  In many practical applications of control theory some constraints on the
state and/or on the control need to be imposed.
  In this paper, we prove controllability results for semilinear parabolic
equations under positivity constraints on the control, when the time horizon is
long enough. As we shall see, in fact, the minimal controllability time turns
out to be strictly positive.
  More precisely, we prove a global steady state constrained controllability
result for a semilinear parabolic equation with $C^1$ nonlinearity, without
sign or globally Lipschitz assumptions on the nonlinear term. Then, under
suitable dissipativity assumptions on the system, we extend the result to any
initial datum and any target trajectory.
  We conclude with some numerical simulations that confirm the theoretical
results that provide further information of the sparse structure of constrained
controls in minimal time.
