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Symmetry
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Symmetries in Dynamical Systems and Control Theory
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Symmetries in Dynamical Systems and Control Theory

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 15 May 2026 | Viewed by 3402

Special Issue Editors

Dr. Zhe Zhang

E-Mail Website
Guest Editor
1. School of Electrical Engineering, Guangxi University, Nanning, China
2. College of Electrical and Information Engineering, Hunan University, Changsha, China
Interests: fractional system dynamics; nonlinear control theory and technology; robot visual perception and control
Special Issues, Collections and Topics in MDPI journals
Dr. Hengyu Li

E-Mail Website
Guest Editor
School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China
Interests: robot perception and control; multi-robot cooperative control
Dr. Zhinan Peng

E-Mail Website
Guest Editor
School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
Interests: fractional-order dynamic systems; multiagent systems; reinforcement learning; robot control
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Symmetry in dynamical systems and control theory plays a significant role in system modeling, analysis, and control design. Symmetry assists in identifying and elucidating the system’s fundamental structure while simplifying dynamic analysis, thereby facilitating the design of efficient and resilient controllers. In recent years, with the improvement of mathematical tools and computing power, significant progress has been made in the application research of symmetry in complex dynamical systems, nonlinear control, multi-agent systems, robotics and network systems. This progress can be attributed to improvements in mathematical tools and computing power. This Special Issue aims to gather the latest theoretical achievements and engineering applications related to symmetry in power systems and control theory. We welcome and cordially invite you to engage in discourse on symmetry theory, dynamic analysis methods, symmetry-preserving control, and the utilization of symmetry in robot systems in power systems and control. Original research submissions in areas such as bifurcation and pattern formation under symmetric structures, Lie group and Lie algebra methods, and innovative applications of the combination of artificial intelligence and symmetry are required to promote theoretical innovation and cross-integration in this field.

Dr. Zhe Zhang
Dr. Hengyu Li
Dr. Zhinan Peng
Guest Editors

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 250 words) can be sent to the Editorial Office for assessment.

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. Symmetry is an international peer-reviewed open access monthly 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 2400 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

  • symmetry of the power system
  • control theory
  • robot motion planning and symmetry control
  • symmetry dynamics
  • symmetrical collaboration
  • symmetry maintenance control
  • Lie groups and Lie algebra methods
  • research on the combination of artificial intelligence and symmetry

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (4 papers)

