of the network are all different types of discrete versions of chaotic attractors of three dimensions. Using the V-stability method, we propose a [...] Read more.
In this work, a neural impulsive pinning controller for a twenty-node dynamical discrete complex network is presented. The node dynamics of the network are all different types of discrete versions of chaotic attractors of three dimensions. Using the V-stability method, we propose a criterion for selecting nodes to design pinning control, in which only a small fraction of the nodes is locally controlled in order to stabilize the network states at zero. A discrete recurrent high order neural network (RHONN) trained with extended Kalman filter (EKF) is used to identify the dynamics of controlled nodes and synthesize the control law. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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29 pages, 473 KiB  
Article
Global Mittag–Leffler Stability and Stabilization Analysis of Fractional-Order Quaternion-Valued Memristive Neural Networks
by Grienggrai Rajchakit, Pharunyou Chanthorn, Pramet Kaewmesri, Ramalingam Sriraman and Chee Peng Lim
Mathematics 2020, 8(3), 422; https://doi.org/10.3390/math8030422 - 14 Mar 2020
Cited by 66 | Viewed by 2860
Abstract
This paper studies the global Mittag–Leffler stability and stabilization analysis of fractional-order quaternion-valued memristive neural networks (FOQVMNNs). The state feedback stabilizing control law is designed in order to stabilize the considered problem. Based on the non-commutativity of quaternion multiplication, the original fractional-order quaternion-valued [...] Read more.
This paper studies the global Mittag–Leffler stability and stabilization analysis of fractional-order quaternion-valued memristive neural networks (FOQVMNNs). The state feedback stabilizing control law is designed in order to stabilize the considered problem. Based on the non-commutativity of quaternion multiplication, the original fractional-order quaternion-valued systems is divided into four fractional-order real-valued systems. By using the method of Lyapunov fractional-order derivative, fractional-order differential inclusions, set-valued maps, several global Mittag–Leffler stability and stabilization conditions of considered FOQVMNNs are established. Two numerical examples are provided to illustrate the usefulness of our analytical results. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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10 pages, 763 KiB  
Article
Adaptive Output Synchronization of General Complex Dynamical Network with Time-Varying Delays
by Liangzhi Gan, Shuo Li, Na Duan and Xiangyong Kong
Mathematics 2020, 8(3), 311; https://doi.org/10.3390/math8030311 - 01 Mar 2020
Cited by 4 | Viewed by 1542
Abstract
This paper is concerned with the output synchronization problems for a class of delayed complex dynamical networks. Based on the invariant principle of functional differential equations and Lyapunov stability theory, the feedback controller and parameter update laws are constructed for a large-scale network [...] Read more.
This paper is concerned with the output synchronization problems for a class of delayed complex dynamical networks. Based on the invariant principle of functional differential equations and Lyapunov stability theory, the feedback controller and parameter update laws are constructed for a large-scale network with uncertainties. The general complex delayed network can achieve synchronization by adaptively adjusting their feedback gains. Numerical examples are presented to further verify the effectiveness of the proposed control scheme. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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10 pages, 267 KiB  
Article
pth Moment Stability of a Stationary Solution for a Reaction Diffusion System with Distributed Delays
by Xiongrui Wang, Ruofeng Rao and Shouming Zhong
Mathematics 2020, 8(2), 200; https://doi.org/10.3390/math8020200 - 06 Feb 2020
Cited by 2 | Viewed by 1455
Abstract
In this paper, the Sobolev embedding theorem, Holder inequality, the Lebesgue contrl convergence theorem, the operator norm estimation technique, and critical point theory are employed to prove the existence of nontrivial stationary solution for p-Laplacian diffusion system with distributed delays. Furthermore, by [...] Read more.
In this paper, the Sobolev embedding theorem, Holder inequality, the Lebesgue contrl convergence theorem, the operator norm estimation technique, and critical point theory are employed to prove the existence of nontrivial stationary solution for p-Laplacian diffusion system with distributed delays. Furthermore, by giving the definition of pth moment stability, the authors use the Lyapunovfunctional method and Kamke function to derive the stability of nontrivialstationary solution. Moreover, a numerical example illuminates the effectiveness of the proposed methods. Finally, an interesting further thought is put forward, which is conducive to the in-depth study of the problem. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
12 pages, 273 KiB  
Article
Robust H∞-Control for Uncertain Stochastic Systems with Impulsive Effects
by Zhezhe Xin, Chunjie Xiao, Ting Hou and Xiao Shen
Mathematics 2019, 7(12), 1169; https://doi.org/10.3390/math7121169 - 03 Dec 2019
Cited by 4 | Viewed by 1807
Abstract
Robust stabilization and H ∞ controller design for uncertain systems with impulsive and stochastic effects have been deeply discussed. Some sufficient conditions for the considered system to be robustly stable are derived in terms of linear matrix inequalities (LMIs). In addition, an example [...] Read more.
