  The controllability problem for nonlinear control systems with
one-dimensional control of the form $ dx/dt=a(x)+B(x)\beta(x,u)$ is considered,
where $a(x)$ is an $n$-dimensional vector function, $B(x)$ is an $(n\times
m)$-matrix, and $\beta(x,u)$ is an $m$-dimensional vector function. Under
certain conditions we reduce such system to a system consisting of $m$
subsystems; in each subsystem all equations are linear except of the last one.
We use the controllability function method to give sufficient conditions for
controllability of the considered system. We propose an approach for
construction of controls which transfer an arbitrary initial point to the rest
point in a certain finite time. Each such control is constructed as a
concatenation of a finite number of positional controls (we call it a stepwise
synthesis control). On each step of our approach we choose a new synthesis
control. Our approach essentially uses nonlinearity of a system with respect to
a control. The obtained results are illustrated by examples. In particular, the
problem of the complete stoppage of a two-link pendulum is solved. We also
introduce the class of nonlinear systems which is called the class of staircase
systems that provides the applicability of our approach.
