ABSTRACT
The chapter discusses a new necessary condition for neutralizing bad brackets and considers the case when a bad bracket is not neutralized by lower order brackets, but a pair of bad brackets is balancing each other to possibly give small-time local controllability (STLC). There are many possible approaches to finding conditions for local controllability, leading to different results and requiring different hypotheses. For important engineering applications and for mathematicians the more interesting property is, of course, local controllability. A basic tool to prove general theorems about local controllability as well as optimality, but also to obtain information about the geometry of individual systems where the known theorems fail, are local approximation cones to the attainable set, and the associated families of admissible control variations. The general theorem of H. J. Sussmann applies to the multi-input case, and allows the brackets of type in above statement to be replaced by more general "fixed elements of a group of input symmetries".
