Asymptotic stability of a system of bodies partially filled with ideal fluids under the action of elastic and damping forces

In this paper, the asymptotic stability of a two-dimensional model problem on small motions of a system of open vessels partially filled with ideal homogeneous fluids under the action of elastic and damping forces is investigated. the first and last bodies are attached by springs to two supports with a given law of motion. the trajectory of the system is perpendicular to the direction of gravity, and the damping forces acting on the hydromechanical system are generated by the friction of bodies against a stationary horizontal support. the law of total energy balance is formulated for the described system. in the author's work [ljm] the initial-boundary value problem is reduced to the cauchy problem for a first-order differential equation with operator coefficients in the orthogonal sum of some hilbert spaces. the theorem on the unique solvability of the obtained cauchy problem on the positive semiaxis is proved. sufficient conditions for the existence of a strong solution of the initial boundary value problem describing the evolution of the hydraulic system are found. in the author's work [semi] the spectral properties of this problem are investigated. it is proved that the problem has a discrete spectrum localized in a vertical strip. the asymptotic behavior of the spectrum is investigated. the theorem on the abel-lidsky basis property of root elements of the problem is proved. in this paper, relying on the theorem of v. arendt, c. batty, yu.i. lubich, f. vu, the strong stability of a system consisting of one or two rectangular bodies partially filled with ideal fluids under the action of an elastic damping device is investigated. it is established that the question of asymptotic stability of the system is equivalent to the presence of common eigenvalues of a series of steklov problems with an additional normalization condition. it is proved that in the case of one body, the system is not strongly stable. in the case of two bodies, the system is not strongly stable if and only if the bodies are congruent.