Matrix equations and robust stability of family of matrices

In this dissertaion, the stability problems for a single matrix and a family of matrices are studied. A single condition for the eigenvalues of a single matrix to lie in a rational region are obtained. This condition are expressed as Hurwitz stability of a suitable set of matrices and as an existence of acommon solution to the Lyapunov equations. Hurwitz stability of a polynomial family is expressed as an existence of a common solution to the Lyapunov equations. In this dissertaion,the sector stability is equivalent to the nonsingularity of an extended polytope. For aspecial types of 3x3 matrices the existence problem of a common solution to the Lyapunov equations. The obtained results are illustrated by a number of examples.