On the relation between small-time local controllability and normal self-reachability

Abstract

We introduce a local property of nonlinear systems called the nontangency property and we show that, in the presence of this nontangency property, small-time local controllability by measurable controls implies small-time local controllability by piecewise-constant controls; furthermore, the initial state is normally reachable from itself in arbitrarily small time. The class of systems that are small-time locally controllable and satisfy the nontangency property is shown to contain all real-analytic systems, all smooth systems with the Lie-algebra rank condition, and all locally boundedC 1 systems. Some consequences of small-time normal self-reachability are also discussed.