Page 64 - Textos de Matemática Vol. 36

54 Chapter 2. Proof. Pˆ := P − Prd is a solution of
d Pˆ = −(AT − PrdS)Pˆ − Pˆ(A − SPrd) + PˆSPˆ. dt
Global aspects
(2.51)
(2.52)
From this together with C = I and (2.50) we infer from Theorem 2.5.2 that the ball ||P || < √a remains negative invariant for solutions of (2.51), which yields the negative invariance of ||P − Prd|| < √a for solutions of (2.48).
Remark 2.5.4. i) In control theoretic applications one is interested in the be- havior of solutions of HRDE in reverse time direction, i.e., in subintervals of (−∞, tf ].
ii) From Remark 2.4.11 we infer that the generalized basin of attraction (as t → −∞) of the reverse dichotomic solution Prd is open and dense. There- fore it seems clear to search for negative invariant balls around the reverse dichotomic solution.
iii) If one is interested in positive invariant domains, then it would be advan- tageous to use the dichotomic solution Pd, if it exists.
iv) It should be pointed out here that we need not impose definiteness condi- tions, neither on S nor on Q. This makes the approach applicable also to H∞-type problems.
The following numerical example shows an application to HRDE of H∞
type.
The accompanying Hamilton matrix to (2.51) is ˆA−SPrd −S
H= 0 −(AT−PrdS) .
Example 2.5.5. Let μ = 1 and 2
1−1 10 A= ,S11=
1 0 ,S22= 2 ,
020104
41 1 2
3 0 4 .
, S=S11−μS22= 2222
Q:=
Then {− 1 √22, − 1 √2, 1 √2, 1 √22} is the set of the eigenvalues of the corre-
0 −1
sponding Hamiltonian matrix
H= −Q −AT .
A −S