The Estimates Of Solution Of Discrete Lyapunov Equation And The Trace Bound For The Product Of Real Square Matrices

The Lyapunov matrix equation and the Riccati matrix equation are importantequations which play a fundamental role in a variety of control theory. The solutionbounds of these equations can be applied to deal with many control problems suchas optimal control, time delays, Kalman filter design and robust stability analysis.Therefore, the estimate problem for the solution of these equations has attractedconsiderable attention. At the same time, in the process of estimate of the solutionfor these equations, there is correlated question which is how to obtain the estimateof trace bound for the product of matrices. Many scholars have devoted themselvesto give the estimate of trace bound for the product of matrices.In this paper, we remove some limited condition of estimate by using ma-jorization inequality and some thinking method of matrix theory, and improve andgeneralize some related results.In chapter one, we remove the assumption ofλ₁ (AA^T)＜1 by using the similar-ity transformation, and obtain the estimate of the solution for the discrete Lyapunovmatrix equation by using majorization inequality and some properties of eigenvalue,which improve and generalize the recent results.In chapter two, by using the decomposition of singular value of the matrix andthe correlated properties of trace of matrix, we propose new trace bounds for theproduct of two arbitrary real square matrices in nonsymmetric case. At the sametime, we apply the obtained results to estimate the trace bound for the continueRiccati equation and obtain the better result.