Some bounds in the continuous algebraic Riccati equation

A new lower bound for the trace and maximal eigenvalue and an upper bound for the minimal value of the positive definite solution to the continuous algebraic Riccati equation are derived. To obtain these bounds we used the fact that the spectrum of matrix A - BP can be found on the base of the coefficients of the Riccati equation without solving the equation. This well known property of the Riccati equation has not been yet used to estimate the spectrum of the solution. Our bounds appear to be considerably tighter in some cases. A proposition of improving one of existing lower bound for the trace is given also.