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Research

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18 pages, 4660 KB  
Article
Symmetry Error Cost Function-Based Modular Robot Tracking Control: An Approximate Dynamic Programming Cooperative Game Approach
by Bing Ma, Zebin Ji, Yi Qin, Hucheng Jiang and Tianjiao An
Symmetry 2026, 18(2), 327; https://doi.org/10.3390/sym18020327 - 11 Feb 2026
Cited by 1 | Viewed by 332
Abstract
To address the issue the that traditional positive definite quadratic cost function, which incorporates both state and control variables, tends to approach infinity over an infinite time horizon in tracking problems—thus rendering optimization infeasible—this paper proposes a symmetric error cost function-based approach for [...] Read more.
To address the issue the that traditional positive definite quadratic cost function, which incorporates both state and control variables, tends to approach infinity over an infinite time horizon in tracking problems—thus rendering optimization infeasible—this paper proposes a symmetric error cost function-based approach for the tracking control of modular robots. The dynamic model of the modular robot system is constructed using joint torque feedback technology. By adopting the concept of approximate dynamic programming, each module of the system is treated as a participant in a cooperative game, transforming the trajectory tracking problem into an optimal control formulation. A critic fuzzy network is employed to approximate the system’s cost function, thereby deriving the optimal tracking control policy. The stability of the closed-loop system is demonstrated through the stability theorem, and the effectiveness of the proposed algorithm is verified via an experimental platform. Full article
(This article belongs to the Special Issue Symmetries in Dynamical Systems and Control Theory)
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19 pages, 607 KB  
Article
The Stability of Linear Control Systems on Low-Dimensional Lie Groups
by Víctor Ayala, William Eduardo Valdivia Hanco, Jhon Eddy Pariapaza Mamani and María Luisa Torreblanca Todco
Symmetry 2025, 17(10), 1766; https://doi.org/10.3390/sym17101766 - 20 Oct 2025
Viewed by 733
Abstract
This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where trajectories respect the group’s intrinsic symmetries. Stability notions, such [...] Read more.
This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where trajectories respect the group’s intrinsic symmetries. Stability notions, such as inner asymptotic, inner, and input–output (BIBO) stability, are studied. The qualitative behavior of solutions is shown to depend critically on the spectral decomposition of derivations associated with the drift, and on the algebraic structure of the underlying Lie algebra. We study two classes of examples in detail: Abelian and solvable two-dimensional Lie groups, and the three-dimensional nilpotent Heisenberg group. These settings, while mathematically tractable, retain essential features of non-commutativity, geometric non-linearity, and sub-Riemannian geometry, making them canonical models in control theory. The results highlight the interplay between algebraic properties, invariant submanifolds, and trajectory behavior, offering insights applicable to robotic motion planning, quantum control, and signal processing. Full article
(This article belongs to the Special Issue Symmetries in Dynamical Systems and Control Theory)
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15 pages, 3531 KB  
Article
Cooperative Differential Game-Based Modular Unmanned System Approximate Optimal Control: An Adaptive Critic Design Approach
by Liang Si, Yebao Liu, Luyang Zhong and Yuhan Qian
Symmetry 2025, 17(10), 1665; https://doi.org/10.3390/sym17101665 - 6 Oct 2025
Viewed by 777
Abstract
An approximate optimal control issue for modular unmanned systems (MUSs) is presented via a cooperative differential game for solving the trajectory tracking problem. Initially, the modular unmanned system’s dynamic model is built with the joint torque feedback technique. The moment of inertia of [...] Read more.
An approximate optimal control issue for modular unmanned systems (MUSs) is presented via a cooperative differential game for solving the trajectory tracking problem. Initially, the modular unmanned system’s dynamic model is built with the joint torque feedback technique. The moment of inertia of the motor rotor has positive symmetry. Each MUS module is deemed as a participant in the cooperative differential game. Then, the MUS trajectory tracking problem is transformed into an approximate optimal control problem by means of adaptive critic design (ACD). The approximate optimal control is obtained by the critic network, approaching the joint performance index function of the system. The stability of the closed-loop system is proved through Lyapunov theory. The feasibility of the proposed control algorithm is verified by an experimental platform. Full article
(This article belongs to the Special Issue Symmetries in Dynamical Systems and Control Theory)
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Review

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24 pages, 648 KB  
Review
A Review of Control Sets of Linear Control Systems on Two-Dimensional Lie Groups and Applications
by Víctor Ayala, Jhon Eddy Pariapaza Mamani, William Eduardo Valdivia Hanco and María Luisa Torreblanca Todco
Symmetry 2025, 17(10), 1776; https://doi.org/10.3390/sym17101776 - 21 Oct 2025
Viewed by 837
Abstract
This review article explores the theory of control sets for linear control systems defined on two-dimensional Lie groups, with a focus on the plane R2 and the affine group Aff+(2). We systematically summarize recent advances, [...] Read more.
This review article explores the theory of control sets for linear control systems defined on two-dimensional Lie groups, with a focus on the plane R2 and the affine group Aff+(2). We systematically summarize recent advances, emphasizing how the geometric and algebraic structures inherent in low-dimensional Lie groups influence the formation, shape, and properties of control sets—maximal regions where controllability is maintained. Control sets with non-empty interiors are of particular interest as they characterize regions where the system can be steered between states via bounded inputs. The review highlights key results concerning the existence, uniqueness, and boundedness of these sets, including criteria based on the Ad-rank condition and orbit analysis. We also underscore the central role of the symmetry properties of Lie groups, which facilitate the systematic classification and description of control sets, linking the abstract mathematical framework to concrete, physically motivated applications. To illustrate the practical relevance of the theory, we present examples from mechanics, motion planning, and neuroscience, demonstrating how control sets naturally emerge in diverse domains. Overall, this work aims to deepen the understanding of controllability regions in low-dimensional Lie group systems and to foster future research that bridges geometric control theory with applied problems. Full article
(This article belongs to the Special Issue Symmetries in Dynamical Systems and Control Theory)
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