Asymmetric exact controllability for networks of spatial elastic strings, springs and masses

TL;DR

The paper develops a vector-valued, quasilinear hyperbolic model for networks of nonlinear elastic strings connected by springs with end masses, introducing dynamic boundary conditions at joints. It proves semi-global well-posedness near stretched equilibria and establishes local and global-local exact boundary controllability for star-like networks with $n-1$ boundary controls, while revealing asymmetric controllability spaces induced by masses and network Laplacian structure. The study also analyzes damaged configurations (missing springs), showing the rank condition on the joint Laplacian is sufficient but not necessary for controllability. These results motivate extensions to serial/ring topologies, progressive damage modeling, and a hybrid PDE–ODE framework to unify the dynamics and further analyze stabilization and robustness.

Abstract

We consider networks of elastic strings with end masses, where the coupling is modeled via elastic springs. The model is representative of a network of nonlinear strings, where the strings are coupled to elastic bodies. The coupled system converges to the classical string network model with Kirchhoff and continuity transmission conditions as the spring stiffness terms approach infinity and the masses at the nodes vanish. Due to the presence of point masses at the nodes, the boundary conditions become dynamic, and consequently, the corresponding first-order system of quasilinear balance laws exhibits nonlocal boundary conditions. We demonstrate well-posedness in the sense of semi-global classical solutions \cite{li} (i.e., for arbitrarily large time intervals provided that the initial and boundary data are small enough) and observe extra regularity at the masses as in \cite{WangLeugeringLi2017,WangLeugeringLi2019}. We prove local and global-local exact boundary controllability of a star-like network when control is active at the endpoints of the string-spring-mass system, except for one clamped end. In this case, at multiple nodes, a complex smoothing pattern appears, leading to asymmetric control spaces when the springs and masses are present. Furthermore, the rank of the Laplacian matrix at the junction is crucial for the controllability property, particularly in models containing wave equations with degeneration at dynamic boundaries, which can be interpreted as damage in mechanical vibration systems where parts of the springs are missing.

Figures (9)

• Figure 1: A star-like network consisting of four strings represented in $I\!\!R^3$ (left) and as a diagram (right).
• Figure 2: A consistent indices notation for the global string graph $\mathcal{G}$ and the local spring graph $\mathcal{G}^j$ -- Visualization with a star-like network.
• Figure 3: Diagrams of star-like networks consisting of $n=4$ strings, either with rank $n-1$ Laplacian matrix (left, middle), or with rank $< n-1$ Laplacian matrix (right).
• Figure 4: Star-like network clamped at the node $\mathbf{N}^1$ and controlled at the other simple nodes $\{\mathbf{N}^{i}\}_{i=2}^n$.
• Figure 5: Constructive method with forward, backward process (left), and sidewise Cauchy problem from boundary to boundary (right)

Theorems & Definitions (27)

• Remark 1
• Remark 2
• Definition 1
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8
• Example 1