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## D. Properties of Continuous-Time Riccati Equations

### D.2 Solutions of the CTARE

The Continuous Time Algebraic Riccati Equation (CTARE) has many solutions, because it is a matrix quadratic equation. The solutions can be characterized as follows.

Lemma D.3 Consider the following CTARE:

 (D.2.1)

(i) The CTARE can be expressed as
 (D.2.2)
where is defined in (D.1.8).
(ii) Let be defined so that
 (D.2.3)
where are any partitioning of the (generalized) eigenvalues of such that, if is equal to for same , then for some .
Let
 (D.2.4)
Then is a solution of the CTARE.
##### Proof
(i) This follows direct substitution.
(ii) The form of ensures that
 (D.2.5)
 (D.2.6)

where * denotes a possible nonzero component.
Hence,

 (D.2.7)
 (D.2.8)

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