Local R-controllability to zero of nonlinear algebraic-differential systems

We consider a control system of nonlinear ordinary differential equations unsolved with respect to the derivative of the desired vector function and identically degenerate in the domain of definition. An arbitrarily high index of unsolvability is allowed. The conditions of local R-controllability to zero (zero-controllability within the reachable set) of such system are obtained in terms of the first order linear approximation. In the linear case, it is shown that R-controllability implies local R-controllability to zero.