  We consider bilinear optimal control problems, whose objective functionals do
not depend on the controls. Hence, bang-bang solutions will appear. We
investigate sufficient second-order conditions for bang-bang controls, which
guarantee local quadratic growth of the objective functional in $L^1$. In
addition, we prove that for controls that are not bang-bang, no such growth can
be expected. Finally, we study the finite-element discretization, and prove
error estimates of bang-bang controls in $L^1$-norms.
