Finite-time stabilization of a network of strings

TL;DR

This paper demonstrates that with appropriate damping coefficients, solutions of wave equations on a network of strings stabilize to a constant state in finite time, extending classical results to complex network geometries.

Contribution

It introduces a novel finite-time stabilization method for string networks using damping at internal nodes, with sharp conditions on damping coefficients.

Findings

Solutions become constant after finite time with suitable damping.

Results are sharp for star-shaped networks.

Similar stabilization results hold under Dirichlet or Neumann boundary conditions.

Abstract

We investigate the finite-time stabilization of a tree-shaped network of strings. Transparent boundary conditions are applied at all the external nodes. At any internal node, in addition to the usual continuity conditions, a modified Kirchhoff law incorporating a damping term $\alpha u_t$ with a coefficient $\alpha$ that may depend on the node is considered. We show that for a convenient choice of the sequence of coefficients $\alpha$, any solution of the wave equation on the network becomes constant after a finite time. The condition on the coefficients proves to be sharp at least for a star-shaped tree. Similar results are derived when we replace the transparent boundary condition by the Dirichlet (resp. Neumann) boundary condition at one external node.