[@Xu2006], with stochastic uncertainty.

In many optimal control problems, the application of the Pontryagin maximum principle leads to bang-bang control strategies, and the classical $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ theories were not designed for such a purpose. But the optimal trajectories are in general not robust. Adding needle-like variations is therefore a way to improve robustness, and is the main motivation of this paper. Of course, the method applies to any bang-bang control strategy, not necessarily optimal.

The approach that we suggest in this paper combines an off-line treatment of the control strategies, with a feedback algorithm based on the structure of the control. We emphasize here that this algorithm preserves the bang-bang structure of the control. It consists of applying a nominal control strategy (that needs to be computed *a priori*), and adjusting it in real time, allowing one to track a nominal trajectory. The off-line method takes a solution of the control problem and makes it more robust by adding additional switching times (i.e., *redundancy*), which can be seen as additional degrees of freedom. Note that our analysis is done in the state space, without needing to consider the frequency domain. A key ingredient to the method is the use 