Page 88 - Textos de Matemática Vol. 36

78 Appendix A. Theorem A.1.1. (i) The solution x(·; t0 , x0 ) of the Cauchy problem (A.1),
(A.2), is unique and exists globally; i.e., x(t; t0, x0) is defined for t ∈ R.
(ii) The set L0 of all solutions of the homogeneous differential equation (A.3) is an n-dimensional vector space; each basis {x1,...,xn} of L0 is called a fundamental system (of solutions) of (A.3), and in this case the matrix Φ(t) := (x1,...,xn)(t), t ∈ R, is called a fundamental matrix of (A.3) with the Wronskian W (t) = det Φ(t).
(iii) For t, t0 ∈ R the special fundamental matrix ΦA (t, t0 ) defined (uniquely) by
Φ˙ A(t, t0) := ∂ ΦA(t, t0) = A(t)ΦA(t, t0), ΦA(t0, t0) = In, (A.4) ∂t
is called the transition matrix (or evolution matrix) of (A.1) (or (A.3))
with (iv) The a)
b) c)
d) e)
f)
g)
respect to t0.
transition matrix has the following properties (see e.g., [Broc70])
ΦA(t,τ)ΦA(τ,t0) = ΦA(t,t0) for t,t0,τ ∈ R, hence ΦA(t0, t) = (ΦA(t, t0))−1 for t, t0 ∈ R.
∂ Φ∗A(t, t0) = [ ∂ ΦA(t, t0)]∗ = Φ∗A(t, t0)A∗(t) for t, t0 ∈ R. ∂t ∂t
∂ Φ∗A(t0, t) = −A∗(t)Φ∗A(t0, t) for t, t0 ∈ R,
∂t ∗ ∗
i.e., Φ−A∗ (t, t0) = ΦA(t0, t) or ΦA∗ (t, t0) = Φ−A(t, t0) for t, t0 ∈ R. ∂ ΦA(t0, t) = −ΦA(t, t0)A(t) for t, t0 ∈ R.
∂t
(Abel–Jacobi–Liouville Formula)
t TrA(τ)dτ detΦA(t,t0)=et0 for t,t0∈R,
n
where Tr A(t) = ajj (t) is the trace of A(t).
j=1
If A is constant, then for t,t0 ∈ R,
ΦA(t,t0) = e(t−t0)A :=
(Variation of Constants Formula) The solution of the Cauchy prob-
lem (A.1), (A.2) is given by
x(t) = ΦA(t,t0)x0 +
t t0
ΦA(t,τ)b(τ)dτ; (A.5)
∞ (t−t)j Aj 0 .
j! j=0