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Dynamics and Control Using Functional Interpolation
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Dynamics and Control Using Functional Interpolation

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

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 3224

Special Issue Editors

Prof. Dr. Daniele Mortari

E-Mail Website
Guest Editor
Department of Aerospace Engineering, Texas A&M University, 3141 TAMU, College Station, TX 77843-3141, USA
Interests: attitude and position determination systems; satellite constellations design; sensor data processing; algorithms and linear algebra
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Roberto Furfaro

E-Mail Website
Guest Editor
1. Systems & Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA
2. Aerospace & Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA
Interests: aerospace and mechanical engineering

Special Issue Information

Dear Colleagues,

This Special Issue of Mathematics on “Dynamics and Control Using Functional Interpolation” invites both original and survey articles that bring together new mathematical tools and numerical methods for studying general problems on dynamics or optimal control (direct or indirect methods) based on functional interpolation. This issue is motivated by the recent profusion and success of functional interpolation in aerospace problems (e.g., relative navigation, landing, orbit propagation, homotopy continuation) and its extension to general problems in dynamics and control.

Some topics of interest include numerical stability, convergence and complexity analysis, approximation models, continuation methods, numerical linear algebra, differential and integrodifferential equations (ordinary, partial), optimization, and the use of analytical and/or machine learning approaches.

All submissions must include a discussion of theoretical guarantees or at least justifications for the methods. Articles that explicitly address specific needs of real problems and/or connections to equivalent critical problems are particularly encouraged.

Prof. Dr. Daniele Mortari
Prof. Dr. Roberto Furfaro
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. 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

  • numerical stability
  • convergence and complexity analysis approximation models
  • continuation methods
  • numerical linear algebra
  • differential and integrodifferential equations (ordinary, partial)
  • optimization, and the use of analytical and/or machine-learning approaches

Published Papers (3 papers)

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Research

35 pages, 1568 KiB  
Article
A Theory of Functional Connections-Based hp-Adaptive Mesh Refinement Algorithm for Solving Hypersensitive Two-Point Boundary-Value Problems
by Kristofer Drozd, Roberto Furfaro and Andrea D’Ambrosio
Mathematics 2024, 12(9), 1360; https://doi.org/10.3390/math12091360 - 29 Apr 2024
Viewed by 188
Abstract
This manuscript introduces the first hp-adaptive mesh refinement algorithm for the Theory of Functional Connections (TFC) to solve hypersensitive two-point boundary-value problems (TPBVPs). The TFC is a mathematical framework that analytically satisfies linear constraints using an approximation method called a constrained expression. [...] Read more.
This manuscript introduces the first hp-adaptive mesh refinement algorithm for the Theory of Functional Connections (TFC) to solve hypersensitive two-point boundary-value problems (TPBVPs). The TFC is a mathematical framework that analytically satisfies linear constraints using an approximation method called a constrained expression. The constrained expression utilized in this work is composed of two parts. The first part consists of Chebyshev orthogonal polynomials, which conform to the solution of differentiation variables. The second part is a summation of products between switching and projection functionals, which satisfy the boundary constraints. The mesh refinement algorithm relies on the truncation error of the constrained expressions to determine the ideal number of basis functions within a segment’s polynomials. Whether to increase the number of basis functions in a segment or divide it is determined by the decay rate of the truncation error. The results show that the proposed algorithm is capable of solving hypersensitive TPBVPs more accurately than MATLAB R2021b’s bvp4c routine and is much better than the standard TFC method that uses global constrained expressions. The proposed algorithm’s main flaw is its long runtime due to the numerical approximation of the Jacobians. Full article
(This article belongs to the Special Issue Dynamics and Control Using Functional Interpolation)
24 pages, 969 KiB  
Article
Physics-Informed Neural Networks and Functional Interpolation for Solving the Matrix Differential Riccati Equation
by Kristofer Drozd, Roberto Furfaro, Enrico Schiassi and Andrea D’Ambrosio
Mathematics 2023, 11(17), 3635; https://doi.org/10.3390/math11173635 - 23 Aug 2023
Viewed by 982
Abstract
In this manuscript, we explore how the solution of the matrix differential Riccati equation (MDRE) can be computed with the Extreme Theory of Functional Connections (X-TFC). X-TFC is a physics-informed neural network that uses functional interpolation to analytically satisfy linear constraints, such as [...] Read more.
In this manuscript, we explore how the solution of the matrix differential Riccati equation (MDRE) can be computed with the Extreme Theory of Functional Connections (X-TFC). X-TFC is a physics-informed neural network that uses functional interpolation to analytically satisfy linear constraints, such as the MDRE’s terminal constraint. We utilize two approaches for solving the MDRE with X-TFC: direct and indirect implementation. The first approach involves solving the MDRE directly with X-TFC, where the matrix equations are vectorized to form a system of first order differential equations and solved with iterative least squares. In the latter approach, the MDRE is first transformed into a matrix differential Lyapunov equation (MDLE) based on the anti-stabilizing solution of the algebraic Riccati equation. The MDLE is easier to solve with X-TFC because it is linear, while the MDRE is nonlinear. Furthermore, the MDLE solution can easily be transformed back into the MDRE solution. Both approaches are validated by solving a fluid catalytic reactor problem and comparing the results with several state-of-the-art methods. Our work demonstrates that the first approach should be performed if a highly accurate solution is desired, while the second approach should be used if a quicker computation time is needed. Full article
(This article belongs to the Special Issue Dynamics and Control Using Functional Interpolation)
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18 pages, 643 KiB  
Article
Theory of Functional Connections Extended to Fractional Operators
by Daniele Mortari, Roberto Garrappa and Luigi Nicolò
Mathematics 2023, 11(7), 1721; https://doi.org/10.3390/math11071721 - 04 Apr 2023
Cited by 3 | Viewed by 1518
Abstract
The theory of functional connections, an analytical framework generalizing interpolation, was extended and applied in the context of fractional-order operators (integrals and derivatives). The extension was performed and presented for univariate functions, with the aim of determining the whole set of functions satisfying [...] Read more.
The theory of functional connections, an analytical framework generalizing interpolation, was extended and applied in the context of fractional-order operators (integrals and derivatives). The extension was performed and presented for univariate functions, with the aim of determining the whole set of functions satisfying some constraints expressed in terms of integrals and derivatives of non-integer order. The objective of these expressions was to solve fractional differential equations or other problems subject to fractional constraints. Although this work focused on the Riemann–Liouville definitions, the method is, however, more general, and it can be applied with different definitions of fractional operators just by changing the way they are computed. Three examples are provided showing, step by step, how to apply this extension for: (1) one constraint in terms of a fractional derivative, (2) three constraints (a function, a fractional derivative, and an integral), and (3) two constraints expressed in terms of linear combinations of fractional derivatives and integrals. Full article
(This article belongs to the Special Issue Dynamics and Control Using Functional Interpolation)
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