Ifeife I

By differentiating xk = <&-1(i, ik)x(i) from Eq. (4.2.7) and using the first of Eq. (4.2.6) it can be shown that

with initial conditions

Under certain conditions on A(t), the state transition matrix may be inverted analytically (Battin, 1999).

If the matrix A(t) can be partitioned in the form

where the submatrices have the properties that

" A1

A2 "

_ A3

A4 _

Then $(i, tk) can be similarly partitioned as

$1

$2

$3

$4

r $3

-$3

-$3

$3

The proof follows: Defile

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