Upper and lower matrix bounds of the solutions for the continuous and discrete Lyapunov equations

Solution to dynamic matrix linear systems by operator exponential functions

“…It can be found from (12), (13) and Theorem 6 that the complete solution of the matrix system (7) is simply the sum of zero-input solution and zero-state solution:…”

“…On the other hand, the Lyapunov matrix equations are very closely related to the linear systems [13]. For example, the Lyapunov matrix equations can be used to check the stability [1], [2], and controllability and observerbility [9].…”

Matrix solution bounds of the continuous Lyapunov equation by using a bilinear transformation

“…Therefore, during the past two decades, research for deriving solution bounds including matrix and eigenvalue bounds of the CALE (1) has become an attractive topic. A number of research approaches for this topic have been proposed in the literature [2,3,[5][6][7][8][9][11][12][13][14][15][16][20][21][22]24]. Among those solution bounds, the matrix bounds can define all eigenvalue bounds such as bounds of the extreme eigenvalues, the summation of eigenvalues, the trace, the product of eigenvalues, and the determinant; hence they are the most general findings.…”

“…Among those solution bounds, the matrix bounds can define all eigenvalue bounds such as bounds of the extreme eigenvalues, the summation of eigenvalues, the trace, the product of eigenvalues, and the determinant; hence they are the most general findings. Surveying existing works, matrix bounds of the solution of the CALE (1) have been proposed in [2,3,6,[12][13][14][15]20,22]. However, all of those estimations have some limitations.…”

“…But, how to select the positive constant matrix such that the resulting bounds are the best is also an open problem. In [15], the matrix A must be assumed to be diagonalizable. Furthermore, bounds in [20,22] involve integrations of time functions and hence are troublesome for calculation.…”