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## D. Properties of Continuous-Time Riccati Equations

### D.1 Solutions of the CTDRE

The following lemma gives a useful alternative expression for .

Lemma D.1 The solution, , to the CTDRE (D.0.1), can be expressed as

 (D.1.1)

where and satisfy the following equation:

 (D.1.2)

subject to

 (D.1.3)

##### Proof

We show that , as defined above, satisfies the CTDRE. We first have that

 (D.1.4)

The derivative of can be computed by noting that ; then

 (D.1.5)

from which we obtain

 (D.1.6)

Thus, Equation (D.1.4) can be used with (D.1.2) to yield

 (D.1.7)

which shows that also satisfies (D.0.1), upon using (D.1.1).

The matrix on the right-hand side of (D.1.2), namely,

 (D.1.8)

is called the Hamiltonian matrix associated with this problem.

Next, note that (D.0.1) can be expressed in compact form as

 (D.1.9)

Then, not surprisingly, solutions to the CTDRE, (D.0.1), are intimately connected to the properties of the Hamiltonian matrix.

We first note that has the following reflexive property:

 (D.1.10)

where is the identity matrix in .

Recall that a similarity transformation preserves the eigenvalues; thus, the eigenvalues of are the same as those of . On the other hand, the eigenvalues of and must be the same. Hence, the spectral set of is the union of two sets, and , such that, if , then . We assume that does not contain any eigenvalue on the imaginary axis (note that it suffices, for this to occur, that be stabilizable and that the pair have no undetectable poles on the stability boundary). In this case, can be so formed that it contains only the eigenvalues of that lie in the open LHP. Then, there always exists a nonsingular transformation such that

 (D.1.11)

where and are diagonal matrices with eigenvalue sets and , respectively.

We can use to transform the matrices and , to obtain

 (D.1.12)

Thus, (D.1.2) can be expressed in the equivalent form:

 (D.1.13)

If we partition in a form consistent with the matrix Equation (D.1.13), we have that

 (D.1.14)

The solution to the CTDRE is then given by the following lemma.

Lemma D.2 A solution for Equation (D.0.1) is given by

 (D.1.15)

where

 (D.1.16)

 (D.1.17)

 (D.1.18)

##### Proof

From (D.1.12), we have

 (D.1.19)

Hence, from (D.1.3),

 (D.1.20)

or

 (D.1.21)

or

 (D.1.22)

Now, from (D.1.10),

 (D.1.23)

and the solution to (D.1.13) is

 (D.1.24)

Hence,

 (D.1.25)

Substituting (D.1.25) into (D.1.23) gives the result.

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