Analysis of a time optimal control problem related to the management of a bioreactor^*,**

¹ Departamento de Matemática Aplicada II, E.T.S.I. Telecomunicación, Universidad de Vigo, 36310 Vigo, Spain. lino@dma.uvigo.es; aurea@dma.uvigo.es
² Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15706 Santiago, Spain. fjavier.fernandez@usc.es

Received: 10 February 2009
Revised: 3 September 2009
Revised: 21 January 2010

Abstract

We consider a time optimal control problem arisen from the optimal management of a bioreactor devoted to the treatment of eutrophicated water. We formulate this realistic problem as a state-control constrained time optimal control problem. After analyzing the state system (a complex system of coupled partial differential equations with non-smooth coefficients for advection-diffusion-reaction with Michaelis-Menten kinetics, modelling the eutrophication processes) we demonstrate the existence of, at least, an optimal solution. Then we present a detailed derivation of a first order optimality condition (involving the corresponding adjoint systems) characterizing these optimal solutions. Finally, a numerical example is shown.