Minimum Time Path with Smooth and Non-smooth Slope-Dependent Speed

We solve a problem in slope-dependent path planning, as an illustration for a more general problem in optimal control for a class of hybrid systems, where the modes are determined by the control and states. It is shown that optimal solutions (namely trajectories that follow the boundaries) may be lost if one naively solves the optimal control problem. The correct procedure involves the simultaneous satisfaction of the necessary conditions for different modes, having complementary constraints. We identify optimal path segments in various cases for smooth and non-smooth speed-velocity characteristic. The hybrid character stems from this non-smoothness, which in this paper is lack of differentiability.