External reachability (reachability with pole assignment by p. d. feedback) for implicit descriptions

Even for flat implicit linear systems (i. e. having more state components than state equations), reachability is a well defined concept in terms of the set of state trajectories: it characterizes the property that from any initial state can start a smooth state trajectory which reaches any final state. What can happen, however, for this general class of systems, is that a system with no control input can be completely reachable. We introduce here the notion of "external reachability" which expresses the fact that trajectories can actually be controlled through the input (by proportional and derivative state feedback). Geometric necessary and sufficient conditions are given for external reachability. A new design method is proposed for pole assignment which uses this concept and relies upon right inversion techniques.