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We consider a vibrational system control problem over a finite time horizon.The performance measure of the system is taken to be $p$-mixed $H_2$ norm whichgeneralizes the standard $H_2$ norm. We present an algorithm for efficientcalculation of this norm in the case when the system is parameter dependent andthe number of inputs or outputs of the system is significantly smaller than theorder of the system. Our approach is based on a novel procedure which is notbased on solving Lyapunov equations and which takes into account the structureof the system. We use a characterization of the $H_2$ norm given in terms ofintegrals which we solve using adaptive quadrature rules. This enables us touse recycling strategies as well as parallelization. The efficiency of the newalgorithm allows for an analysis of the influence of various system parametersand different finite time horizons on the value of the $p$-mixed $H_2$ norm. Weillustrate our approach by numerical examples concerning an $n$-mass oscillatorwith one damper.

Finite time horizon mixed control of vibrational systems