Extended projection conditions for some structured LMI problems

The article investigates the extension of the projection Lemma to a particular case of linear matrix inequalities (LMIs). Throughout the article, the LMIs considered have a special two-block-diagonal structure in the optimization variable. Such a system of LMIs arises naturally in certain optimization problems: e.g. the synthesis of a common state-feedback for two different plants and some special matrix dilation problems. The feasibility condition is derived using convex analysis and it is demonstrated that the extended projection conditions are nonlinear in the optimisation variables