Trim turnpikes for discrete optimal control problems with symmetries

• Applicants: Dr. Sofya Maslovskaya; Professorin Dr. Sina Ober-Blöbaum
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 571894560

This project investigates trim turnpikes in discrete optimal control systems with symmetries, a novel research direction that has yet to be explored. It encompasses several key topics, including discrete optimal control theory, geometric reduction, turnpike theory and structure-preserving methods. Over the past decade, the turnpike property has been extensively studied in optimal control problems (OCPs) across various settings: finite and infinite-dimensional, deterministic and stochastic, as well as continuous- and discrete-time frameworks. The turnpike property is particularly important for efficiently computing long-term optimal solutions, for simplifying control strategies in Model Predictive Control (MPC), and for ensuring the robustness and stability of certain problems. While much of the research has focused on steady-state turnpikes, where solutions converge to an equilibrium point, many optimal control problems exhibit non-static turnpikes, including periodic, linear, manifold and trim turnpikes. The latter are related to symmetries in optimal control problems, which usually appear in mechanics due to the relation between symmetries and conservation laws. For this broader class of problems, several fundamental questions remain open. In the proposed project, we address the following points. (i) We set up the frame and tools needed to analyze the turnpike property in the context of discrete optimal control systems. (ii) We translate reduction techniques for control systems with symmetries in the discrete setting, then (iii) we extend the state-control-turnpike theorem based on dissipativity assumption to discrete optimal control systems with symmetries. (iv) We analyze in the same context the turnpike property from the Pontryagin Maximum Principle point of view in order to derive a state-adjoint turnpike property in the discrete setting. (v) We establish a trim turnpike property in the continuous setting for symmetric OCPs with general boundary conditions. (vi) An equivalent result in the discrete setting will then be proven, building upon the analytical developments from (i)-(iv). (vii) Having now well established (trim) turnpike in continuous and discrete settings, we study the preservation of the (trim) turnpike through discretization of the continuous problem. (viii) The specific class of mechanical control systems, one of the main motivations of this project, will be considered.

• DFG Programme: Research Grants