Destabilizing Heegaard splittings of knot exteriors

We give a necessary and sufficient condition for Heegaard splittings of knot exteriors to admit destabilizations. As an application, we show the following: let K1 and K2 be a pair of knots which is introduced by Morimoto as an example giving degeneration of tunnel number under connected sum. The Heegaard splitting of the exterior of K1#K2 derived from certain minimal unknotting tunnel systems of K1 and K2 is stabilized.