The paper deals with a dynamic object the mapping of which is given by the following differential inclusion: \[ \dot x(t)\in F(t,(x(t)), \quad x(0)=x_ 0\leqno(*) \] where \(F\) is assumed to be a set-valued map on \(R\times K\) with nonempty convex compact values in a finite-dimensional space \(X\), integrably bounded, measurable in \(t\) and upper semicontinuous in \(x\), \(K\) is a nonempty closed subset of \(X\), \(x_ o\in K\). (\(*\)) is assumed to be viable, i.e. \(x(t)\in K\) for each \(t\geq 0\). After defining a sequence of upper semicontinuous approximations of \(F\) and after passing to the limit a viable solution of the original problem is determined. The exemplified solution of a control problem simplifies the understanding of the ideas essentially and makes the work easily accessible for practical applications.