We analyze $$L^2$$ -regularization of a class of linear-quadratic optimal control problems with an additional $$L^1$$ -control cost depending on a parameter $$\beta $$ . To deal with this nonsmooth problem we use an augmentation approach known from linear programming in which the number of control variables is doubled. It is shown that if the optimal control for a given $$\beta ^*\ge 0$$ is bang-zero-bang, the solutions are continuous functions of the parameter  $$\beta $$ and the regularization parameter  $$\alpha $$ . Moreover we derive error estimates for Euler discretization.