Optimal Control Problems for Fractional Differential Equations

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: 31 May 2024 | Viewed by 4793

Special Issue Editor


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Guest Editor
Laboratoire LAMIA, Université des Antilles, Campus Fouillole, 97159 Pointe-à-Pitre, Guadeloupe, France
Interests: optimal control; controllability; partial differential equations; fractional partial differential equations

Special Issue Information

Dear Colleagues,

During these last four decades, fractional differential equations have gained interest in the modelling of real-world phenomena. Acting on models in order to achieve a desired goal is the aim of optimal control. The aim of this Special Issue is to report recent developments in the optimal control of fractional differential equations and inequalities involving time fractional and space fractional derivatives, including generalized fractional derivatives. This Special Issue will accept high-quality papers containing original research results and survey articles of exceptional merit on the optimal control of models in bounded and unbounded domains, as well as on networks.

Prof. Dr. Gisèle M. Mophou
Guest Editor

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Keywords

  • fractional differential equations
  • optimal control
  • fractional differential inequalities
  • time fractional and space fractional derivatives
  • generalized fractional derivatives
  • optimal control on bounded and unbounded domain
  • optimal control on network
  • Rieman-Liouville fractional derivative
  • Caputo fractional derivative
  • generalized fractional derivative
  • fractional Laplace operator
  • optimality systems
  • linear fractional differential equations
  • nonlinear fractional differential equations

Published Papers (4 papers)

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Research

11 pages, 305 KiB  
Article
Distributed Control for Non-Cooperative Systems Governed by Time-Fractional Hyperbolic Operators
by Hassan M. Serag, Areej A. Almoneef, Mahmoud El-Badawy and Abd-Allah Hyder
Fractal Fract. 2024, 8(5), 295; https://doi.org/10.3390/fractalfract8050295 - 16 May 2024
Viewed by 469
Abstract
This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which [...] Read more.
This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system’s state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed. Full article
(This article belongs to the Special Issue Optimal Control Problems for Fractional Differential Equations)
20 pages, 1369 KiB  
Article
Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls
by Günter Leugering, Vaibhav Mehandiratta and Mani Mehra
Fractal Fract. 2024, 8(3), 129; https://doi.org/10.3390/fractalfract8030129 - 22 Feb 2024
Viewed by 1109
Abstract
We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on [...] Read more.
We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method. Full article
(This article belongs to the Special Issue Optimal Control Problems for Fractional Differential Equations)
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27 pages, 565 KiB  
Article
Optimal Control for Neutral Stochastic Integrodifferential Equations with Infinite Delay Driven by Poisson Jumps and Rosenblatt Process
by Dimplekumar Chalishajar, Ramkumar Kasinathan and Ravikumar Kasinathan
Fractal Fract. 2023, 7(11), 783; https://doi.org/10.3390/fractalfract7110783 - 26 Oct 2023
Cited by 1 | Viewed by 1090
Abstract
In this paper, we investigate the optimal control problems for a class of neutral stochastic integrodifferential equations (NSIDEs) with infinite delay driven by Poisson jumps and the Rosenblat process in Hilbert space involving concrete-fading memory-phase space, in which we define the advanced phase [...] Read more.
In this paper, we investigate the optimal control problems for a class of neutral stochastic integrodifferential equations (NSIDEs) with infinite delay driven by Poisson jumps and the Rosenblat process in Hilbert space involving concrete-fading memory-phase space, in which we define the advanced phase space for infinite delay for the stochastic process. First, we introduce conditions that ensure the existence and uniqueness of mild solutions using stochastic analysis theory, successive approximation, and Grimmer’s resolvent operator theory. Next, we prove exponential stability, which includes mean square exponential stability, and this especially includes the exponential stability of solutions and their maps. Following that, we discuss the existence requirements of an optimal pair of systems governed by stochastic partial integrodifferential equations with infinite delay. Then, we explore examples that illustrate the potential of the main result, mainly in the heat equation, filter system, traffic signal light systems, and the biological processes in the human body. We conclude with a numerical simulation of the system studied. This work is a unique combination of the theory with practical examples and a numerical simulation. Full article
(This article belongs to the Special Issue Optimal Control Problems for Fractional Differential Equations)
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16 pages, 1167 KiB  
Article
The Improved Stochastic Fractional Order Gradient Descent Algorithm
by Yang Yang, Lipo Mo, Yusen Hu and Fei Long
Fractal Fract. 2023, 7(8), 631; https://doi.org/10.3390/fractalfract7080631 - 18 Aug 2023
Viewed by 1387
Abstract
This paper mainly proposes some improved stochastic gradient descent (SGD) algorithms with a fractional order gradient for the online optimization problem. For three scenarios, including standard learning rate, adaptive gradient learning rate, and momentum learning rate, three new SGD algorithms are designed combining [...] Read more.
This paper mainly proposes some improved stochastic gradient descent (SGD) algorithms with a fractional order gradient for the online optimization problem. For three scenarios, including standard learning rate, adaptive gradient learning rate, and momentum learning rate, three new SGD algorithms are designed combining a fractional order gradient and it is shown that the corresponding regret functions are convergent at a sub-linear rate. Then we discuss the impact of the fractional order on the convergence and monotonicity and prove that the better performance can be obtained by adjusting the order of the fractional gradient. Finally, several practical examples are given to verify the superiority and validity of the proposed algorithm. Full article
(This article belongs to the Special Issue Optimal Control Problems for Fractional Differential Equations)
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