Weakly dissipative linear dynamical systems and a quadratic Keldysh pencil

In a Hilbert space, we consider a second-order differential equation with unbounded operator coefficients modeling small motions of a dynamical system with weak energy dissipation. A theorem on the existence and uniqueness of a classical solution is stated. The corresponding spectral problem is reduced to the study of an elliptic quadratic pencil, which, in turn, can be reduced to a “modified” Keldysh pencil. Depending on the asymptotics of the spectrum of the main operator of the problem (the potential energy operator) and the subordination coefficient of the energy dissipation operator, we prove that the corresponding root function system of the linearized problem is a $2$-fold Bari basis, Riesz basis, or Abel–Lidskii basis with parentheses. By way of application, we consider the problem on a quadratic Sturm–Liouville pencil as well as the general spectral problem generated by the problem on small motions of a system of hinged pendulums with cavities completely or partly filled with ideal incompressible fluids and with friction in the hinges.