Markov-Dubins problem with control in a triangle

We consider a model of a wheeled robot that moves forward on a plane (without obstacles). The robot is controlled by two drive wheels that rotate with positive speeds limited by a given maximum value. It is assumed that the wheel speeds can be changed instantaneously. By transitioning from speed control of the drive wheels to control through linear and angular velocity, the problem of fast performance on the motion group SE(2) with control on a triangle is formulated for this robot model, refining the classical Markov-Dubins problem. The Pontryagin maximum principle is applied to the problem, and the vertical subsystem of the Hamiltonian system is investigated. Possible types of extremal controls (relay, singular, mixed) are described, all of which are defined using piecewise constant functions with values at the vertices of the triangle that define the set of admissible controls. For each extremal control, an upper bound on the cut time is obtained (through a restriction on the number of switches along the optimal trajectory). The optimal synthesis is reduced to searching through a finite set of candidates for optimality. For each type of control, expressions for narrowing the exponential mapping on trajectories suspicious of optimality are calculated. A program has been developed for constructing a sphere (a set of points reachable by optimal trajectories in a fixed time) and for comparing it with the sphere in the Markov-Dubins problem.