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DFG project: mathematical theories to counteract turbulences

DFG project: mathematical theories to counteract turbulences

With his work, Passau-based researcher Dr Andrii Mironchenko may be able to fundamentally improve our understanding of complex systems. In a DFG-funded project, he is analysing feedback effects between infinitely dimensional systems and creating theories that may be able to provide greater stability in various situations, such as flying.

Our world is complex. Strictly speaking, every existing system is infinitely dimensional - such as air currents in the atmosphere, the flow of blood in our cells, chemical reactions, the dynamic of organisms living in groups, the flexible parts of robots and aircraft: these are all examples of infinitely dimensional systems. With the help of differential equations, mathematics can describe how these things are regulated.

Yet this is not all. These infinitely dimensional systems interact with one another and thus in turn create complex networks that behave in non-linear fashion: organisms living in groups form ecosystems, for example; chemical reactions consist of many subreactions containing their own chemical processes; satellites and aircraft are comprised of many flexible parts, supported by numerous sensors.

The stability of complex, interlinked systems

How can these complex systems be controlled with a small number of sensors and few possibilities to influence the respective system? How can one design an efficient control system - despite imprecise sensor data, disturbances and turbulence? When is a complex network stable?

Dr Andrii Mironchenko, graduate research assistant to Professor Fabian Wirth - holder of the Chair of Dynamical Systems at the University of Passau - is taking a step towards answering these questions. In the project "Robust stabilisation of interlinked, infinitely dimensional systems with peripheral connections", he is using mathematical models to determine when complex, interlinked systems are stable.

For example, Dr Mironchenko is analysing how to minimise the influence of turbulence when flying. Dr Mironchenko: "We focus on certain disturbances we basically know nothing about. However, we want these mechanisms to function properly even under these conditions, so that we can nevertheless achieve our objectives. That is precisely what this project is about."

Reducing the influence of turbulence when flying

The mathematician's work could make the development work performed by engineers considerably easier. This is because his theoretical considerations may allow them to dispense with numerous sophisticated tests that require expensive sensors. Instead, all it might take is just a few tests, as the sensors can already be attuned much more accurately. "We provide the tools to allow engineers to develop robust, high-performance mechanisms," said Dr Mironchenko.

Stabilising a nonlinear control system is a fundamental problem in this area of mathematics. Tools developed for this purpose can generally be applied in other areas as well. For instance, they might be used to analyse the behaviour of large crowds of people in real-world settings as well as in the digital world, such as in social networks.

Principal Investigator(s) at the University Dr. Andrii Mironchenko (Lehrstuhl für Mathematik mit Schwerpunkt Dynamische Systeme)
Source of funding
DFG - Deutsche Forschungsgemeinschaft > DFG - Sachbeihilfe
DFG - Deutsche Forschungsgemeinschaft > DFG - Sachbeihilfe
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