On the Optimality of Dubins Paths across Heterogeneous Terrain

We derive optimality conditions for the paths of a Dubins vehicle when the state space is partitioned into two patches with different vehicle's forward velocity. We recast this problem as a hybrid optimal control problem and solve it using optimality principles for hybrid systems. Among the optimality conditions, we derive a "refraction" law at the boundary of the patches which generalizes the so-called Snell's law of refraction in optics to the case of paths with bounded maximum curvature.

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