

Process stabilization and closedloop control of nonlinear diffusionreaction developments in structured populations


Marek Niezgódka (ICM University of Warsaw) and HansGeorg Bock (IWR, University of Heidelberg)


The project is carried out by Grzegorz Dudziuk at University of Warsaw
 
Heterogeneous populations that evolve over networkshaped domains are characteristic components of a large variety of realworld 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 intercomponent 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 timedependent, with material properties of the physical medium.
This type of problems leads to mathematical models in the form of spacedependent nonlinear dynamical systems,
driven by nonclassical diffusionreaction mechanisms and defined over timedependent 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 farreaching refinements over traditional relatively weakly nonlinear parabolic diffusionreaction 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
illposedness of the latter in the case of widely used ultrasound techniques,
(3) construction and implementation of the numerical algorithms for closedloop 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: 271293.
[2] K.H. Hoffmann, M. Niezgódka and J. Sprekels, (1990) Nonlinear Anal.: Theory, Meth. and Appl., 15: 955976.
[3] Z. Szymańnska and M. Zylicz, (2009) J. Theor. Biol. to appear.




