- Solving optimization problems modelled by partial differential equations (PDEs) with additional constraints on the controls and/or states is one of the most challenging problems in the context of real-world applications. In their successful treatment the interplay betw een optimization techniques and numerical simulation plays a key role. In particular, the desire to scale the results and algorithms to the current level of numerical and machine learning techniques requires the development of rigorous and sophisticated mathematical apparatus. Within this project, we aim to address a list of infinite-dimensional constrained optimization problems by developing and exploring novel numerical and data driven algorithms. It is planned to provide rigorous approximation results which will guarantee the robustness (with respect to the dimension of the problem) of the algorithms on both the qualitative as well as quantitative level. The main scientific activities of the project will focus on the follow ing highly relevant topics, attracting significant attention from a theoretical, numerical and applicational point of view :
1. Optimal control problems for evolution systems.
2. Spectral analysis of Gramian operators. Lyapunov and Riccati equation.
3. Numerical resolvent calculus based on reduced order models.
4. Passive control of vibrational systems.
The study of each topic will not be done in an isolated manner and we expect interaction and synergy effects. The planned results will be of interest to the control theory community (including both mathematicians and engineers), the PDE and numerical linear algebra community, civil and mechanical engineers, as well as to researchers involved in scientific computing and operator theory. Special attention will be given to the career development of young researchers.The tools and techniques we will use to achieve these goals include, but are not limited to, greedy algorithms, model reduction, bicriterial synthesis, Carleman inequalities, homogenisation, H-measures, dissipativity theory, rational Krylov methods, martingales convergence theory, Pontryagin principle, Lyapunov stability theory, semi-classical analysis, harmonic analysis and Lie groups. The team members already have had important contributions to some of these topics, and some even contributed to their inauguration.