Extension of the bang-bang principle for piecewise affine dynamics under constraints

As an extension of the classical bang-bang principle for linear systems, we show that for non-autonomous regional optimal control problems with state constraints, the bang-bang principle does not hold globally even when the dynamics is continuous with respect to the state variable. However, we show that there exist minimal time trajectories to reach a target with extreme controls at the loci where the dynamics is differentiable. We give two examples which exhibit singular arcs exactly at the loci of non-differentiability, i.e. at the boundaries of the regions.