On the exact controllability of hyperbolic magnetic Schrödinger equations
In this paper, we address the exact controllability problem for the hyperbolic magnetic Schr√∂dinger equation, which plays an important role in the research of electromagnetics. Typical techniques, such as Hamiltonian induced Hilbert spaces and pseudodifferential operators are introduced. By choosing an appropriate multiplier, we proved the observability inequality with sharp constants. In particular, a genuine compactness-uniqueness argument is applied to obtain the minimal time. In the final analysis, a suitable boundary control is constructed by the systematic Hilbert Uniqueness Method introduced by J. L. Lions. Compared with the micro-local discussion in Bardos et al. (1992), we do not require the coefficients belong to C‚àû. Actually, C1 is already sufficient for the vector potential of the hyperbolic electromagnetic equation.