Page 14 - Textos de Matemática Vol. 36

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If we apply the transformation y = y
Chapter 1. Introduction
to equation (1.1),
(1.4)
(1.5)
then we obtain the linear system
y˙ = A˜(t)y + B˜(t)u,
where B˜ = TB and
˜˜ ˙−1 −1A11−A12KA12
A = A(K, t) = T T + T AT = 0 KA + A . 12 22
This yields the first decoupling step if the Riccati differential equation (1.3) admits a solution in a prescribed time interval; in this case the correspond- ing decoupling transformation is called Riccati transformation. Notice that the analogous decoupling of (1.1) is achieved by the transformation
In 0 y(t)= −P(t) I x,
m
where P is a solution of the RDE
P˙ = −P A11(t) + A22(t)P + A21(t) − P A12(t)P.
In the second step, in order to achieve a system in block diagonal form, we use another transformation z = Uy, where
L(t) I
with the inverse U(t) = 0 differential equation
−L(t)
U(t) := −1 In
, (1.6)
In 0
(n+m)×(n+m)
L˙ = (A11 − A12K)L − L(A22 + KA12) − A12. Then the system (1.1) decouples and we obtain
˜ ˜
z˙ = A(t)z + U(t)B(t)u,
(1.7)
(1.8)
I
m
where
˜˜ ˙−1 ˜−1A11−A12K 0
m
A = A(K, t) = UU + UAU = 0 KA + A . 12 22
For any solution K(t) of (1.3) we generally infer that for all t the spectrum σ(A˜) ˜ ˜
of A(K, t) (or A(K, t)) decomposes into two sets, i.e.,
σ(A˜(K, t)) = σ(A11(t) − A12(t)K(t)) ∪ σ(K(t)A12(t) + A22(t)).
x1 = Tx = T x
y1 22
∈ R
and L being a solution of the Sylvester