# American Institute of Mathematical Sciences

2013, 10(4): 1067-1094. doi: 10.3934/mbe.2013.10.1067

## 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  October 2012 Revised  April 2013 Published  June 2013

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.
Citation: Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Parametrization of the attainable set for a nonlinear control model of a biochemical process. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1067-1094. doi: 10.3934/mbe.2013.10.1067
##### References:

show all references

##### References:
 [1] Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751 [2] Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 799-818. doi: 10.3934/dcdsb.2010.13.799 [3] Chih-Yuan Chen, Shin-Hwa Wang, Kuo-Chih Hung. S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2589-2608. doi: 10.3934/cpaa.2014.13.2589 [4] Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607 [5] M. A. Efendiev. On the compactness of the stable set for rate independent processes. Communications on Pure & Applied Analysis, 2003, 2 (4) : 495-509. doi: 10.3934/cpaa.2003.2.495 [6] Igor Pažanin, Marcone C. Pereira. On the nonlinear convection-diffusion-reaction problem in a thin domain with a weak boundary absorption. Communications on Pure & Applied Analysis, 2018, 17 (2) : 579-592. doi: 10.3934/cpaa.2018031 [7] Zhong Wan, Jingjing Liu, Jing Zhang. Nonlinear optimization to management problems of end-of-life vehicles with environmental protection awareness and damaged/aging degrees. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-23. doi: 10.3934/jimo.2019046 [8] Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997 [9] Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 93-112. doi: 10.3934/dcdsb.2011.15.93 [10] Chi-Cheung Poon. Blowup rate of solutions of a degenerate nonlinear parabolic equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5317-5336. doi: 10.3934/dcdsb.2019060 [11] Juan Dávila, Louis Dupaigne, Marcelo Montenegro. The extremal solution of a boundary reaction problem. Communications on Pure & Applied Analysis, 2008, 7 (4) : 795-817. doi: 10.3934/cpaa.2008.7.795 [12] Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Dirichlet problems with a crossing reaction. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2749-2766. doi: 10.3934/cpaa.2014.13.2749 [13] Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593 [14] Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115 [15] Pierre Fabrie, Elodie Jaumouillé, Iraj Mortazavi, Olivier Piller. Numerical approximation of an optimization problem to reduce leakage in water distribution systems. Mathematical Control & Related Fields, 2012, 2 (2) : 101-120. doi: 10.3934/mcrf.2012.2.101 [16] Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327 [17] Kien Ming Ng, Trung Hieu Tran. A parallel water flow algorithm with local search for solving the quadratic assignment problem. Journal of Industrial & Management Optimization, 2019, 15 (1) : 235-259. doi: 10.3934/jimo.2018041 [18] Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281 [19] R. S. Johnson. A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1497-1522. doi: 10.3934/cpaa.2012.11.1497 [20] Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065

2018 Impact Factor: 1.313