Stabilization of the Schrödinger Equation

E. Machtyngier and E. Zuazua

Abstract: We study the stabilization problem for Schrödinger equation in a bounded domain in two different situations. First, the boundary stabilization problem is considered. Dissipative boundary conditions are introduced. By using multiplier techniques and constructing energy functionals well adapted to the system, the exponential decay in $H^1$ is proved. On the other hand, the internal stabilization problem is considered. When the damping term is effective on a neighborhood of the boundary, the exponential decay in $L^2$ is proved by multiplier techniques. These results extend to Schrödinger equation recent results on the stabilizability of wave and plate equations.

Keywords: Schrödinger equation; stabilization; dissipative boundary conditions; internal damping; multipliers; perturbed energy functionals.

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