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Symmetry
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Symmetry and Control of Discrete and Continuous Systems
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Symmetry and Control of Discrete and Continuous Systems

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

Deadline for manuscript submissions: closed (30 April 2022) | Viewed by 3481

Special Issue Editors

Prof. Dr. Marco Favretti

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Guest Editor
Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, 35121 Padova, Italy
Interests: maximum entropy principle; statistical ecology; geometric control theory and nonholonomic mechanics
Prof. Dr. Franco Cardin

E-Mail Website
Guest Editor
Department of Mathematics “Tullio Levi-Civita”, University of Padua, via Trieste 63, Padua, Italy
Interests: discrete and continuous dynamical systems; symplectic geometry and topology; calculus of variations; Hamilton-Jacobi equation
Prof. Dr. Andrea Giacobbe

E-Mail Website
Guest Editor
Department of Mathematics Università degli Studi di Catania, 95131 Catania, Italy
Interests: geometry of Hamiltonian and non-holonomic systems; dynamics of epidemiological systems; stability in fluid dynamics

Special Issue Information

Dear colleagues,

Physical theories derived from variational principle symmetries are a key feature because they reveal the existence of conserved quantities (first integrals of the motion). This aspect has been fully investigated using modern geometric mechanics from the first ancestral results (Noether theorem) to more recent and fruitful notions (Souriau’s moment map, Marsden reduction, Hannay–Berry holonomy, to cite a few). The presence of symmetries reduces the dimensionality of the system.

In a different vein, a variational theory may account for externally imposed constraints on the system evolution by the Lagrange multipliers method. The geometry of constrained variational principles is described in terms of Lagrangian (or Legendre) submanifolds, and it has a pivotal role for the geometric formulation of thermodynamics and statistical mechanics via the maximum entropy principle. Moreover, constrained variational principles are at the core of the modern geometric formulation of control theory for mechanical systems, nonholonomic mechanics, optimal transport theory, and finite exact reduction. We think that the interplay between variational principles, conserved quantities and active control is a fruitful framework for theoretical development and the design of technological applications. We therefore invite the scientific community to contribute to this issue with both theoretical and applied papers.

Prof. Dr. Marco Favretti
Prof. Dr. Franco Cardin
Prof. Dr. Andrea Giacobbe 
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 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. 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

  • Variational formulation of physical theories
  • Geometric control
  • Symplectic reduction
  • Optimal transport
  • Geometric thermodynamics

Published Papers (3 papers)

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Research

23 pages, 1862 KiB  
Article
A Pareto–Pontryagin Maximum Principle for Optimal Control
by Alberto Lovison and Franco Cardin
Symmetry 2022, 14(6), 1169; https://doi.org/10.3390/sym14061169 - 06 Jun 2022
Cited by 3 | Viewed by 1540
Abstract
In this paper, an attempt to unify two important lines of thought in applied optimization is proposed. We wish to integrate the well-known (dynamic) theory of Pontryagin optimal control with the Pareto optimization (of the static type), involving the maximization/minimization of a non-trivial [...] Read more.
In this paper, an attempt to unify two important lines of thought in applied optimization is proposed. We wish to integrate the well-known (dynamic) theory of Pontryagin optimal control with the Pareto optimization (of the static type), involving the maximization/minimization of a non-trivial number of functions or functionals, Pontryagin optimal control offers the definitive theoretical device for the dynamic realization of the objectives to be optimized. The Pareto theory is undoubtedly less known in mathematical literature, even if it was studied in topological and variational details (Morse theory) by Stephen Smale. This reunification, obviously partial, presents new conceptual problems; therefore, a basic review is necessary and desirable. After this review, we define and unify the two theories. Finally, we propose a Pontryagin extension of a recent multiobjective optimization application to the evolution of trees and the related anatomy of the xylems. This work is intended as the first contribution to a series to be developed by the authors on this subject. Full article
(This article belongs to the Special Issue Symmetry and Control of Discrete and Continuous Systems)
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17 pages, 1301 KiB  
Article
Rearrangement of Energy Levels between Energy Super-Bands Characterized by Second Chern Class
by Dmitrii Sadovskii and Boris Zhilinskii
Symmetry 2022, 14(2), 183; https://doi.org/10.3390/sym14020183 - 18 Jan 2022
Cited by 2 | Viewed by 1201
Abstract
We generalize the dynamical analog of the Berry geometric phase setup to the quaternionic model of Avron et al. In our dynamical quaternionic system, the fast half-integer spin subsystem interacts with a slow two-degrees-of-freedom subsystem. The model is invariant under the 1:1:2 weighted [...] Read more.
We generalize the dynamical analog of the Berry geometric phase setup to the quaternionic model of Avron et al. In our dynamical quaternionic system, the fast half-integer spin subsystem interacts with a slow two-degrees-of-freedom subsystem. The model is invariant under the 1:1:2 weighted SO(2) symmetry and spin inversion. There is one formal control parameter in addition to four dynamical variables of the slow subsystem. We demonstrate that the most elementary qualitative phenomenon associated with the rearrangement of the energy super-bands of our model consists of the rearrangement of one energy level between two energy superbands which takes place when the formal control parameter takes the special isolated value associated with the conical degeneracy of the semi-quantum eigenvalues. This qualitative phenomenon is of topological origin, and is characterized by the second Chern class of the associated semi-quantum system. The correspondence between the number of redistributed energy levels and the second Chern number is confirmed through a series of examples. Full article
(This article belongs to the Special Issue Symmetry and Control of Discrete and Continuous Systems)
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8 pages, 256 KiB  
Article
Hamilton–Jacobi Homogenization and the Isospectral Problem
by Lorenzo Zanelli
Symmetry 2021, 13(7), 1196; https://doi.org/10.3390/sym13071196 - 02 Jul 2021
Viewed by 896
Abstract
We consider the homogenization theory for Hamilton–Jacobi equations on the one-dimensional flat torus in connection to the isospectrality problem of Schrödinger operators. In particular, we link the equivalence of effective Hamiltonians provided by the weak KAM theory with the class of the corresponding [...] Read more.
We consider the homogenization theory for Hamilton–Jacobi equations on the one-dimensional flat torus in connection to the isospectrality problem of Schrödinger operators. In particular, we link the equivalence of effective Hamiltonians provided by the weak KAM theory with the class of the corresponding operators exhibiting the same spectrum. Full article
(This article belongs to the Special Issue Symmetry and Control of Discrete and Continuous Systems)
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