Robust stabilization and H ∞ controller design for uncertain systems with impulsive and stochastic effects have been deeply discussed. Some sufficient conditions for the considered system to be robustly stable are derived in terms of linear matrix inequalities (LMIs). In addition, an example with simulations is given to better demonstrate the usefulness of the proposed H ∞ controller design method. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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18 pages, 306 KiB  
Article
Existence, Uniqueness and Exponential Stability of Periodic Solution for Discrete-Time Delayed BAM Neural Networks Based on Coincidence Degree Theory and Graph Theoretic Method
by Manickam Iswarya, Ramachandran Raja, Grienggrai Rajchakit, Jinde Cao, Jehad Alzabut and Chuangxia Huang
Mathematics 2019, 7(11), 1055; https://doi.org/10.3390/math7111055 - 04 Nov 2019
Cited by 45 | Viewed by 2483
Abstract
In this work, a general class of discrete time bidirectional associative memory (BAM) neural networks (NNs) is investigated. In this model, discrete and continuously distributed time delays are taken into account. By utilizing this novel method, which incorporates the approach of Kirchhoff’s matrix [...] Read more.
In this work, a general class of discrete time bidirectional associative memory (BAM) neural networks (NNs) is investigated. In this model, discrete and continuously distributed time delays are taken into account. By utilizing this novel method, which incorporates the approach of Kirchhoff’s matrix tree theorem in graph theory, Continuation theorem in coincidence degree theory and Lyapunov function, we derive a few sufficient conditions to ensure the existence, uniqueness and exponential stability of the periodic solution of the considered model. At the end of this work, we give a numerical simulation that shows the effectiveness of this work. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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17 pages, 291 KiB  
Article
Multi Fractals of Generalized Multivalued Iterated Function Systems in b-Metric Spaces with Applications
by Sudesh Kumari, Renu Chugh, Jinde Cao and Chuangxia Huang
Mathematics 2019, 7(10), 967; https://doi.org/10.3390/math7100967 - 14 Oct 2019
Cited by 16 | Viewed by 2406
Abstract
In this paper, we obtain multifractals (attractors) in the framework of Hausdorff b-metric spaces. Fractals and multifractals are defined to be the fixed points of associated fractal operators, which are known as attractors in the literature of fractals. We extend the results [...] Read more.
In this paper, we obtain multifractals (attractors) in the framework of Hausdorff b-metric spaces. Fractals and multifractals are defined to be the fixed points of associated fractal operators, which are known as attractors in the literature of fractals. We extend the results obtained by Chifu et al. (2014) and N.A. Secelean (2015) and generalize the results of Nazir et al. (2016) by using the assumptions imposed by Dung et al. (2017) to the case of ciric type generalized multi-iterated function system (CGMIFS) composed of ciric type generalized multivalued G-contractions defined on multifractal space C ( U ) in the framework of a Hausdorff b-metric space, where U = U 1 × U 2 × ⋯ × U N , N being a finite natural number. As an application of our study, we derive collage theorem which can be used to construct general fractals and to solve inverse problem in Hausdorff b-metric spaces which are more general spaces than Hausdorff metric spaces. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
19 pages, 1179 KiB  
Article
Dynamic Properties of Foreign Exchange Complex Network
by Xin Yang, Shigang Wen, Zhifeng Liu, Cai Li and Chuangxia Huang
Mathematics 2019, 7(9), 832; https://doi.org/10.3390/math7090832 - 09 Sep 2019
Cited by 60 | Viewed by 4774
Abstract
The foreign exchange (FX) market, one of the important components of the financial market, is a typical complex system. In this paper, by resorting to the complex network method, we use the daily closing prices of 41 FX markets to build the dynamical [...] Read more.
