Delays are ubiquitous in engineering and natural sciences, e.g., communication delays in control devices and complex networks or maturation and gestation delays in population dynamics. The inclusion of past history in the time evolution adds nontrivial complexities with respect to ordinary systems, balancing the advantage of dealing with more realistic models. Equations involving delays generate dynamical systems of infinite dimension, asking for advanced tools and methods in the background mathematical analysis, the numerical treatment, the development, design and optimization of control strategies. Eventually, the comprehension of fundamental issues like stability of equilibria and other invariants is crucial, especially for varying or uncertain parameters.
The school brings together strong and up-to-date contributions in the field of time delay systems, concerning analytical, numerical and application aspects, under the paradigm of control theory. It aims at discussing the most recent advances in these different contexts, focusing also on their interdisciplinary connections. Analysis, methods, control strategies and applications will be illustrated, starting from rapid introductions of the basics, reaching a state-of-the-art level by evolving the classic approaches into modern perspectives.
Attention will be devoted to the framework of semigroups, leading to a proper theory of linearized stability, where characteristic equations and Laplace transforms will be discussed for the sake of asymptotic analysis. An overview will thus be offered on efficient discretization schemes to reduce the dimension of the stability question, focusing on reliable and fast converging techniques. In parallel, the influence of model and control parameters on delayed dynamics will be investigated and characterized efficiently via continuation methods.
Moreover, the design problem for time-delay plants will be adequately addressed. Indeed, delays are crucial elements of engineering systems, e.g., regenerative delays in machine tool vibrations, leading often to unpredictable dynamics. Stabilization in the presence of feedback delay and sensory uncertainties is a challenging task and different control concepts will be discussed, together with an overview of limitations induced by delays in control loops. Recent control methodologies will be introduced, such as direct optimization approaches for synthesizing controllers. Concerning parameters, studying bifurcations of linear systems requires attention to the location of the rightmost roots, for which Weierstrass-based algorithms and frequency-sweeping tests will be proposed. Besides the design of controllers, considered the detrimental effects of delays in the plant model or in the feedback loops, the “dual problem” of intentionally using the delays in control law in order to improve the system behavior will be addressed. Also model reduction for large-scale systems will be discussed.
The school is primarily addressed to PhD students and post-docs in the fields connected to dynamical systems involving time delays and their control theory and numerical analysis, ranging from mathematics to engineering and physics. The lectures are also suited for young and senior researchers in the above or neighboring fields, from academia or private R&D centers, interested in gaining a compact yet comprehensive overview of dynamical systems with delay, from their mathematical background to control and computation.