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Exact boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary controls on a general bounded domain. (English) Zbl 1475.93059

Summary: In this paper, by means of a suitable projection of the system, we introduce the concept of a presynchronized system and propose a new method based on Fredholm’s theory of compact operators to characterize the space of exactly synchronizable state by groups. First, under the minimum rank condition of a Kalman matrix, we show the necessity of the conditions of compatibility for the coupling matrices involved in the system on a general bounded domain. Thus the results in [T. Li et al., in: Contemporary computational mathematics – a celebration of the 80th birthday of Ian Sloan. In 2 volumes. Cham: Springer. 837–868 (2018; Zbl 1405.93039)] on the necessity of the conditions of compatibility for the coupling matrices can be established from a special parallelepiped domain to a general smooth bounded domain. Next, we establish the equivalence between the minimality of Kalman’s rank condition and the independence of the presynchronized system with respect to applied boundary controls. This method has a general character and can be applied to other problems.

MSC:

93C20 Control/observation systems governed by partial differential equations
35L53 Initial-boundary value problems for second-order hyperbolic systems

Citations:

Zbl 1405.93039
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References:

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