The foreign exchange (FX) market, one of the important components of the financial market, is a typical complex system. In this paper, by resorting to the complex network method, we use the daily closing prices of 41 FX markets to build the dynamical networks and their minimum spanning tree (MST) maps by virtue of a moving window correlation coefficient. The properties of FX networks are characterized by the normalized tree length, node degree distributions, centrality measures and edge survival ratios. Empirical results show that: (i) the normalized tree length plays a role in identifying crises and is negatively correlated with the market return and volatility; (ii) 83% of FX networks follow power-law node degree distribution, which means that the FX market is a typical heterogeneous market, and a few hub nodes play key roles in the market; (iii) the highest centrality measures reveal that the USD, EUR and CNY are the three most powerful currencies in FX markets; and (iv) the edge survival ratio analysis implies that the FX structure is relatively stable. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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23 pages, 372 KiB  
Article
Hybrid Control Scheme for Projective Lag Synchronization of Riemann–Liouville Sense Fractional Order Memristive BAM NeuralNetworks with Mixed Delays
by Grienggrai Rajchakit, Anbalagan Pratap, Ramachandran Raja, Jinde Cao, Jehad Alzabut and Chuangxia Huang
Mathematics 2019, 7(8), 759; https://doi.org/10.3390/math7080759 - 19 Aug 2019
Cited by 119 | Viewed by 4525
Abstract
This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop [...] Read more.
This sequel is concerned with the analysis of projective lag synchronization of Riemann–Liouville sense fractional order memristive BAM neural networks (FOMBNNs) with mixed time delays via hybrid controller. Firstly, a new type of hybrid control scheme, which is the combination of open loop control and adaptive state feedback control is designed to guarantee the global projective lag synchronization of the addressed FOMBNNs model. Secondly, by using a Lyapunov–Krasovskii functional and Barbalet’s lemma, a new brand of sufficient criterion is proposed to ensure the projective lag synchronization of the FOMBNNs model considered. Moreover, as special cases by using a hybrid control scheme, some sufficient conditions are derived to ensure the global projective synchronization, global complete synchronization and global anti-synchronization for the FOMBNNs model considered. Finally, numerical simulations are provided to check the accuracy and validity of our obtained synchronization results. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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10 pages, 337 KiB  
Article
Global Stability of Fractional Order Coupled Systems with Impulses via a Graphic Approach
by Bei Zhang, Yonghui Xia, Lijuan Zhu, Haidong Liu and Longfei Gu
Mathematics 2019, 7(8), 744; https://doi.org/10.3390/math7080744 - 15 Aug 2019
Cited by 7 | Viewed by 2599
Abstract
Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between [...] Read more.
Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between fractional-order system and integer-order system ends the paper. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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16 pages, 370 KiB  
Article
Practical Stability with Respect to h-Manifolds for Impulsive Control Functional Differential Equations with Variable Impulsive Perturbations
by Gani Stamov, Ivanka Stamova, Xiaodi Li and Ekaterina Gospodinova
Mathematics 2019, 7(7), 656; https://doi.org/10.3390/math7070656 - 21 Jul 2019
Cited by 11 | Viewed by 2428
Abstract
The present paper is devoted to the problems of practical stability with respect to h-manifolds for impulsive control differential equations with variable impulsive perturbations. We will consider these problems in light of the Lyapunov–Razumikhin method of piecewise continuous functions. The new results [...] Read more.
The present paper is devoted to the problems of practical stability with respect to h-manifolds for impulsive control differential equations with variable impulsive perturbations. We will consider these problems in light of the Lyapunov–Razumikhin method of piecewise continuous functions. The new results are applied to an impulsive control cellular neural network model with variable impulsive perturbations. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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10 pages, 254 KiB  
Article
Robust Synchronization of Fractional-Order Uncertain Chaotic Systems Based on Output Feedback Sliding Mode Control
by Chao Song, Shumin Fei, Jinde Cao and Chuangxia Huang
Mathematics 2019, 7(7), 599; https://doi.org/10.3390/math7070599 - 05 Jul 2019
Cited by 68 | Viewed by 3123
Abstract
This paper mainly focuses on the robust synchronization issue for drive-response fractional-order chaotic systems (FOCS) when they have unknown parameters and external disturbances. In order to achieve the goal, the sliding mode control scheme only using output information is designed, and at the [...] Read more.
