On the Uniqueness of the Almost Stabilizing Solution of the Modified Algebraic Riccati Equation

In this paper, a modified algebraic Riccati equation (MARE), which arises in the modified optimal control and filtering, the networked control and estimation, and the discrete-time network synchronization problems, is investigated in the case of a positive semi-definite weighting matrix with no detectability assumption. The new approach of studying the uniqueness of the positive definite solution is proposed for analyzing the uniqueness of the almost stabilizing solution. First, for the MARE with an anti-Schur system matrix and an invertible input matrix, it is established that there exists a unique positive definite solution, which is stabilizing and is the unique almost stabilizing and positive semi-definite solution. Furthermore, regarding the MARE with a two-dimensional system matrix and a zero weighting matrix, the matrix solutions are obtained explicitly, and the uniqueness of the almost stabilizing solution is demonstrated. In addition, numerical examples are provided to illustrate the theoretical results.