Time Minimization During Simultaneous Longitudinal and Rotational Motion of a Chaplygin Sleigh

The problem of minization of the time of motion of a Chaplygin sleigh in the horizontal plane is analyzed. The force directed along the linear velocity vector and the torque are considered as control inputs. The use of the Pontryagin maximum principle allows for establishing that three types of control are possible: regular control, taking the boundary values (bang-bang) for the force and the torque, first-order singular control for the force, and second-order singular control for the torque. If the control is singular for the force on the entire interval, the solution of the optimization problem is not unique. The second-order control segment can be connected with nonsingular control only by means of the “chattering” regime. A simpler suboptimal control is proposed that contains a finite number of switchings. Examples are considered for the specified end states and partially specified end states. Analysis of possible combinations of singular and regular arcs of the trajectory is performed.