This paper mainly focuses on the robust synchronization issue for drive-response fractional-order chaotic systems (FOCS) when they have unknown parameters and external disturbances. In order to achieve the goal, the sliding mode control scheme only using output information is designed, and at the same time, the structures of a sliding mode surface and a sliding mode controller are also constructed. A sufficient criterion is presented to ensure the robust synchronization of FOCS according to the stability theory of the fractional calculus and sliding mode control technique. In addition, the result can be applied to identical or non-identical chaotic systems with fractional-order. In the end, we build two practical examples to illustrate the feasibility of our theoretical results. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
15 pages, 400 KiB  
Article
Global Stability of a Markovian Jumping Chaotic Financial System with Partially Unknown Transition Rates under Impulsive Control Involved in the Positive Interest Rate
by Ruofeng Rao
Mathematics 2019, 7(7), 579; https://doi.org/10.3390/math7070579 - 28 Jun 2019
Cited by 16 | Viewed by 2098
Abstract
The intrinsic instability of the financial system itself results in chaos and unpredictable economic behavior. To gain the globally asymptotic stability of the equilibrium point with a positive interest rate of the chaotic financial system, pulse control is sometimes very necessary and is [...] Read more.
The intrinsic instability of the financial system itself results in chaos and unpredictable economic behavior. To gain the globally asymptotic stability of the equilibrium point with a positive interest rate of the chaotic financial system, pulse control is sometimes very necessary and is employed in this paper to derive the globally exponential stability of financial system. It should be pointed out that the delayed feedback model brings an essential difficulty so that the regional control method has to be adopted. In this paper, the author firstly employs impulsive control, regional control, the Lyapunov function technique, and variational methods to derive the stochastically globally asymptotic stability criterion of the economic balance point with a positive interest rate for a delayed feedback financial system with Markovian jumping and partially unknown transition rates. Besides, the mathematical induction method and the proof by contradiction are applied synthetically to deduce the globally exponential stability of the equilibrium point with a positive interest rate for the impulsive financial system without time-delays. Moreover, numerical examples illustrate that under suitable data conditions on the two main criteria mentioned above, the interest rates are positive decimals when the financial system reaches stability, which means better economic significance. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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16 pages, 545 KiB  
Article
Anti-Synchronization of a Class of Chaotic Systems with Application to Lorenz System: A Unified Analysis of the Integer Order and Fractional Order
by Liang Chen, Chengdai Huang, Haidong Liu and Yonghui Xia
Mathematics 2019, 7(6), 559; https://doi.org/10.3390/math7060559 - 19 Jun 2019
Cited by 9 | Viewed by 2674
Abstract
The paper proves a unified analysis for finite-time anti-synchronization of a class of integer-order and fractional-order chaotic systems. We establish an effective controller to ensure that the chaotic system with unknown parameters achieves anti-synchronization in finite time under our controller. Then, we apply [...] Read more.
The paper proves a unified analysis for finite-time anti-synchronization of a class of integer-order and fractional-order chaotic systems. We establish an effective controller to ensure that the chaotic system with unknown parameters achieves anti-synchronization in finite time under our controller. Then, we apply our results to the integer-order and fractional-order Lorenz system, respectively. Finally, numerical simulations are presented to show the feasibility of the proposed control scheme. At the same time, through the numerical simulation results, it is show that for the Lorenz chaotic system, when the order is greater, the more quickly is anti-synchronization achieved. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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Review

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28 pages, 569 KiB  
Review
Review of Some Control Theory Results on Uniform Stability of Impulsive Systems
by Bin Liu, Bo Xu, Guohua Zhang and Lisheng Tong
Mathematics 2019, 7(12), 1186; https://doi.org/10.3390/math7121186 - 04 Dec 2019
Cited by 11 | Viewed by 2737
Abstract
This paper aims to review some uniform stability results for impulsive systems. For the review, we classify the models of impulsive systems into time-based impulsive systems and state-based ones, including continuous-time impulsive systems, discrete-time impulsive systems, stochastic impulsive systems, and impulsive hybrid systems. [...] Read more.
This paper aims to review some uniform stability results for impulsive systems. For the review, we classify the models of impulsive systems into time-based impulsive systems and state-based ones, including continuous-time impulsive systems, discrete-time impulsive systems, stochastic impulsive systems, and impulsive hybrid systems. According to these models, we review, respectively, the related stability concepts and some representative results focused on uniform stability, including the results on uniform asymptotic stability, input-to-state stability (ISS), KLL -stability (uniform stability expressed by KLL -functions), event-stability, and event-ISS. And we formulate some questions for those not fully developed aspects on uniform stability at each subsection. Full article
(This article belongs to the Special Issue Impulsive Control Systems and Complexity)
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