Accurate approximated solution to the differential inclusion based on the ordinary differential equation

UDC 517.9 Many problems in applied mathematics can be transformed and described by the differential inclusion $\dot x\in f(t, x)-N_Qx$ involving $N_Qx,$ which is a normal cone to a closed convex set $Q \in \mathbb R^n$ at $x\in Q.$ The Cauchy problem of this inclusion is studied in the paper. Since the change of $x$ leads to the change of $N_Qx,$ solving the inclusion becomes extremely complicated. In this paper, we consider an ordinary differential equation containing a control parameter $K.$ When $K$ is large enough, the studied equation gives a solution approximating to a solution of the inclusion above. The theorem about the approximation of these solutions with arbitrary small error (this error can be controlled by increasing $K$) is proved in this paper.