STABILITY AND ASYMPTOTIC BEHAVIOR IN OPTIMAL CONTROL PROBLEM

Optimal control is an important branch of applied mathematics with numerous applications in applied sciences, including macroeconomics, engineering, and operations research. It can be viewed as both a generalization of the theory of the Calculus of Variations and as a constrained infinite- dimensional optimization, where the objective is to find an optimal control law for a given system. This 4--year bilateral project between TU Wien (Austria) and IRMAR, INSA Rennes and LMBA- Brest (France), is rooted in this rich area of mathematics. Based on the complementary expertise of the involved research teams, the project proposes significant advances in topics related to Partial Differential Equations (PDE), Variational Analysis, and Dynamics. The project specifically focuses on the regularity properties of the Pontryagin map as they relate to stability around a reference trajectory. These issues are significant for the efficient resolution of several nonconvex control problems. To address this, the project proposes in parallel novel numerical approaches utilizing indirect methods and new optimization techniques. In a second stage, the project endeavors to explore infinite-horizon control problems by rigorously studying alternative methods for averaging running costs over infinite periods. Since infinite-horizon control problems are often non-ergodic, that is, the behavior of a specific system over time cannot determine the average behavior, there is a need to develop new, complex mathematical tools (like advanced averaging techniques) to figure out the best possible outcome for decision-making scenarios, where existing mathematical methods are insufficient. The project also endeavors to connect two mathematical areas: the study of PDEs (used to describe motion or energy) and Variational Analysis (measuring the way that functions change and using it to optimize certain quantities, such as energy or distance). By linking these ideas, the goal is to determine when a problem has a single, unique solution (aka value function of the control problem), and to connect this understanding with well-developed theories, such as the slope-determination, the theory of Hamilton-Jacobi equations or the weak-KAM theory, in the ergodic case, about how systems behave over time. In summary, this proposal is both theoretical and applied in nature and involves the joint supervision of two PhD students and one postdoctoral researcher.