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Process stabilization and closed-loop control of nonlinear diffusion-reaction developments in structured populations
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Marek Niezgódka (ICM University of Warsaw) and Hans-Georg Bock (IWR, University of Heidelberg)
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The project is carried out by Grzegorz Dudziuk at University of Warsaw
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Heterogeneous populations that evolve over network-shaped domains are characteristic components of a large variety of real-world systems.
The related mathematical models exhibit a structure of high complexity, as:
(1) they are nonlinear and distributed in space, with dynamic couplings between individual governing equations in the evolutionary systems,
(2) they are nonlocal in space and time (the latter referring to specific memory properties), because of the role of inherent mutual interaction mechanisms,
(3) they show multiscale nature, with macroscopic behaviour on the level of the medium and inter-component interactions, while mesoscopic intrinsic kinetics
for their single components (individual members and, possibly, some aggregates of the latter),
(4) the domains on which the populations evolve not only are geometrically complex but also their topology may be
challenging in as much it gets time-dependent, with material properties of the physical medium.
This type of problems leads to mathematical models in the form of space-dependent nonlinear dynamical systems,
driven by non-classical diffusion-reaction mechanisms and defined over time-dependent domains [1]. In particular,
the domains are themselves system variables, hence contributing to a free boundary nature of the underlying models [2].
Applied to the problems of medical diagnostics and a corresponding decision making on drug delivery patterns,
in particular for the blood circulation system and selected organs, the above category of models is highly prospective
as they allow for far-reaching refinements over traditional relatively weakly nonlinear parabolic diffusion-reaction systems [3].
The Ph.D. project will focus on:
(1) qualitative analysis of fundamental dynamic properties of the systems in the form specific for a class of real applications to medical diagnostics and
therapy planning,
(2) construction of the algorithms for system stabilization, including an analysis of the data assimilation problems that assume visual acquisition and structural
ill-posedness of the latter in the case of widely used ultrasound techniques,
(3) construction and implementation of the numerical algorithms for closed-loop control of the dynamical systems
in the above context, assuming the activation of either boundary, distributed source or, in addition, remote type.
[1] A. Damlamian and N. Kenmochi, (1980) Hiroshima Math. J., 10: 271-293.
[2] K.-H. Hoffmann, M. Niezgódka and J. Sprekels, (1990) Nonlinear Anal.: Theory, Meth. and Appl., 15: 955-976.
[3] Z. Szymańnska and M. Zylicz, (2009) J. Theor. Biol. to appear.
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