## Monotonicity of maximal solutions of algebraic Riccati equations.(English)Zbl 0583.15007

It is shown that if $$X_+$$ is a solution of the Riccati equation (1) $$XBB^*X-A^*X-XA-Q=0,\ldots$$, (where the pair (A,B) is stabilizable, and Q is Hermitian) such that Re $$\lambda$$ (A-BB$${}^*X_+)\leq 0$$, and if $$X_ 1$$ is any solution of $$X_ 1B_ 1B^*\!_ 1X_ 1-A^*\!_ 1X_ 1-X_ 1A_ 1-Q_ 1=0$$, then $\left( \begin{matrix} Q\\ A\end{matrix} \begin{matrix} A^*\\ -BB^*\end{matrix} \right)\geq \left( \begin{matrix} Q_ 1\\ A_ 1\end{matrix} \begin{matrix} A^*\!_ 1\\ -B_ 1B^*\!_ 1\end{matrix} \right)$ implies $$X_+\geq X_ 1$$. Based on this monotonicity result it is further shown that a) $$X_+\geq X$$ for all other solutions X of (1), b) $$X_+$$ is unique, and c) if in addition $$Q\geq 0$$, then $$X_+\geq 0$$.
Reviewer: M.E.Sezer

### MSC:

 15A24 Matrix equations and identities

### Keywords:

maximal solutions; stabilizability; Riccati equation; Hermitian
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### References:

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