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# Optimal control of a diffusion/reaction/switching system

• We consider an optimal control problem involving the use of bacteria for pollution removal where the model assumes the bacteria switch instantaneously between active and dormant states, determined by threshold sensitivity to the local concentration $v$ of a diffusing critical nutrient; compare [7], [3], [6] in which nutrient transport is convective. It is shown that the direct problem has a solution for each boundary control $ψ = ∂v/∂n$ and that optimal controls exist, minimizing a combination of residual pollutant and aggregated cost of the nutrient.
Mathematics Subject Classification: Primary: 49J21; Secondary: 34K34, 35K40.

 Citation:

•  [1] T. C. Hazen, Cometabolic Bioremediation (Ch. 7, pp. 2505-2514) and In Situ Groundwater Bioremediation (Ch. 13, pp. 2583-2596) in Handbook of Hydrocarbon Lipid Microbiology, (Timmins et al., eds) Springer, NY, 2010. [2] M. A. Krasnosel'skĭi and A. V. Pokrovskiĭ, Systems with Hysteresis, Translated from the Russian by Marek Niezgódka. Springer-Verlag, Berlin, 1989.doi: 10.1007/978-3-642-61302-9. [3] S. Lenhart, T. I. Seidman and J. Yong, Optimal control of a bioreactor with modal switching, Math. Models Methods in Appl. Sci., 11 (2001), 933-949.doi: 10.1142/S0218202501001185. [4] R. D. Norris, et al., Handbook of Bioremediation, Lewis Publishers, Boca Raton, 1994. [5] T. I. Seidman, Switching systems: Thermostats and periodicity, Math. Res. Report 83-07, UMBC, 1983. http://userpages.umbc.edu/ seidman/ss_83.pdf [6] T. I. Seidman, A 1-dimensional bioremediation model with modal switching, in Control of Distributed Parameter and Stochastic Systems, (S. Chen, X. Li, J. Yong, X.Y. Zhou, eds.) pp. 127-131, Kluwer Acad. Publ., Norwell, 1999. [7] T. I. Seidman, A convection/reaction/switching system, Nonlinear Anal. - TMA, 67 (2007), 2060-2071.doi: 10.1016/j.na.2006.08.050. [8] T. I. Seidman, Some aspects of modeling with discontinuities, Int'l. J. Evolution Eqns., 3 (2009), 419-433. [9] G. Stampacchia, Equations Elliptiques Du Second Ordre á Coefficients Discontinues, (French) Séminaire de Mathématiques Supérieures, No. 16 (été, 1965); Les Presses de l'Université de Montréal, Montreal, Que. 1966 326 pp. [10] E. Venkataramani and R. Ahlert, Role of cometabolism in biological oxidation of synthetic compounds, Biotechnology and Bioengineering, 27 (1985) 1306-1311.doi: 10.1002/bit.260270906. [11] A. Visintin, Differential Models of Hysteresis, Applied Mathematical Sciences, 111. Springer-Verlag, Berlin, 1994.

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