Estimates of the phase trajectories of controlled systems with multi-valued impulses

We consider a controlled system for the differential equation $\dot{x}(t)=f(t,x(t),u(t), \xi), \ \ t \in [a,b] , \ \ x(a)=\mathrm{x},$ where the parameter $\xi$ is an element of some given metric space, the control $u$ satisfies the constraint $u(t)\in U(t,x(t), \xi), \ \ t \in [a,b].$ It is assumed that at each given moment of time $t_k\in (a,b)$ a solution $x:[a,b]\to \mathbb{R}^n$ (a phase trajectory) suffers  discontinuity, the magnitude of which belongs to a non-empty compact set $I_k( x(t_k))\subset \mathbb{R}^n,$ and is an  absolutely continuous function on intervals $(t_{k-1},t_k]$. The  control function is assumed to be measurable. A theorem on estimating the distance from a given piece-wise absolutely continuous function $y:[a,b]\to \mathbb{R}^n$  to the set of phase trajectories for all initial values from a neighborhood of a vector $x_0$ and for all parameters from a neighborhood of a point $\xi_0$ is proven. It is assumed that for the given initial value $\mathrm{x}=x_0$ of the solution and for the value $\xi=\xi_0$ of the parameter, the set of phase trajectories is a priori limited. The proven theorem allows, by selecting the function $y$, to obtain an approximate solution of the controlled system, as well as an estimate of the error of such solution.