New estimates for solutions of Lyapunov equations

The estimation problem for the solution of the Riccati and Lyapunov matrix equations has attracted considerable attention in the past two decades. In most works, the lower and upper bounds for the following quantities of the solution of the Riccati and Lyapunov matrix equations are obtained: the smallest eigenvalue the largest eigenvalue, the trace, the determinant, the partial summation of eigenvalues, the partial product of eigenvalues, the solution itself. Komaroff (1990, 1988, 1992) used majorisation techniques to obtain some very excellent estimates for the partial summation and partial product of the solution of Lyapunov matrix equations. Mrabti and Hmamed (1992) presented a unified approach using the delta operator technique to obtain lower bound estimates for the solution of both the continuous-time and discrete-time Lyapunov matrix equations. A common assumption for most of results is that A+A´ is negative definite, i.e., λ₁(A+A´) < O. This is obviously restrictive, because the stability of A does not guarantee this assumption. In this paper, the authors remove this assumption and give some more general estimates for the continuous-time Lyapunov matrix differential and algebraic equations