Controlled differential equations with a parameter and with multivalued impulses

We study the Cauchy problem for a controlled differential system with a parameter which is an element of some metric space $\Xi$ containing phase constraints on the control. It is assumed that at the given time instants $t_{k},$ $k=1,2,\ldots, p,$ the solution $x$ is continuous from the left and suffers a discontinuity, the value of which is $x(t_k + 0)-x(t_k),$ belongs to some non-empty compact set of the space $\mathbb{R}^{n}.$ The notions of an admissible pair of this controlled impulsive system are introduced. The questions of continuity of admissible pairs are considered. Definitions of a priori boundedness and a priori collective boundedness on a given set $S \times K$ (where $S\subset \mathbb{R}^n $ is a set of initial values, $K \subset \Xi$ is a set of parameter values) of the set of phase trajectories are considered. It is proved that if at some point $(x_0, \xi) \in \mathbb{R}^n \times \Xi $ the set of phase trajectories is a priori bounded, then it will be a priori bounded in some neighborhood of this point.