This project aims at establishing a strong group in mathematical control theory within the Croatian research network that will be able to address cutting-edge issues emerging from real-life problems.
Control theory is an interdisciplinary field combining engineering and mathematics that deals with the behaviour of dynamical systems (usually modelled by differential equations) with inputs. It is a well-studied area with intensive research. Results from control theory have been successfully applied to many concrete problems in fields such as electrical and mechanical engineering, transportation, communication, medicine, biology etc. Although the existing theory is able to provide answers to a number of fundamental and practical questions, many important problems are still unresolved. In order to model and control (physical or economic) systems as closely as possible, there is an increasing demand for a sophisticated mathematical apparatus.
The project will focus on the highly relevant topics attracting significant attention from theoretical, numerical and applicational points of view. The scientific activities of the project will be divided into 6 working packages:
Control of parameter dependent problems
Control of vibrational systems
Adaptive Dual Control under realistic communications
The cost of controlling hyperbolic partial differential equations in a multiscale setting
Optimal design and small amplitude homogenisation
Modeling, system analysis and control of autonomous systems
Within these working packages we aim to cover a range of important issues in modern control, including the following topics: the extension of the greedy control results in several directions, extension of the results for the homogeneous vibrational systems to the systems with external input, the interplay between adaptive dual control and degraded information, estimates for the cost of null-controllability for hyperbolic PDEs on multiscale domains, optimal design result for two phase composites by means of small amplitude homogenisation techniques up to the third (and higher) order expansion, and application of novel control concepts on models representing real-life systems as well as controllability studies which will be useful in the design process of this systems.
Participation of engineers will enable integral and interdisciplinary approach to the phenomena of interest, and will pave the path for successful applications, while the inclusion of four eminent scientist as consultants/experts will help keeping the project relevant and successful. The results will be applied to a number of problems including aerial delivery, human-robot cooperation and traffic control. This choice of problems and applications is influenced by importance in the current international research as well as the background and expertise of members of our team.
The goal of the project is to provide new theoretical methods and results, analyse the corresponding numerical issues, develop new computational software and verify applications of the newly developed methods to systems based on real-life scenarios, thereby bridging the gap between theoretical considerations and industrial applications.
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.