An algorithmic solution of parameter-varying Linear Matrix Inequalities in Toeplitz form

In this paper the necessary and sufficient conditions are given for the solution of a system of parameter varying linear inequalities of the form ( ) ( ) t t ≥ A x b for all t T ∈ , where T is an arbitrary set, x is the unknown vector, ( ) t A is a known triangular Toeplitz matrix and ( ) t b is a known vector. For every t T ∈ the corresponding inequality defines a polyhedron, in which the solution should exist. The solution of the linear system is the intersection of the corresponding polyhedrons for every t T ∈ . A decomposition method has been developed, which is based on the successive reduction of the initial system of inequalities by reducing iteratively the number of variables and by considering an equivalent system of inequalities.