Optimal Bilinear control restricted to the three-dimensional chemo-repulsion model with potential production

In this paper we study the following three-dimensional parabolic-parabolic chemo-repulsion model with potential production, logistic reaction and bilinear control, defined in $Q=(0,T)\times\Omega$: \begin{equation*}\label{eq0} \left\{ \begin{array}{rcl} \partial_tu-\Delta u&=&\nabla\cdot(u\nabla v)+r\,u-\mu\, u^p,\\ \partial_tv-\Delta v+v&=&u^p+f\,v\, 1_{\Omega_c}, \end{array} \right. \end{equation*} where $1< p<+\infty$, $r,\mu\geq 0$, and $f=f(t,x)$ is the control function acting on a subdomain $(0,T)\times \Omega_c$, with $\Omega_c\subseteq\Omega$. This system is endowed with initial and non-flux boundary conditions. We prove the existence of global weak solutions of this controlled problem when $f\in L^{5/2}(0,T;L^{5/2}(\Omega_c))$, analyzing the role of the diffusion and the logistic terms to get energy estimates. In particular, the logistic competition term $\mu \, u^p$ is necessary only for $p>5/3$. Secondly, if $f\in L^{5/2}(0,T;L^{5/2+}(\Omega_c))$, any weak solution $(u,v)$ satisfying the regularity criterion $u\in L^{5p/2}(Q)\cap L^{10/3}(Q)$ is in fact more regular, arriving in particular to $u,\nabla v\in L^5(Q)$ for $p\le 2$ and $u,\nabla v\in L^{5(p-1)}(Q)$ for $p> 2$ which is the critical regularity to solve a related optimal bilinear control problem. In fact, this setting let us to prove the existence of global optimal solutions, and the differentiability of the control-to-state mapping via the Implicit Function Theorem in Banach spaces. Then, we can identify the gradient of the (reduced) cost with respect to the control solving the adjoint problem by duality. In particular, we derive first-order necessary optimality conditions for local optimal solutions.