Parametrization of the attainable set for a nonlinear control model of a biochemical process

Ellina Grigorieva; Evgenii Khailov; Andrei Korobeinikov

doi:10.3934/mbe.2013.10.1067

Article Contents

2013, Volume 10, Issue 4: 1067-1094. Doi: 10.3934/mbe.2013.10.1067

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Parametrization of the attainable set for a nonlinear control model of a biochemical process

1.
Department of Mathematics and Computer Sciences, Texas Woman's University, Denton, TX 76204
2.
Department of Computer Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992
3.
Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona

Received: September 30, 2012

Revised: March 31, 2013

Published: May 2013

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Abstract

In this paper, we study a three-dimensional nonlinear model of a controllable reaction $ [X] + [Y] + [Z] \rightarrow [Z] $, where the reaction rate is given by a unspecified nonlinear function. A model of this type describes a variety of real-life processes in chemical kinetics and biology; in this paper our particular interests is in its application to waste water biotreatment. For this control model, we analytically study the corresponding attainable set and parameterize it by the moments of switching of piecewise constant control functions. This allows us to visualize the attainable sets using a numerical procedure.
These analytical results generalize the earlier findings, which were obtained for a trilinear reaction rate (which corresponds to the law of mass action) and reported in [18,19], to the case of a general rate of reaction. These results allow to reduce the problem of constructing the optimal control to a straightforward constrained finite dimensional optimization problem.

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Mathematics Subject Classification: Primary: 49J15, 49N90; Secondary: 93C10, 93C95.

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