Well-posedness, regularity and exact controllability for the problem of transmission of the Schrödinger equation

Allag, I.; Rebiai, S.

doi:10.1090/S0033-569X-2013-01351-0

Authors: I. Allag and S. E. Rebiai
Journal: Quart. Appl. Math. 72 (2014), 93-108
MSC (2000): Primary 35J10, 93C20, 93C25, 93D15, 93B05, 93B07
DOI: https://doi.org/10.1090/S0033-569X-2013-01351-0
Published electronically: November 13, 2013
MathSciNet review: 3185134
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Abstract: In this paper, we shall study the system of transmission of the Schrödinger equation with Dirichlet control and colocated observation. Using the multiplier method, we show that the system is well-posed with input and ouput space $U=L^{2}(\Gamma )$ and state space $X=H^{-1}(\Omega ).$ The regularity of the system is also established, and the feedthrough operator is found to be zero. Finally, the exact controllability of the open-loop system is obtained by proving the observability inequality of the dual system.

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