ABSTRACT
The present paper is devoted to the mathematical investigation of the initialboundary value problem that describes small vibrations of an ideal filament which is fixed at one end and carries a heavy load at the other end. In current mathematical literature, there exists many works in which the motions of different types of filaments (both linear and nonlinear) have been investigated from purely theoretical and computational points of view (see [5, 7-15, 20-22] and references therein). In this paper, we study the problem which can be considered as a generalization of
the results discussed in the monograph [13]. The author of [13], D.R. Merkin, has started from the general ideas of mechanics and then has derived static and dynamic equations of filaments, has formulated numerous boundary-value problems, and has presented solutions for some of them. Many technological applications have been given in [13] and several computer algorithms have been discussed. In our work, we consider a more general case of a filament with spatially nonhomogeneous parameters and a distributed viscous damping, and we answer several questions that have not been raised in [13] or other sources. For example, we present here a precise asymptotics of the spectrum for a loaded filament and give a rigorous description of such properties of the eigenstates as completeness, minimality, and in the forthcoming paper, we will prove the Riesz basis property of eigenstates in an appropriate state space of the system. The latter results will be instrumental for the solutions of different problems on boundary and distributed controllability of a filament. Such questions are beyond the scope of monograph [13]. In connection with our future applications of the spectral results to controllability problems, we would like to mention paper [6]. An important engineering model of an “overhead crane,” i.e., a motorized platform moving along a horizontal bench with a flexible cable attached to the platform is considered in [6]. The cable is supposed to be homogeneous and to have no damping. A boundary control is proposed that guarantees uniform exponential stability. In their proofs, the authors of [6] have introduced the Lyapunov function (the generalized energy) and have derived the estimate from above in the exponential form. Our goal is different; namely, we would like not only to prove an exact controllability result for the system, but also to give an explicit expression for the control law in the terms of the spectral characteristics of the problem. Finally, we would like to mention two interesting papers [16, 17] by V. Pivovarchik, in which the author has addressed the question of the reconstruction of the string parameters based on the spectral characteristics of the appropriate Sturm-Liouville problem.
