On the Synchronizable System
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
On the Synchronizable System∗ Zhen LEI1
Tatsien LI2
Bopeng RAO3
Abstract In this paper, the synchronizable system is defined and studied for a coupled system of wave equations with the same wave speed or with different wave speeds. Keywords Synchronizable system, Synchronization solution, Coupled system of wave equations, Exact boundary synchronization 2000 MR Subject Classification 35L05, 35L53, 93B05, 93C20
1 Introduction In the study of the exact boundary synchronization for a coupled system of wave equations with boundary controls, it is necessary to assume that the system possesses synchronization solutions after all boundary controls are eliminated (see [2]), however, it is certainly not the case that all the systems satisfy this property. From this point of view, we call a system to be a synchronizable system if it has synchronization solutions. In this paper, we will examine the characters of the synchronizable system, namely, to answer the following questions: What system is a synchronizable system? And, for a given synchronizable system, how to get all the synchronization solutions? Namely, how to find all the initial data with synchronization property, such that the corresponding solutions of the system are synchronization ones. A special and important situation is that all the initial data with synchronization property give the corresponding synchronization solutions; otherwise, one should find all suitable initial data with synchronization property, such that the corresponding solutions are synchronization ones. In other words, we should precisely study the existence and the construction of synchronization solutions. In order to solve these problems, in this paper we first consider in Section 2 a coupled system of wave equations with the same wave speed together with Dirichlet boundary controls (see [1, 2]), then examine the case with different wave speeds in Section 3. Finally, some possible generalizations will be given in Section 4 for the further study. Manuscript received June 19, 2020. of Mathematical Sciences, Fudan University, Shanghai 200433, China; Shanghai Key Laboratory for Contemporary Applied Mathematics; Nonlinear Mathematical Modeling and Methods Laboratory. E-mail: [email protected] 2 Corresponding author. School of Mathematical Sciences, Fudan University, Shanghai 200433, China; Shanghai Key Laboratory for Contemporary Applied Mathematics; Nonlinear Mathematical Modeling and Methods Laboratory. E-mail: [email protected] 3 Institut de Recherche Math´ ematique Avanc´ ee, Universit´ e de Strasbourg, 67084 Strasbourg, France. E-mail: [email protected] ∗ Project supported by the National Natural Science Foundation of China (Nos. 11831011, 11725102). 1 School
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Z. Lei, T. T. Li and B. P. Rao
2 A Coupled System of Wave Equations with the Same Wave Speed Let Ω ⊂ Rn be a bounded domain with smooth boundary Γ. Consider the following homogeneous coupled system of wa
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