A control problem for a cross-diffusion system in a nonhomogeneous medium

We study a model, motivated by a bioremediation process, describing a cross-diffusion movement of a bacteria population b attracted by a chemoattractant signal c, in a nonhomogeneous stratified medium with n layers. We assume that this reaction-diffusion process is characterized by a low rate of degradation and a low diffusion coefficient of the chemoattractant, expressed in the model by a small parameter ϵ. The model consists of n systems of nonlinear parabolic equations with transmission conditions between layers. We prove a global-in-time solution for the asymptotic model setup with respect to the small parameter of the problem, for arbitrarily large initial data. Next, we deal with the control problem focusing mainly on the reduction of the chemoattractant concentration, by acting upon the initial distribution of the bacteria population b0. To this end, we prove the existence of a solution to the control problem and determine the optimality conditions.