Control of a nonlinear system along a trajectory under disturbance conditions

We consider two control problems along a given trajectory with a disturbance, which is the second player in a differential game. The dynamics of the first problem are described by a nonlinear system of first-order differential equations, and the dynamics of the second problem are described by a nonlinear system of second-order differential equations. Control is performed by means of a piecewise constant controller, the set of values of which is finite. The goal of control is to move as close as possible to a finite trajectory under any disturbance action. In the first problem, the trajectory is determined by the solution of an auxiliary system with simple motion. In the second problem, the trajectory is determined by the solution of an auxiliary controlled system of second-order differential equations. In the first problem, it is shown that, for any neighborhood of the specified trajectory, there exists a piecewise constant pursuer control that guarantees motion in this neighborhood under any disturbance actions, from the initial point of the trajectory to the neighborhood of the final point of the trajectory. In the second problem, the motion is also ensured as close as possible to an arbitrary finite trajectory of the auxiliary system, both the phase trajectory of the original system and the velocity trajectory. Hence, in the second problem, we prove soft capture, in which, in addition to bringing the phase coordinates to any pre-defined neighborhood of zero, the velocity is also brought to the same pre-defined neighborhood of zero.