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: