Existence of Homogeneous Polynomial Solutions for Parameter-Dependent Linear Matrix Inequalities with Parameters in the Simplex

This paper presents some general results concerning the existence of homogeneous polynomial solutions to parameter-dependent linear matrix inequalities whose coefficients are continuous functions of parameters lying in the unit simplex. These results are useful in the context of robust analysis and synthesis of parameter-dependent feedback gains (gain-scheduling) for uncertain linear systems in polytopic domains. A result showing the generality of the class of static gains with homogeneous polynomial dependence and a result dealing with the solutions of parameter-dependent linear matrix inequalities with slowly time-varying parameters are also given