Exponential stability of a tree-shaped string network and its reduced-order discretization via frequency-domain multiplier

Abstract In this paper, we consider the stability and discretization of a tree-shaped string network, which models flexible space structures like large deployable antennas. We first show that the continuous system, described by coupled wave equations with boundary feedback, is exponentially stable. To achieve this, we design frequency-domain multipliers and use only two controllers-fewer than in previous studies. Next, we discretize the system using a reduced-order finite difference scheme and construct discrete multipliers to prove that the semi-discrete system preserves exponential stability. Numerical simulations support the theoretical findings. This work extends the frequency-domain multiplier method to discrete partial differential equation systems with network structures and demonstrates that exponential stability can be maintained after discretization.