Minimization of the trace of the solution of Lyapunov equation connected with damped vibrational systems

Our aim is to optimize the damping of a linear vibrating system. As the optimality criterion we use the one where the penalty function is given as the average total energy over all initial states of unit energy, which is equal to the trace of the corresponding Lyapunov solution multiplied by a matrix corresponding to the chosen measure on the set of initial states. We solve this optimization problem and show that the optimal damping matrix corresponds to the so-called modal critical damping. AMS subject classifications: 15A24, 70J99