  In this article, we consider vibrational systems with semi-active damping
that are described by a second-order model. In order to minimize the influence
of external inputs to the system response, we are optimizing some damping
values. As minimization criterion, we evaluate the energy response, that is the
$\cH_2$-norm of the corresponding transfer function of the system. Computing
the energy response includes solving Lyapunov equations for different damping
parameters. Hence, the minimization process leads to high computational costs
if the system is of large dimension. We present two techniques that reduce the
optimization problem by applying the reduced basis method to the corresponding
parametric Lyapunov equations. In the first method, we determine a reduced
solution space on which the Lyapunov equations and hence the resulting energy
response values are computed approximately in a reasonable time. The second
method includes the reduced basis method in the minimization process. To
evaluate the quality of the approximations, we introduce error estimators that
evaluate the error in the controllability Gramians and the energy response.
Finally, we illustrate the advantages of our methods by applying them to two
different examples.
