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Optimal damping of the infinite-dimensional vibrational systems: commutative case

In this paper we treat the case of abstract vibrational system of the form $M\ddot{; ; ; ; ; ; x}; ; ; ; ; ; +C\dot{; ; ; ; ; ; x}; ; ; ; ; ; +x=0$, where the positive semi- definite selfadjoint operators $M$ and $C$ commute. We explicitly calculate the solution of the corresponding Lyapunov equation which enables us to obtain the set of optimal damping operators, thus extending already known results in the matrix case.

optimal damping; modal damping; Lyapunov equation

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