发表机构
University of California, San Diego（加州大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对末端无负载、由移动小车悬挂的重链，提出反步变换实现快速镇定，通过奇异核的Frobenius分支求解，使位移、速度、应变指数收敛至零。
AI 中文摘要
我们考虑一个从移动小车悬挂、末端无负载的重链的边界镇定问题。由于张力在自由端消失，波速在该处也趋于零；在黎曼坐标下，模型退化为一个$2\ imes2$双曲系统，其中耦合是奇异的，且自由端的反射在域内产生而非由边界条件决定。我们构造了一个Volterra反步变换，将该系统映射到具有任意指定速率的均匀阻尼以及小车处弹性约束的同一链条上，从而使得位移、速度和应变指数收敛到零。奇异核方程通过在自由端选取其有界的Frobenius分支来求解，该分支替代了缺失的边界数据，四个核由全局收敛的幂级数生成。该变换在能量空间上有界可逆，其逆变换通过反转指定的衰减速率获得。该结果将针对圆盘和球上抛物型方程发展的径向反步结构推广到了退化双曲系统。
英文摘要
We consider boundary stabilization of a heavy chain hanging from a moving trolley with no tip load. Because the tension vanishes at the free end, the wave speed vanishes there; in Riemann coordinates the model becomes a degenerate $2\times2$ hyperbolic system in which the coupling is singular and the free-end reflection is generated in the domain rather than by a boundary condition. We construct a Volterra backstepping transformation that maps this system to the same chain with uniform damping of an arbitrarily prescribed rate and an elastic restraint at the trolley, yielding exponential convergence of displacement, velocity, and strain to zero. The singular kernel equations are solved by selecting their bounded Frobenius branch at the free end, which replaces the missing boundary datum, and the four kernels are generated by a globally convergent power series. The transformation is boundedly invertible on the energy space, with inverse obtained by reversing the prescribed decay rate. The result extends the radial backstepping structure developed for parabolic equations on disks and balls to a degenerate hyperbolic system.
