  In this paper, we study an optimal control problem for a coupled non-linear
system of reaction-diffusion equations with degenerate diffusion, consisting of
two partial differential equations representing the density of cells and the
concentration of the chemotactic agent. By controlling the concentration of the
chemical substrates, this study can guide the optimal growth of cells. The
novelty of this work lies on the direct and dual models that remain in a weak
setting, which is uncommon in the recent literature for solving optimal control
systems. Moreover, it is known that the adjoint problems offer a powerful
approach to quantifying the uncertainty associated with model inputs. However,
these systems typically lack closed-form solutions, making it challenging to
obtain weak solutions. For that, the well-posedness of the direct problem is
first well guaranteed. Then, the existence of an optimal control and the
first-order optimality conditions are established. Finally, weak solutions for
the adjoint system to the non-linear degenerate direct model, are introduced
and investigated.
