COMPLETE CONTROLLABILITY OF PERTURBED LINEAR-CONTROL SYSTEMS, REVISITED

Let L1(I, R(n,n)) x M(I, R(n,m)) be the space of all pairs (A, B), where A and B are measurable functions from a compact interval I to R(n,n) and R(n,m), respectively, and A is Lebesgue integrable. Also, let this space be endowed with the topology of the L1-norm with respect to A and the topology of convergence in measure with respect to B. Then, the set of all pairs (A, B), for which the corresponding linear control system (S) x = A(t)x + B(t)u(t), a.e. t epsilon-I, is completely controllable on I, is shown to be open in L1(I, R(n,n)) x M (I, R(n,m)). It is also proved that, given any (A, B) epsilon L1(I, R(n,n)) x M (I, R(n,m)), (S) can be made completely controllable by means of an arbitrarily small perturbation of B in the L-infinity-norm. These results are extensions of the analogous ones given by Dauer in the case when also B is integrable. Also, it is observed that a well-known complete controllability criterion due to Conti works in the present case.