This paper studies control problems for elliptic systems. A typical examples is to minimize the functional \(J(y,u)=\frac12\| y- z\|^ 2_ 2+\frac12\| u\|^2_2\) for some \(z\) in \(L^2(\Omega)\), over all \(u\) in some admissible subset of \(L^2(\Omega)\) and \(y\) in a convex subset of \(L^2(\Omega)\) with \(y\) determined from \(u\) by the boundary value problem \(\Delta y-y=u\) in \(\Omega\), \(y=0\) on \(\partial\Omega\). It is relatively easy to obtain coupled optimality conditions for \(y\) and \(u\), that is conditions in which \(u\) and \(y\) are coupled by the boundary value problem, but the author is also able to obtain decoupled optimality conditions. His method, which handles far more general problems than the simple example here, is based on penalization; for our simple example, the penalized functional is \[ J_ \varepsilon(y,u)=J(y,u)+\frac12 \varepsilon\int_\Omega(\Delta y-y- u)^2\,dx. \]