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Mathematics
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Advances in Nonlinear Analysis and Control
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Advances in Nonlinear Analysis and Control

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (29 February 2024) | Viewed by 1328

Special Issue Editor

Prof. Dr. Vasilii A. Gromov

E-Mail Website
Guest Editor
School of Data Analysis and Artificial Intelligence, National Research University Higher School of Economics, 109028 Moscow, Russia
Interests: bifurcation theory; non-linear boundary problems for PDEs; numerical methods; complex systems; global models; language as a system; chaotic time series

Special Issue Information

Dear Colleagues,

Ilya Prigogine, a Nobel Prize winner, once said: ‘The world is nonlinear, temporal, and stochastic’. Linear systems tend to be just linearized nonlinear (unknown) ones; we should necessarily verify whether the linearization procedure is adequate for given parameter values.

Nonlinear phenomena are responsible complex for systems behavior. We believe that nonlinearity constitutes the core of complexity, and nonlinear processes, their genesis, evolution, and termination, form the subject matter of this Special Issue.

In the present Issue, we would like to bring together studies on both theoretical aspects and real-world applications of nonlinearity; we are interested in natural and humanitarian science applications. Eugene Wigner, the other Nobel Prize winner, wrote about ‘the unreasonable effectiveness of mathematics in the natural sciences’. We believe that pure and applied mathematics constitute the same field of science, and that complex systems theory bridges the gap between them.   

It would be our pleasure to see your papers on complex systems, nonlinear science, self-organized systems, chaos theory, bifurcation theory, catastrophe theory, complexity, numerical methods, global models, nonlinear phenomena in natural and social sciences, language as a complex system, chaotic time series, AI and nonlinear processes, nonlinear control, and ‘nonlinear’ philosophy, to name just a few.

Prof. Dr. Vasilii A. Gromov
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

  • complex systems
  • nonlinear science
  • self-organized systems
  • chaos theory
  • bifurcation theory
  • catastrophe theory
  • complexity
  • numerical methods
  • global models
  • inverse problems
  • nonlinear phenomena in natural and social sciences
  • language as a complex system
  • chaotic time series
  • AI and nonlinear processes
  • nonlinear control
  • ‘nonlinear’ philosophy

Published Papers (2 papers)

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18 pages, 413 KiB  
Article
Near-Optimal Tracking Control of Partially Unknown Discrete-Time Nonlinear Systems Based on Radial Basis Function Neural Network
by Jiashun Huang, Dengguo Xu, Yahui Li and Yan Ma
Mathematics 2024, 12(8), 1146; https://doi.org/10.3390/math12081146 - 10 Apr 2024
Viewed by 358
Abstract
This paper proposes an optimal tracking control scheme through adaptive dynamic programming (ADP) for a class of partially unknown discrete-time (DT) nonlinear systems based on a radial basis function neural network (RBF-NN). In order to acquire the unknown system dynamics, we use two [...] Read more.
This paper proposes an optimal tracking control scheme through adaptive dynamic programming (ADP) for a class of partially unknown discrete-time (DT) nonlinear systems based on a radial basis function neural network (RBF-NN). In order to acquire the unknown system dynamics, we use two RBF-NNs; the first one is used to construct the identifier, and the other one is used to directly approximate the steady-state control input, where a novel adaptive law is proposed to update neural network weights. The optimal feedback control and the cost function are derived via feedforward neural network approximation, and a means of regulating the tracking error is proposed. The critic network and the actor network were trained online to obtain the solution of the associated Hamilton–Jacobi–Bellman (HJB) equation within the ADP framework. Simulations were carried out to verify the effectiveness of the optimal tracking control technique using the neural networks. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Control)
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20 pages, 20733 KiB  
Article
Chaotic Path-Planning Algorithm Based on Courbage–Nekorkin Artificial Neuron Model
by Dmitriy Kvitko, Vyacheslav Rybin, Oleg Bayazitov, Artur Karimov, Timur Karimov and Denis Butusov
Mathematics 2024, 12(6), 892; https://doi.org/10.3390/math12060892 - 18 Mar 2024
Viewed by 651
Abstract
Developing efficient path-planning algorithms is an essential topic in modern robotics and control theory. Autonomous rovers and wheeled and tracked robots require path generators that can efficiently cover the explorable space with minimal redundancy. In this paper, we present a new path-planning algorithm [...] Read more.
Developing efficient path-planning algorithms is an essential topic in modern robotics and control theory. Autonomous rovers and wheeled and tracked robots require path generators that can efficiently cover the explorable space with minimal redundancy. In this paper, we present a new path-planning algorithm based on the chaotic behavior of the Courbage–Nekorkin neuron model with a coverage control parameter. Our study aims to reduce the number of iterations required to cover the chosen investigated area, which is a typical efficiency criterion for this class of algorithms. To achieve this goal, we implemented a pseudorandom bit generator (PRBG) based on a Courbage–Nekorkin chaotic map, which demonstrates chaotic behavior and successfully passes all statistical tests for randomness. The proposed PRBG generates a bit sequence that can be used to move the tracked robot in four or eight directions in an operation area of arbitrary size. Several statistical metrics were applied to evaluate the algorithm’s performance, including the percentage of coverage of the study area and the uniformity of coverage. The performance of several competing path-planning algorithms was analyzed using the chosen metrics when exploring two test areas of the sizes 50 × 50 cells and 100 × 100 cells, respectively, in four and eight directions. The experimental results indicate that the proposed algorithm is superior compared to known chaotic path-planning methods, providing more rapid and uniform coverage with the possibility of controlling the covered area using tunable parameters. In addition, this study revealed the high dependence of the coverage rate on the starting point. To investigate how the coverage rate depends on the choice of chaotic map, we implemented six different PRBGs using various chaotic maps. The obtained results can be efficiently used for solving path-planning tasks in both real-life and virtual (e.g., video games) applications. Full article
(This article belongs to the Special Issue Advances in Nonlinear Analysis and Control)
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