Problem on strings system vibrations on a star-shaped graph with a nonlinear condition at a node

Abstract

We consider a system of $n$ strings located in the equilibrium position along a geometrical star-graph. We suppose that the edges of the graph have the same lengths and the graph is oriented to the node. We study the case when the initial velocity of each string is zero. The initial shape of each string is defined by means of given functions on the edges. We assume that at the boundary vertices the strings are fixed. We study the oscillatory process for the case, when the node point of the string system is located inside the motion limiter. At the same time we suppose that the limiter can move in the direction perpendicular to the graph plane. While the limiter does not touch with the node point of the string system, the transmission condition holds (the Kirchoff condition). Once the limiter touches the node, their joint motion begins and an additional restriction for the sign of the sum of derivatives at the node appears. Thus, at the node, a hysteresis type condition is satisfied.

In the work we obtain a representation for the solution and prove its existence. For a particular
case we consider a case on periodic oscillations of the node point of the string system. We solve a problem on boundary control of the oscillatory process under the assumption that the oscillation time does not exceed the length of the string.