Bilinear optimal control of a one-dimensional degenerate parabolic equation with a nonlocal term

In this paper, a one-dimensional degenerate parabolic equation involving a multiplicative control with a nonlocal space term represented by an integral kernel is considered. After some well-posedness results, we derive several properties of the control-to-state mapping. We establish the existence of at least one optimal control and derive the first and the second conditions for optimality, giving the local uniqueness of an optimal control. Finally, we prove that we may expect the global uniqueness of the optimal control under additional assumptions.