Institute of Systems Dynamics and Control Theory,

Siberian Branch of RAS,

134 Lermontov Str.,

664033 Irkutsk,

RUSSIA

E-mail: irteg@icc.ru

**Application of the Computer Algebra System in Investigation
of D-Stability of Matrices**

**Abstract:**

Algorithms for investigation of quadratic matrices, which possess the
property of D-stability, are discussed. In the general case, it is impossible
to write down constructive (i.e. verifiable within a finite number of steps)
D-stability conditions; dif-ferent authors have obtained only some necessary
and some sufficient conditions. Necessary and sufficient D-stability conditions
for the 2nd and 3rd order matrices have been known long ago. The problem
of necessary and sufficient D-stability conditions for 4th order matrices
has not been solved completely: likewise in the general case, some necessary
and some sufficient (rather strong) conditions are known. Our paper discusses
other necessary D-stability conditions for the 4th order matrices, which
are expressed via matrix elements. It is shown that these conditions are
necessary and sufficient for the matrices of some definite type. Examples
of parametric analysis of matrices are given; the results give evidence
of closeness of the proposed necessary conditions to the sufficient ones.