On the solutions of polynomial matrix inequalities

Abstract: The problem of solving linear, bilinear and polynomial matrix inequalities is currently a subject of interest in control theory and in the linear case, a number of packages have been developed. The theoretical study of the existence of solutions for the bilinear and polynomial cases remains a difficult problem. In this paper, the more general polynomial matrix inequality problem is studied. By using a Lie series-like linearization technique, the problem can be reduced to a set of scalar linear inequalities together with a number of compatibility conditions which, in logarithmic space, are also linear. This means that the existence problem becomes much easier, and reduces to the identification of the range space of a linear operator. The problem has a nice geometrical interpretation as the intersection of a curvilinear cone and some linear hyperplanes.