Stepwise synthesis of constrained controls for single input nonlinear systems of special form

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.