  The article examines a linear-quadratic Neumann control problem that is
governed by a non-coercive elliptic equation. Due to the non-self-adjoint
nature of the linear control-to-state operator, it is necessary to
independently study both the state and adjoint state equations. The article
establishes the existence and uniqueness of solutions for both equations, with
minimal assumptions made about the problem's data. Next, the regularity of
these solutions is studied in three frameworks: Hilbert-Sobolev spaces,
Sobolev-Slobodecki\u\i{} spaces, and weighted Sobolev spaces. These regularity
results enable a numerical analysis of the finite element approximation of both
the state and adjoint state equations. The results cover both convex and
non-convex domains and quasi-uniform and graded meshes. Finally, the optimal
control problem is analyzed and discretized. Existence and uniqueness of the
solution, first-order optimality conditions, and error estimates for the finite
element approximation of the control are obtained. Numerical experiments
confirming these results are included. A significant highlight is that the
discretization error estimates known from the literature, are improved even for
the coercive case.
