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det P = 4 √ 7 = 0 P −1 AP <br />
w = P −1 z ˙w = P −1 ˙z = P −1 AP w <br />
˙w1<br />
˙w2<br />
<br />
=<br />
4 + √ 7 0<br />
0 4 − √ 7<br />
w1<br />
˙wi = λiwi wi(t) = Cieλ<strong>it</strong> i = 1, 2 <br />
z = P w <br />
√ √ <br />
z1(t) 1 + 7 1 − 7 C1e<br />
z(t) = =<br />
z2(t) 2 2<br />
λ1t<br />
<br />
= C1e λ1t<br />
u1 + C2e λ2t<br />
u2.<br />
C2e λ2t<br />
z0 z0 = C1u1 + C2u2 C1 C2 <br />
z0 <br />
<br />
C1<br />
= P −1 <br />
C1<br />
z0, z0 = P .<br />
<br />
T (t) =<br />
C2<br />
e λ1t 0<br />
0 e λ2t<br />
w2<br />
<br />
,<br />
C2<br />
<br />
, T −1 <br />
e−λ1τ 0<br />
(τ) =<br />
0 e−λ2τ <br />
= T (−τ).<br />
T (t)T −1 (τ) = T (t − τ) z0 ∈ R 2 z(t) =<br />
P T (t)P −1 z0 ˙z = Az z(0) = z0<br />
<br />
<br />
Φ(t) = P T (t)P −1 Φ −1 (τ) = P T −1 (τ)P −1 R(t, τ) = Φ(t)Φ −1 (τ) =<br />
P T (t)T −1 (τ)P −1 = P T (t − τ)P −1 <br />
<br />
t<br />
u(t) = R(t, 0)z0 + R(t, τ)b(τ) dτ = P T (t)P −1 t<br />
z0 + P T (t − τ)P −1<br />
<br />
τ<br />
e−τ <br />
dτ<br />
0<br />
= P T (t)P −1 z0 +<br />
t<br />
0<br />
= P T (t)P −1 z0 + P T (t)P −1<br />
t<br />
= P T (t)P −1<br />
<br />
z0 + P<br />
P T (t)P −1 P T (−τ)P −1<br />
<br />
τ<br />
t<br />
0<br />
0<br />
T (−τ)P −1<br />
P T (−τ)P −1<br />
<br />
τ<br />
τ<br />
e −τ<br />
<br />
0<br />
e−τ <br />
dτ<br />
e−τ <br />
<br />
dτ<br />
dτ .<br />
<br />
y ′ = Ay + b(t) y ′ = Ay A ∈ LK(Y ) <br />
t <br />
A <br />
<br />
<br />
y(0) = y0 ϕ(t) = etAy0 <br />
e tA = (tA) j<br />
<br />
j!<br />
j∈N<br />
u ∈ Y u = 0 λ ∈ K ϕ(t) = e λt u y ′ = Ay λ <br />
A u A n <br />
λ1, ..., λn <br />
e tA = [e λ1t u1....e λnt un][u1...un] −1
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