# Some remarks on the separability of damped systems and non-self-adjoint eigenvalue problems

• Eberhard Brommundt
Part of the International Centre for Mechanical Sciences book series (CISM, volume 1)

## Abstract

We introduce
$$\left. {\begin{array}{*{20}{c}} {{u_1}\left( {\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{X} ,t} \right) = u\left( {\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{X} ,t} \right)} \\ {{u_2}\left( {\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{X} ,t} \right) = \dot u\left( {\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{X} ,t} \right)} \end{array}} \right\}\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{u} = \left( {\begin{array}{*{20}{c}} {{u_1}} \\ {{u_2}} \end{array}} \right)$$
(A)
into the equation (*) of section 10.1 and obtain
$$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{L} \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{u} = \underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{\mu } \dot u,$$
(B)
where
$$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{L} = \left( {\begin{array}{*{20}{c}} 0&1 \\ { - L}&{ - b} \end{array}} \right),\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{\mu } = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&\mu \end{array}} \right).$$

## Keywords

Boundary Condition Eigenvalue Problem Complex Conjugate Dimensional Case Order System