On the semicontinuity properties of the set of trajectories of the nonlinear control systems with integral constraints on the control functions

In this paper the control system described by the ordinary nonlinear differential equation is considered. The admissible control functions are chosen from the closed ball of radius r in the space Lp, p∈[1,∞], centred at the origin. The behaviour of the set of trajectories at p = 1 is studied. It is proved that the set of trajectories is the Hausdorff right upper semicontinuous and Vietoris right lower semicontinuous at p = 1. It is illustrated that it is not Hausdorff right lower semicontinuous at p = 1.