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Mathematics
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Set-Based Methods for Differential Equations and Applications
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Set-Based Methods for Differential Equations and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (30 June 2023) | Viewed by 6344

Special Issue Editor

Prof. Dr. Julien Alexandre dit Sandretto

E-Mail Website
Guest Editor
Computer Science and System Engineering Department, ENSTA, Paris, France
Interests: interval analysis; constraint programming; ordinary differential equations; validated methods; control synthesis; parameter estimation

Special Issue Information

Dear Colleagues,

This Special Issue of Mathematics aims to serve as a platform for the publication of new set-based methods and approaches in the field of differential equations. Theoretical and computer science aspects are welcome, such as set abstraction, interval analysis, ellipsoid techniques, and so on. Moreover, applications using set-based methods with differential equations are also strongly encouraged.

Possible topics of interest include the following:

  • Set abstraction: intervals, zonotopes, polytopes, ellipsoids, etc.
  • Differential equations: ordinary, algebraic, partial, etc.
  • Applications: parameter identification, control synthesis, safety verification, etc.

Prof. Dr. Julien Alexandre dit Sandretto
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

  • Ordinary differential equations
  • Interval analysis
  • Validated methods
  • Control synthesis
  • Parameter estimation

Published Papers (4 papers)

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Research

16 pages, 557 KiB  
Article
Reachset Conformance and Automatic Model Adaptation for Hybrid Systems
by Hendrik Roehm, Alexander Rausch and Matthias Althoff
Mathematics 2022, 10(19), 3567; https://doi.org/10.3390/math10193567 - 29 Sep 2022
Cited by 1 | Viewed by 949
Abstract
Model-based verification uses a model to reason about the correctness of a real system. This requires the model and the system to be conformant, such that verification results on the model can be transfered to the real system. Especially for hybrid systems, which [...] Read more.
Model-based verification uses a model to reason about the correctness of a real system. This requires the model and the system to be conformant, such that verification results on the model can be transfered to the real system. Especially for hybrid systems, which combine discrete and continuous behavior, defining and checking conformance is a difficult task. In this work, we present reachset conformance for hybrid systems that transfers safety properties from a model to the real system. We show how a model can be adapted to be conformant to measurements of a real system and demonstrate this for a real autonomous vehicle. The obtained reachset conformant model can be used for the verification of safety-critical properties, such as collision avoidance. Full article
(This article belongs to the Special Issue Set-Based Methods for Differential Equations and Applications)
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14 pages, 450 KiB  
Article
Set-Based B-Series
by Julien Alexandre dit Sandretto
Mathematics 2022, 10(17), 3165; https://doi.org/10.3390/math10173165 - 02 Sep 2022
Cited by 1 | Viewed by 967
Abstract
B-series were defined to unify the formalism of solutions for ordinary differential equations defined by series. Runge–Kutta schemes can be seen as truncated B-series, similar to Taylor series. In the prolific domain of reachability analysis, i.e., the process of computing the set of [...] Read more.
B-series were defined to unify the formalism of solutions for ordinary differential equations defined by series. Runge–Kutta schemes can be seen as truncated B-series, similar to Taylor series. In the prolific domain of reachability analysis, i.e., the process of computing the set of reachable states for a system, many techniques have been proposed without obvious links. In the particular case of uncertain initial conditions and/or parameters in the definition of differential equations, set-based approaches are a natural and elegant method to compute reachable sets. In this paper, an extension to B-series is proposed to merge these techniques in a common formalism—named set-based B-series. We show that the main properties of B-series are preserved. A validated technique, based on Runge–Kutta methods, able to compute such series, is presented. Experiments are provided in order to illustrate the proposed approach. Full article
(This article belongs to the Special Issue Set-Based Methods for Differential Equations and Applications)
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18 pages, 1856 KiB  
Article
Reliability Assessment of an Unscented Kalman Filter by Using Ellipsoidal Enclosure Techniques
by Andreas Rauh, Stefan Wirtensohn, Patrick Hoher, Johannes Reuter and Luc Jaulin
Mathematics 2022, 10(16), 3011; https://doi.org/10.3390/math10163011 - 21 Aug 2022
Cited by 3 | Viewed by 1645
Abstract
The Unscented Kalman Filter (UKF) is widely used for the state, disturbance, and parameter estimation of nonlinear dynamic systems, for which both process and measurement uncertainties are represented in a probabilistic form. Although the UKF can often be shown to be more reliable [...] Read more.
The Unscented Kalman Filter (UKF) is widely used for the state, disturbance, and parameter estimation of nonlinear dynamic systems, for which both process and measurement uncertainties are represented in a probabilistic form. Although the UKF can often be shown to be more reliable for nonlinear processes than the linearization-based Extended Kalman Filter (EKF) due to the enhanced approximation capabilities of its underlying probability distribution, it is not a priori obvious whether its strategy for selecting sigma points is sufficiently accurate to handle nonlinearities in the system dynamics and output equations. Such inaccuracies may arise for sufficiently strong nonlinearities in combination with large state, disturbance, and parameter covariances. Then, computationally more demanding approaches such as particle filters or the representation of (multi-modal) probability densities with the help of (Gaussian) mixture representations are possible ways to resolve this issue. To detect cases in a systematic manner that are not reliably handled by a standard EKF or UKF, this paper proposes the computation of outer bounds for state domains that are compatible with a certain percentage of confidence under the assumption of normally distributed states with the help of a set-based ellipsoidal calculus. The practical applicability of this approach is demonstrated for the estimation of state variables and parameters for the nonlinear dynamics of an unmanned surface vessel (USV). Full article
(This article belongs to the Special Issue Set-Based Methods for Differential Equations and Applications)
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20 pages, 6612 KiB  
Article
Proving Feasibility of a Docking Mission: A Contractor Programming Approach
by Auguste Bourgois, Simon Rohou, Luc Jaulin and Andreas Rauh
Mathematics 2022, 10(7), 1130; https://doi.org/10.3390/math10071130 - 01 Apr 2022
Cited by 1 | Viewed by 1800
Abstract
Recent advances in computational power, algorithms, and sensors allow robots to perform complex and dangerous tasks, such as autonomous missions in space or underwater. Given the high operational costs, simulations are run beforehand to predict the possible outcomes of a mission. However, this [...] Read more.
Recent advances in computational power, algorithms, and sensors allow robots to perform complex and dangerous tasks, such as autonomous missions in space or underwater. Given the high operational costs, simulations are run beforehand to predict the possible outcomes of a mission. However, this approach is limited as it is based on parameter space discretization and therefore cannot be considered a proof of feasibility. To overcome this limitation, set-membership methods based on interval analysis, guaranteed integration, and contractor programming have proven their efficiency. Guaranteed integration algorithms can predict the possible trajectories of a system initialized in a given set in the form of tubes of trajectories. The contractor programming consists in removing the trajectories violating predefined constraints from a system’s tube of possible trajectories. Our contribution consists in merging both approaches to allow for the usage of differential constraints in a contractor programming framework. We illustrate our method through examples related to robotics. We also released an open-source implementation of our algorithm in a unified library for tubes, allowing one to combine it with other constraints and increase the number of possible applications. Full article
(This article belongs to the Special Issue Set-Based Methods for Differential Equations and Applications)
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