AEI-DFG: Lagrangian and Hamiltonian field theories - geometry, discretization, optimal control

• Applicant: Professorin Dr.-Ing. Sigrid Leyendecker
• Subject Area: Mathematics ; Mechanics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541579394

Dynamical systems are the core of evolutionary problems and pervade the entire literature in applied mathematics for both finite and infinite dimensional systems and from continuous to discrete evolution. These are typically classified into two main categories: time-continuous and time-discrete. In the first case, time is a continuous variable and the dynamical system under study is described by differential equations. In the second case, time is a discrete variable and the dynamical system is described by difference equations. In order to obtain quantitative results for the former it is often necessary to transform these time-continuous systems into corresponding time-discrete systems that a computer is able to handle in a manner that preserves all or part of the properties of the original system. This discretization process is what geometric numerical integration is concerned with. Moreover, for dynamical systems that evolve in time and also in spatial dimensions, commonly described by partial differential equations (PDEs), it is necessary to discretize these systems, at least in space. In this project we plan to study fundamental problems on geometric and numerical analysis of dynamical systems, particularly Lagrangian and Hamiltonian field theories and optimal control of PDEs, that require the combination of the continuous and discrete points of view in order to tackle problems with impact in the real-world, as is the case of control of elastic systems. Our interdisciplinary proposal fits into the area of mathematics and mechanics, in the discipline known as geometric mechanics, where techniques of differential geometry, applied mathematics and mathematical physics are used. The proposal is focused on developing relevant aspects of the geometry of dynamical systems, PDEs and their numerical implementation. It goes beyond theoretical developments, including relevant applications to engineering. The overall objective of this project is two-fold. On the one hand, we aim to deepen our understanding of the geometry behind field theories and optimal control problems involved with these. On the other hand, we intend to develop and study structure preserving integrators for field theoretical systems with the goal of applying these to optimal control problems of elastic systems. In particular, we will develop variationally partitioned RK and Runge-Kutta Munthe-Kaas (RKMK) methods that are naturally devoid of any kind of locking phenomena. We want to derive a field-theoretical discrete version of the Hamilton-Pontryagin principle that will allow us to derive arbitrary order methods and provide means to generalize these results to a broader set of systems, such as forced and constrained systems and optimal control problems.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partner: Professor Dr. David Martin de Diego