On the attainable sets of control systems with p-integrable controls

It is well known that, for a control system, under suitable assumptions, the closure of the attainable set does not change if we consider p-integrable controls for different p. This is an interesting problem and has not been studied in depth, whether or not the attainable set changes when p changes. We show that, for a linear system, the attainable sets may be different for different p. In the two-dimensional case, we prove that the number of indices for which the attainable sets change is finite. Moreover, we show that, for a class of systems, the attainable sets are the same, when the time duration is large enough.