On a controllability problem in special control classes (Q1348538)

A controlled object of the form \[ \dot x = f(x, u), \qquad x(0) = 0, \quad \tag{1} \] where \(x\in \mathbb{R}^n\), \(u\in U \subset \mathbb{R}^r\), and \(U\) is convex compact in \(\mathbb{R}^r\) containing the origin, is considered. It is supposed that the vector \(n\)-dimensional function \(f(x,u)\) is continuous in \((x,u)\) and continuously differentiable with respect to \(x\). Let \(D(t)\) be the reachable set at time \(t\) for all possible measurable \(u(t)\), \(t \in \Delta\), where \(\Delta = [0, T].\) One of the important problems of control theory is the next one: to determine sufficient conditions for which the relation \[ 0\in \operatorname {int} {\mathbf D}(T) \tag{2} \] is true. The method of covering by a nonlinear mapping is used to prove the relation (2) and to get a convex compact \(K\) as an estimate from inside of the attainable set. Here the next inclusion \[ 0\in \operatorname {int} K \subset K \subset {D}(t) \] is true. Such estimates from inside represent a great interest for control theory and for differential games.