REACHABILITY OF INTERIOR STATES BY PIECEWISE-CONSTANT CONTROLS

In order to show the futility of attempting to derive general regulari ty theorems for optimal controls in smooth - but not real-analytic - c ontrol systems, H.J. Sussmann has demonstrated how, given a Lebesgue-i ntegrable function mu. [0,1] --> R, one can always exhibit a smooth, s ingle-input control system (say on R(3)) with the property that there exist states p and q for which mu is the unique control that steers p to q. An examination of Sussmann's construction reveals that the traje ctory corresponding to ii which joins p and q evolves on the boundary of the attainable set from p. It is natural to ask whether a similar p athology can occur for points in the interior of the attainable set. I n this paper we modify Sussmann's construction and show that, given a Lebesgue-integrable function mu:[0,1] --> R, one can always exhibit a smooth, two-input control system on R(3) for which there exist states p and q such that q is interior to the attainable set from p and if u, upsilon are controls that steer p to q on the time interval [0,T], th en T>1 and u must agree with mu on [0,1]. However, in this constructio n it is seen that as soon as the trajectory dips into the interior of the attainable set the controls no longer have to agree with any pre-a ssigned ''bad'' control, and in fact can be taken to be piecewise cons tant. The main result of this paper shows that this phenomenon is not specific to our example, but occurs in general. Namely, we prove that every point in the interior of the attainable set of a C-1 control sys tem is reachable by a trajectory corresponding to controls that are pi ecewise constant on the time interval for which the trajectory is inte rior to the attainable set.