Local Exact Controllability of the One-Dimensional Nonlinear Schrödinger Equation in the Case of Dirichlet Boundary Conditions

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS SIAM Journal on Control and Optimization Pub Date : 2024-09-13 DOI:10.1137/23m1556034

Alessandro Duca, Vahagn Nersesyan

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Abstract

SIAM Journal on Control and Optimization, Ahead of Print.
Abstract. We consider the one-dimensional nonlinear Schrödinger equation with bilinear control. In the present paper, we study its local exact controllability near the ground state in the case of Dirichlet boundary conditions. To establish the controllability of the linearized equation, we use a bilinear control acting through four directions: three Fourier modes and one generic direction. The Fourier modes are appropriately chosen so that they satisfy a saturation property. These modes allow one to control approximately the linearzied Schrödinger equation. Then we show that the reachable set for the linearized equation is closed. This is achieved by representing the solution operator as a sum of two linear continuous mappings: one is surjective (here the control in generic direction is used) and the other is compact. A mapping with dense and closed image is surjective, so we conclude that the linearized Schrödinger equation is exactly controllable. The local exact controllability of the nonlinear equation then follows by the inverse mapping theorem.

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迪里夏特边界条件情况下一维非线性薛定谔方程的局部精确可控性

SIAM 控制与优化期刊》，提前印刷。摘要我们考虑具有双线性控制的一维非线性薛定谔方程。在本文中，我们研究了在 Dirichlet 边界条件下，该方程在基态附近的局部精确可控性。为了建立线性化方程的可控性，我们使用了通过四个方向起作用的双线性控制：三个傅立叶模式和一个一般方向。对傅立叶模式进行了适当选择，使其满足饱和特性。这些模式可以近似控制线性薛定谔方程。然后，我们证明线性化方程的可达集是封闭的。这可以通过将解算子表示为两个线性连续映射之和来实现：一个是弹射映射（这里使用的是通用方向控制），另一个是紧凑映射。具有密集和封闭图像的映射是可射的，因此我们得出结论，线性化薛定谔方程是精确可控的。根据逆映射定理，非线性方程的局部精确可控性也就水到渠成了。

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来源期刊

SIAM Journal on Control and Optimization 数学-应用数学

CiteScore

4.00

自引率

4.50%

发文量

143

审稿时长

12 months

期刊介绍： SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition. The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.