We propose diverse upper bounds for the solution matrix of the continuous algebraic Riccati matrix equation (CARE) by building the equivalent form of the CARE and using some matrix inequalities and linear algebraic techniques. Finally, numerical example is given to demonstrate the effectiveness of the obtained results in this work as compared with some existing results in the literature. These new bounds are less restrictive and provide more efficient results in some cases.

In many areas of optimal control [

For example, consider the following linear system such that

When

It is assumed that the pair

The problem of estimating solution bounds for the algebraic Riccati and Lyapunov matrix equations has widely been considered in the recent years, since these equations are widely used in many fields of control system analysis and design. A number of works have reported numerical algorithms to get the exact solution of the mentioned equations [

The existing results obtained during 1974–1994 have been summarized by Kwon et al. [

Let

The following lemmas are used to prove the main result of this paper.

Let

For any matrix

For any symmetric matrices

Let

Let

Let

The following matrix inequality:

The positive semidefinite solution

This eigenvalue upper bound (

Zhang and Liu in [

Assume that

By adding and subtracting

The inequality (3.5) in [

It is well known that most of the studies in the literature have focused to derive the bounds for the maximum and minimum eigenvalues, the trace, and the determinant for the solution of the CARE (

By using Theorem

Assume that

By establishing the more general form than the matrix identity used in Theorem

Let

By adding and subtracting

This builds the proof.

Note that for the upper bound (

From Theorem

The positive semidefinite solution

Applying Lemma

The solution

Let

By the use of the equality (

The proof is finished.

According to Theorem

The positive semidefinite solution

Substituting

The positive semidefinite solution

As considered a diverse matrix identity, in the case that the matrix

If the positive definite matrix

When the term

Thus, the proof is established.

Let

Consider (

This concludes the proof of the theorem.

The solution

Chen and Lee in [

In this section, we will give a numerical example to demonstrate the effectiveness of the proposed results of this paper.

Using Theorem

By a simple computation, we have

In this paper, new upper matrix bounds for the solution of the CARE are improved by using some linear algebraic techniques and matrix inequalities. A numerical example is given to show that the solution upper bounds presented in this paper are sharper than some results in the literature.

The authors would like to thank the editor and the reviewers for the very helpful comments and suggestions to improve the presentation of this study. This study has been supported by the Coordinatorship of Selçuk University’s Scientific Research Projects (BAP) and The Scientific and Technical Research Council of Turkey (TUBITAK).