Partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls
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Partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls Chenmu WANG,
Yanyan WANG
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
c Higher Education Press 2020
Abstract In this paper, we propose the concept of partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls, and make a deep discussion on it. We analyze the relationship between the partial approximate boundary synchronization and the partial exact boundary synchronization, and obtain sufficient conditions to realize the partial approximate boundary synchronization and necessary conditions of Kalman’s criterion. In addition, with the help of partial synchronization decomposition, a condition that the approximately synchronizable state does not depend on the sequence of boundary controls is also given. Keywords Coupled system of wave equations, partial approximate boundary synchronization, partially approximately synchronizable state MSC 93B05, 93C20 1
Introduction
Based on the work of Li and Rao [1,2,4,5,7] about the exact boundary synchronization, Wang [11] proposed the concept of partial exact boundary synchronization (by groups). When the domain satisfies the multiplier geometrical condition, in order to realize the partial exact boundary synchronization by p-groups for a coupled system of wave equations composed of N state variables, without the requirement of synchronization for the first m state variables, while the last (N −m) state variables are synchronizable by p-groups, the number of boundary controls should not be less than (N − m − p).
Received February 9, 2020; accepted June 23, 2020 Corresponding author: Chenmu WANG, E-mail: [email protected]
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Chenmu WANG, Yanyan WANG
In this paper, we consider the partial approximate boundary synchronization. From [3,6], we know that the approximate boundary synchronization can not only reduce the requirement of multiplier geometrical condition of the domain, but also reduce the number of boundary controls. This paper will concern the partial approximate boundary synchronization on the basis of the research work of approximate boundary synchronization [3,6] and the partial exact boundary synchronization. We will consider when the partial approximate boundary synchronization can be realized, and what is the property of the corresponding partially approximately synchronizable state. Consider the following coupled system of wave equations with Dirichlet boundary controls: 00 U − ∆U + AU = 0 in (0, +∞) × Ω, U =0 on (0, +∞) × Γ0 , (1) U = DH on (0, +∞) × Γ1 , with the initial condition ˆ0 , U ˆ1 ) t = 0 : (U, U 0 ) = (U where, “ 0 ” stands for the time derivative, ∆ =
in Ω, Pn ∂ 2
i=1 ∂x2i
(2) is the n-dimensional
(u(1) , . . . , u(N ) )T
Laplacian operator, U = denotes the state variable, Ω is a bounded domain with smooth boundary Γ, Γ = Γ0 ∪ Γ1 with Γ0 ∩ Γ1 = ∅ and mes(Γ1 ) 6= 0; and H = (h(1) , . . . , h(M ) )T (M 6 N ) is the boundary
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