Local stabilization and controllability of a class of nontriangular nonlinear systems

The problem of local continuous (possibly nonsmooth) static state feedback asymptotic stabilization and small time local controllability of single-input nonlinear systems is considered. First, a specific class of nontriangular systems is introduced and characterized in a coordinate independent manner. Next, for this class of systems sufficient conditions for local continuous static state feedback asymptotic stabilizability and necessary and sufficient conditions for small time local controllability are obtained. As a consequence, it is shown that for the above class of nontriangular systems it holds that small time local controllability implies local continuous feedback asymptotic stabilizability. Another consequence is the sufficient geometric conditions for local continuous static state feedback asymptotic stabilizability, that may be checked in arbitrary coordinates using Lie brackets of vector fields on the right-hand side of a given single-input system.