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In this paper, we study the following parabolic chemo-repulsion withnonlinear production model: $$ \left\{ \begin{array}{rcl} \partial_tu-\Deltau&=&\nabla\cdot(u\nabla v),\\ \partial_tv-\Delta v+v&=&u^p+fv\, 1_{\Omega_c}.\end{array} \right. $$ This problem is related to a bilinear control problem,where the state $(u,v)$ is the cell density and the chemical concentrationrespectively, and the control $f$ acts in a bilinear form in the chemicalequation. For $2D$ domains, we first consider the case of quadratic signalproduction ($p=2$), proving the existence and uniqueness of global strong statesolution for each control, and the existence of global optimum solution.Afterwards, we deduce the optimality system for any local optimum via aLagrange multiplier Theorem, proving regularity of the Lagrange multipliers.Finally, we consider the case of signal production $u^p$ with $1<p<2$.

On a bi-dimensional chemo-repulsion model with nonlinear production