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

In this paper we treat the case of an abstract vibrational system of the form Mx″+Cx′+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