An extension of the Levy-Desplanque theorem and some stability conditions for matrices with uncertain entries

Sufficient conditions for Hurwitz stability and for the "degree of stability" of a family of complex matrices with uncertain entries in bounded sets in the complex plane, are derived. The Levy-Desplanque theorem is extended in two directions: the requirement for strict diagonal dominance is alleviated and the (alleviated) theorem is made applicable to families of matrices with uncertain entries. Also, sufficient conditions for Schur stability and for Schur "degree of stability" of a family of real interval matrices are derived. All the above sufficient conditions, as well as the Levy-Desplanque theorem extension, are remarkable in their simplicity to carry out and in the rich variety of possibilities of using them.