Abstract
We study in this paper a wave–Schrödinger transmission system for its stability. By analyzing carefully Green’s functions for the infinitesimal generator of the semigroup associated with the system under consideration, we obtain a useful resolvent estimate on this generator which can be applied to derive the decaying property. Our study is inspired by L. Lu & J.-M. Wang [Appl. Math. Lett., 54:7–14, 2016] whose energy decay result is improved upon in our paper. Our method, different from the one used in the previous reference, can be adapted to study stability problems for other 1-D transmission systems.
Highlights
Thanks to its wide applicability, the Schrödinger equation i∂tu + Δu + f (∇u, u, x, t) = 0, (x, t) ∈ Rn × R, where Δ = n j=1∂2 ∂ x2j is the Laplacian onRn, has been receiving extensive attention from the mathematical control community; see [2,3,4,5,6,7,8] and the references cited therein
With the aid of the idea of Green’s functions, we provide in Sect. 2 an explicit formulae for the resolvent R(iγ ; A)
4 Concluding comments and an open question By analyzing carefully Green’s functions for boundary value problems associated with ordinary differential equations (i.e., (2.1)), we prove that the infinitesimal generator of the semigroup associated with system (1.1) satisfies the resolvent estimate (1.9), thereby proving that the energy of system (1.1) decays polynomially
Summary
The systems described by the Schrödinger equation have received extensive studies for their stability in the past three decades. Machtyngier and Zuazua [4] studied the boundary and internal stabilization problem via the multiplier method (the main idea has originated from stability studies for wave equations). Zuazua [2] provided a nice survey on the recent studies on the control properties for the Schrödinger equation. This paper is devoted to the study of the stabilization of the Schrödinger equation via a damped wave equation through a common end point. System (1.1) was recently studied by Lu and Wang [1] with the intension to understand better the transmission of dissipation effect from a damped wave equation to a dampingfree Schrödinger equation where the energy can be exchanged by (1.1).
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