scholarly journals Approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary conditions

2021 ◽  
Author(s):  
Ta-Tsien Li ◽  
Bopeng Rao
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Coupled System ◽  
Algebraic Characterization ◽  
Robin Boundary Conditions ◽  
Boundary Controllability ◽  
Boundary Controls ◽  
Robin Boundary

In this paper, we first give an algebraic characterization of uniqueness of continuation for a coupled system of wave equations with coupled Robin boundary conditions. Then, the approximate boundary controllability and the approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary controls are developed around this fundamental characterization.

scholarly journals Exact boundary controllability and exact boundary synchronization for a coupled system of wave equations with coupled Robin boundary controls

2020 ◽  
Author(s):  
Tatsien LI ◽  
Xing LU ◽  
Bopeng Rao
Keyword(s):  
Boundary Conditions ◽  
Mixed Problem ◽  
Neumann Boundary ◽  
Coupled System ◽  
Exact Boundary Controllability ◽  
Boundary Controllability ◽  
Exact Boundary ◽  
Boundary Controls ◽  
Robin Boundary ◽  
Exact Boundary Synchronization

In this paper, we consider the exact boundary controllability and the exact boundary synchronization (by groups) for a coupled system of wave quations with coupled Robin boundary controls. Owing to the difficulty coming from the lack of regularity of the solution, we confront a bigger challenge than that in the case with Dirichlet or Neumann boundary controls. In order to overcome this difficulty, we use the regularity results of solutions to the mixed problem with Neumann boundary conditions by Lasiecka and Triggiani ([6]) to get the regularity of solutions to the mixed problem with coupled Robin boundary conditions. Thus we show the exact boundary controllability of the system, and by a method of compact perturbation, we obtain the non-exact boundary controllability of the system with fewer boundary controls on some special domains. Based on this, we further study the exact boundary synchronization (by groups) for the same system, the determination of the exactly synchronizable state (by groups), as well as the necessity of the compatibility conditions of the coupling matrices.

scholarly journals Extremal Domains and Pólya-type Inequalities for the Robin Laplacian on Rectangles and Unions of Rectangles

2019 ◽  
Author(s):  
Pedro Freitas ◽  
James B Kennedy
Keyword(s):  
Boundary Conditions ◽  
Robin Boundary Conditions ◽  
Bounded Domains ◽  
Fixed Domain ◽  
Boundary Parameter ◽  
Eigenvalues Of The Laplacian ◽  
Robin Laplacian ◽  
Robin Boundary ◽  
Original Approach

Abstract We investigate the question of whether the eigenvalues of the Laplacian with Robin boundary conditions can satisfy inequalities of the same type as those in Pólya’s conjecture for the Dirichlet and Neumann Laplacians and, if so, what form these inequalities should take. Motivated in part by Pólya’s original approach and in part by recent analogous works treating the Dirichlet and Neumann Laplacians, we consider rectangles and unions of rectangles and show that for these two families of domains, for any fixed positive value $\alpha$ of the boundary parameter, Pólya-type inequalities do indeed hold, albeit with an exponent smaller than that of the corresponding Weyl asympotics for a fixed domain. We determine the optimal exponents in both cases, showing that they are different in the two situations. Our approach to proving these results includes a characterization of the corresponding extremal domains for the $k^{\textrm{}}$th eigenvalue in regions of the $(k,\alpha )$-plane, which in turn supports recent conjectures on the nature of the extrema among all bounded domains.

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