  This work concentrates on a class of optimal control problems for semilinear
parabolic equations subject to control constraint of the form
$\|u(t)\|_{L^1(\Omega)} \le \gamma$ for $t \in (0,T)$. This limits the total
control that can be applied to the system at any instant of time. The
$L^1$-norm of the constraint leads to sparsity of the control in space, for the
time instants when the constraint is active. Due to the non-smoothness of the
constraint, the analysis of the control problem requires new techniques.
Existence of a solution, first and second order optimality conditions, and
regularity of the optimal control are proved. Further, stability of the optimal
controls with respect to $\gamma$ is investigated on the basis of different
second order conditions.
