Special Issue

The Investigation of Various Kinds of Regularity Mappings of Optimal Control Problem

• Submission Deadline: Mar. 01, 2024
• Status: Submission Closed
• Lead Guest Editor: Khelifa Djendel
Mathematical concepts often involve intricate mappings in their formulations, analytical investigations, and numerical analysis. Addressing optimization problems with constraints typically requires a solution finding procedure that incorporates approximation methods and computational algorithms. However, a critical question arises: does this procedure guarantee that the obtained solution is close to the exact solution, and can any solution of the optimization problem be approached by the generated procedure?
Approximation methods can be viewed as special perturbation versions of the original problem, introducing different kinds of regularity in set-valued mappings that map perturbations to solutions in the optimization problem. Consequently, the original problem can be replaced by a system of necessary optimality conditions, constructed using equations and inclusions. This necessitates investigating the regularity properties and stability analysis of the set-valued mappings (the system of first-order optimality conditions). In recent year, the optimal control problem for ordinary differential equations (ODEs) has made significant progress, enabling us to tackle previously insurmountable challenges. However, refining and perfecting this area, especially in the context of partial differential equations (PDEs), remains a pressing need.
The investigation of the optimal control problem for ODEs can be extended to explore relevant mathematical characteristics and properties in the context of PDEs. While research into various types of stability and regularity mappings has seen substantial advancements, new problems arising in science, engineering, and economics, coupled with emerging mathematical approaches, present attractive and worthy challenges for further exploration. Despite notable progress, there exists a dearth of quantitative regularity and stability results for the optimal control problem in PDEs, similar to the recently obtained results in ODEs. The disparity is primarily attributed to essential differences between the existence of optimal control for PDEs and ODEs, notably the compact property of the corresponding space. As PDEs involve more complex analysis tools, they demand a specialized approach.
Therefore, the special issue is particularly proposed to provide the opportunity for the development of several aspects of the regularity theory of optimal control problem for ODEs and PDEs. It will bring together scholars and researchers in relevant areas to debate the most recent advances, new research methodologies, and potential research topics. All original papers related to the investigation of regularity properties and stability analysis of the optimal control and optimization problems are most welcome. The potential topics of interest include, but are not limited to:
1. 1. Parabolic and Elleptic optimal control problem.
2. 2. Optimal control problem with boundary conditions, with delay.
3. 3. Regularity properties of mappings in optimal control.
4. 4. On the strong metric sub-regularity of the optimality mapping in mathematical programming, calculus variations, and optimal control problem.
5. 5. Stability analysis in optimal control problem with ODE and PDE (with integral equations as well).
6. 6. Finite element error analysis and solution stability of optimal control problems.

Keywords:

1. Optimal control problem.
2. Partial differential equations “ODEs”and”PDEs”.
3. Stability analysis.
4. Regularity properties.
5. Pontryagin maximum principle.
6. Sufficient and necessary optimality conditions.
7. Optimality mapping.
• Khelifa Djendel

Department of Mathematics and Informatics, University of Bordj Bou Arreridj., BordjBouArreridj, Algeria

Guest Editors
• Heng Zhou

Department of Computer and Information Science and Engineering, University of Florida, Gainesville, United States

Department of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China

• Yassine El Gantouh

Department of Mathematics, Zhejiang Normal University, Jinhua, China

• Zhongcheng Zhou

School of Mathematics and Statistics，Southwest University, Chongqing, China

• Shuai Liu

School of Electrical and Electronic Engineering, Shandong University, Jinan, China

• Abdselam Silem

Department of Physics, Zhejiang Normal University, Jinhua, China

• Walid Remili

Mathematics and Computer Science., University of BordjBouArreridj, BordjBouArreridj, Algeria

Published Articles
• Alemie Assefa

Issue: Volume 11, Issue 3, September 2023
Pages: 45-56