Interception of matrix inequalities solution sets in problems synthesis of static feedback controllers

The article considers some possible descriptions of Lyapunov matrix inequalities solutions sets in the problems of synthesis of static state and static output feedback controllers for linear continuous stationary controlled systems. For each inequality with a given matrix of the system, in general, there is its own set of solutions, which is a set of positive definite matrices of the Lyapunov function. The conditions for the solvability of Lyapunov inequalities in different bases are shown, depending on the selected non-degenerate transformation matrices. If we perform a similarity transformation for the matrix of the system, then for the solvability of the Lyapunov inequality, it is sufficient that in the corresponding basis the matrix of the Lyapunov function is congruent to the matrix of the same name in the original basis. It is shown that in some cases, the matrix inequalities solutions sets for a static state and static output feedback controllers may coincide, which allows us to reduce the static output feedback controller synthesis task to the static state feedback controller synthesis task. The conditions for the implementation of such cases imply a certain structure of the input and output matrices, as well as the presence of one stable diagonal block of nonzero dimension in the original matrix of the linear system.