Extremals in the Markov – Dubins Problem with Control on a Triangle

We formulate a time-optimal problem for a differential drive robot with bounded positive velocities of the driving wheels. This problem is equivalent to a generalization of the classical Markov – Dubins problem with an extended domain of control. We classify all extremal controls via the Pontryagin maximum principle. Some optimality conditions are obtained; therefore, the optimal synthesis is reduced to the enumeration of a finite number of possible solutions.