  We consider de Finetti's problem for spectrally one-sided L\'evy risk models
with control strategies that are absolutely continuous with respect to the
Lebesgue measure. Furthermore, we consider the version with a constraint on the
time of ruin. To characterize the solution to the aforementioned models, we
first solve the optimal dividend problem with a terminal value at ruin and show
the optimality of threshold strategies. Next, we introduce the dual Lagrangian
problem and show that the complementary slackness conditions are satisfied,
characterizing the optimal Lagrange multiplier. Finally, we illustrate our
findings with a series of numerical examples.
