Chin. Ann. Math. Ser. B 41(3), 2020, 325–334 DOI: 10.1007/s11401-020-0201-1 Chinese Annals of Mathematics, Series B c  The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2020 Exact Boundary Controllability for the Spatial Vibration of String with Dynamical Boundary Conditions * Yue WANG 1 G¨ unter LEUGERING 2 Tatsien LI 3 Abstract This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure. Keywords Spatial vibration of a string, Exact boundary controllability, Dynamical boundary condition 2000 MR Subject Classification 35L05, 35L72, 93B05 1 Introduction In [9] we introduced a quasilinear wave equation with dynamical boundary conditions to describe the lateral vibration of an elastic string with masses on the corresponding ends. The exact boundary controllability of this system was realized using the constructive method given in [1, 3–4]. In this paper, we will consider the spatial vibration in R 3 of an elastic string with masses on the ends. The model is shown in Section 2 (see [2, 5–6, 8]). The main target of this paper is to establish controllability results for an elastic string governed by a system of quasilinear wave equations with dynamical boundary conditions. The main results are shown in Section 3 and proved in Section 5, which are based on the theory of semi-global C 2 solution to the corresponding non-local mixed problem (see Section 4). 2 Spatial Vibration of an Elastic String In order to consider the dynamical behavior of a single elastic string in three-dimensional space, which is not limited to small deformation, we first establish its dynamical equations. Manuscript received September 14, 2018. Revised September 30, 2018. 1 Department of Mathematics, Friedrich-Alexander University, Erlangen 91058, Germany; School of Mathematical Sciences, Fudan University, Shanghai 200433, China. E-mail: yue wang@fudan.edu.cn 2 Department of Mathematics, Friedrich-Alexander University, Erlangen-Nuremberg, Cauerstrasse 11, 91058 Erlangen, Germany. E-mail: guenter.leugering@fau.de 3 School of Mathematical Sciences, Fudan University, Shanghai 200433, China; Shanghai Key Laboratory for Contemporary Applied Mathematic; Nonlinear Mathematical Modeling and Methods Laboratory. E-mail: dqli@fudan.edu.cn * This work was supported by the National Natural Science Foundation of China (No. 11831011).