  We consider sensitivity of a semidefinite program under perturbations in the
case that the primal problem is strictly feasible and the dual problem is
weakly feasible. When the coefficient matrices are perturbed, the optimal
values can change discontinuously as explained in concrete examples. We show
that the optimal value of such a semidefinite program changes continuously
under conditions involving the behavior of the minimal faces of the perturbed
dual problems. In addition, we determine what kinds of perturbations keep the
minimal faces invariant, by using the reducing certificates, which are produced
in facial reduction. Our results allow us to classify the behavior of the
minimal face of a semidefinite program obtained from a control problem.
