Lee C-H: Eigenvalue upper bounds of the solution of the continuous Riccati equation. Linear Algebra and Its Applications 1996, 235: 247вЂ“259. Kouikoglou VS, Tsourveloudis NC: Eigenvalue bounds on the solutions of coupled Lyapunov and Riccati equations. Kim SW, Park PG: Matrix bounds of the discrete ARE solution. Kwon WH, Moon YS, Ahn SC: Bounds in algebraic Riccati and Lyapunov equations: a survey and some new results. Kang T, Kim BS, Lee JG: Spectral norm and trace bounds of algebraic matrix Riccati equations. IEEE Transactions on Automatic Control 1988, 33: 1088вЂ“1091. Mori T: Comments on A matrix inequality associated with bounds on solutions of algebraic Riccati and Lyapunov equations. IEEE Transactions on Automatic Control 1987, 32: 739вЂ“740. Saniuk JM, Rhodes IB: A matrix inequality associated with bounds on solutions of algebraic Riccati and Lyapunov equations. Wang SD, Kuo T-S, HsГј C: Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equation. Ni M-L: Existence condition on solutions to the algebraic Riccati equation. Prentice-Hall, Englewood Cliffs, NJ, USA 1997.

10.1109/TAC.1968.1098852Дavies R, Shi P, Wiltshire R: New upper solution bounds for perturbed continuous algebraic Riccati equations applied to automatic control. IEEE Transactions on Automatic Control 1968, 13: 150вЂ“159. Kleinman DL, Athans M: The design of suboptimal linear time-varying systems. John Wiley & Sons, New York, NY, USA 1972. Kwakernaak K, Sivan R: Linear Optimal Control Systems. In 1995, Lasserre showed the following given any matrix then the following. In symmetric case, a number of works have been proposed for the trace of matrix products (), and is the tightest among the parallel results. Therefore, considering its applications, it is important to discuss trace bounds for the product of two matrices. Moreover, in terms of, we know that an interpretation of is that is the average value of the optimal cost as varies over the surface of a unit sphere. Thus, a number of works have been presented by researchers to evaluate the bounds and trace bounds for the solution of the ARE (see ). In practice, it is hard to solve the ARE, and there is no general method unless the system matrices are special and there are some methods and algorithms to solve (1.4) however, the solution can be time-consuming and computationally difficult, particularly as the dimensions of the system matrices increase. To guarantee the existence of the positive definite solution to (1.4), we will make the following assumptions: the pair is stabilizable, and the pair is observable. With and being positive definite and positive semidefinite matrices, respectively.

Trace of a matrix