Contributions to kernel methods in systems and control

Machine learning is increasingly used in systems and control, which is motivated by increasingly challenging control, simulation and analysis problems, abundant data and computing resources, as well as impressive theoretical and methodological advances in machine learning. The established class of kernel methods is of particular interest in this context, due to their rich theory, efficient and reliable algorithms, and modularity, and indeed kernel methods are increasingly used in systems and control. This thesis contributes to this flourishing field, focusing on two exemplary and complementary topics. First, many learning-based control approaches are based on combining uncertainty bounds for Gaussian process (GP) regression with robust control methods. We revisit the foundations of this domain by consolidating, improving, and carefully evaluating the required uncertainty bounds. As an application, we demonstrate how they can be combined with modern robust controller synthesis, leading to learning-enhanced robust control with rigorous control-theoretic and statistical guarantees. We furthermore discuss a severe practical limitation of these approaches, the a priori knowledge of an upper bound on the reproducing kernel Hilbert space (RKHS) norm of the target function, and propose to combine geometric assumptions together with kernel machines as a promising alternative. Second, we initiate a new research direction by combining kernels with mean field limits as appearing in kinetic theory. Motivated by learning problems on large-scale multiagent systems, we introduce mean field limits of kernels, and provide an extensive theory for the resulting RKHSs. This is used in turn in the analysis of kernel-based statistical learning in the mean field limit, which not only is a novel form of large-scale limit in theoretical machine learning, but provides also a solid foundation for applications in kinetic theory. Finally, using the theory of reproducing kernels, we establish the first existence result for the mean field limit of very general discrete-time multiagent systems, and use this in mean field optimal control. In summary, in this thesis we improve and refine existing uses of kernel methods in systems and control, helping to consolidate the area of learning-based control and pushing it further towards practical applications, and we introduce novel uses of kernels and their theory in systems and control, with many interesting directions for future work.