ON DISCRETE-TIME DIAGONAL AND D-STABILITY

This paper introduces four discrete-time analogs of different types of matrix stability: diagonal, simultaneous, vertex, and D-stability, the last three being defined in terms of a certain (associated) polytope of matrices. The diagonal stability of any vertex of this polytope is shown to imply its simultaneous stability and hence D-stability of the vertices. It is shown, by a counterexample, that D-stability, as in the continuous-time case, is not equivalent to diagonal stability. Several important classes for which this equivalence is true are identified. It is shown that simultaneous stability of the vertices is equivalent to the simultaneous stability of the whole polytope, and it is conjectured that this equivalence holds without the requirement of simultaneity. Some other conjectures relating the four types of stability are made, and it is shown that in the 2 X 2 case all four are equivalent.