Exact Boundary Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls

The approximately synchronizable state for a kind of coupled system of wave equations

“…Li and Rao then gave a necessary condition of Kalman type for the approximate boundary synchronization. 3-6 Furthermore, Li, Lu, and Rao extended the above results to a kind of coupled system of wave equations with Neumann boundary controls in Li et al, 7 Lu 8 and Li et al, 12 where similar results are obtained; however, because of the lack of regularity of the solution, the consideration of corresponding problems becomes more complicated, and the synchronizable space has to be changed.…”

“…Li and Rao then gave a necessary condition of Kalman type for the approximate boundary synchronization. 3-6 Furthermore, Li, Lu, and Rao extended the above results to a kind of coupled system of wave equations with Neumann boundary controls in Li et al, 7 Lu 8 and Li et al, 12 where similar results are obtained; however, because of the lack of regularity of the solution, the consideration of corresponding problems becomes more complicated, and the synchronizable space has to be changed.For a kind of coupled system of wave equations, the discussion on the exactly synchronizable state is almost complete in the study of the exact boundary synchronization, but for the approximate boundary synchronization, the resolution of corresponding problems is still not entirely clear and needs to be explained in details. In this paper, we will mainly study the determination of approximately synchronizable state for the approximate boundary synchronization by groups under Dirichlet boundary controls.Math Meth Appl Sci.wileyonlinelibrary.com/journal/mma…”

“…Under the condition of C p -compatibility (12), assume that the coupled system (1) is approximately synchronizable by p-groups in the pinning sense, we will determine the corresponding approximately synchronizable state by p-groups.…”

“…Under the condition of C p -compatibility (12), taking the inner product with C p on both sides of (1) and (2) and setting W = C p U, we get the following reduced problem from the original problems (1) and (2):…”

“…Noticing (13), we have Lemma 3. 3 Under the condition of C p -compatibility (12), the approximate boundary synchronization by p-groups for system (1) in the consensus sense is equivalent to the approximate boundary null controllability for the reduced system (17).…”