This project aims at making a breakthrough contribution in optimal control and decision making for nonlinear processes that take place on hierarchical network structures and are dynamic in time and/or space. While setting has a wide range of potential domains of applicability with high societal relevance: thermal, electric, or fluid dynamics in energy networks, traffic and logistics, disease spreading dynamics, or cell signalling in biomedicine. This project focuses on energy scarcity control tasks and will pursue the following objectives: To contribute new theory, to develop numerical approximation methods, to implement algorithmic methods in software, and to conduct proof-of-concept studies. Research in the young field of mixed-integer optimal control (MIOC) has recently seen increased momentum together with numerical approximation algorithms and control theory. Despite initial successes, key questions remain unsolved because of a lack of analytical understanding, a lack of tractable formulations, the unavailability of efficient large-scale solvers, or the insufficiency of existing implementations. This project focuses on pivotal but poorly understood topics: decomposition, relaxation, and combinatorial integral approximation; domains admitting homo- genization and limiting processes using weak topologies; tractable approximations of direct costs of decisions; efficient distributed and parallel nonlinear solvers; and robustness of approximate nonlinear decision policies under uncertainty. Due to non-trivial interactions emerging in theory and the unavailability of comprehensive algorithms, these topics cannot be suitably handled by merely combining the respective states of the art. A focused effort to decisively extend MIOC to optimal decision policies for dynamics on hierarchical networks is therefore a timely endeavour that will help to address the challenging demands of practitioners. Proof-of-concept studies for energy scarcity control scenarios in power systems, in heat/cooling, and in mobility will assess the applicability of the solutions proposed.
