Linear-quadratic control problems with L1-control cost

We analyze a class of linear‐quadratic optimal control problems with an additional L¹‐control cost depending on a parameter β. 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 β* 0 is bang‐zero‐bang and the switching function has a stable structure, the solutions are Lipschitz continuous functions of the parameter β. We also show that in this case the optimal controls for β ^* and a β 0 with | β − β ^* | sufficiently small coincide except on a set of measure O(β). Finally, we use the augmentation approach to derive error estimates for Euler discretizations.