维格纳方程的最优控制问题

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Applied Mathematics Pub Date : 2024-03-08 DOI:10.1137/22m1515033

Omar Morandi, Nella Rotundo, Alfio Borzì, Luigi Barletti

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引用次数: 0

摘要

SIAM 应用数学杂志》第 84 卷第 2 期第 387-411 页，2024 年 4 月。摘要。维格纳准密度函数允许对统计量子力学进行相空间表述，这在理论研究和应用中具有根本性的重要意义。这项研究通过对描述准密度函数时间演化的维格纳方程的最优控制问题的表述和分析，为这些任务做出了贡献。为此，考虑了两种可能的控制机制，并相应地对受控非均质 Wigner 方程的加权 Sobolev 空间进行了详细分析。此外，还报告了有关最佳控制的存在性、控制到状态图的可微分性以及集合成本函数的可微分性的进一步理论结果，从而推导出了表征所寻求的最佳控制的最优性系统。

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An Optimal Control Problem for the Wigner Equation

SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 387-411, April 2024.
Abstract. The Wigner quasi-density function allows a phase-space formulation of statistical quantum mechanics that is of fundamental importance in theoretical investigation and in applications. This work contributes to these tasks with the formulation and analysis of an optimal control problem for the Wigner equation, which describes the time evolution of the quasi-density function. For this purpose, two possible control mechanisms are considered, and, correspondingly, a detailed analysis in weighted Sobolev spaces for the controlled nonhomogeneous Wigner equation is presented. Further theoretical results are reported concerning existence of optimal controls and differentiability of the control-to-state map and of the ensemble cost functional, which allows the derivation of the optimality system that characterizes the optimal controls sought.

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来源期刊

SIAM Journal on Applied Mathematics 数学-应用数学

CiteScore

3.60

自引率

0.00%

发文量

审稿时长

12 months

期刊介绍： SIAM Journal on Applied Mathematics (SIAP) is an interdisciplinary journal containing research articles that treat scientific problems using methods that are of mathematical interest. Appropriate subject areas include the physical, engineering, financial, and life sciences. Examples are problems in fluid mechanics, including reaction-diffusion problems, sedimentation, combustion, and transport theory; solid mechanics; elasticity; electromagnetic theory and optics; materials science; mathematical biology, including population dynamics, biomechanics, and physiology; linear and nonlinear wave propagation, including scattering theory and wave propagation in random media; inverse problems; nonlinear dynamics; and stochastic processes, including queueing theory. Mathematical techniques of interest include asymptotic methods, bifurcation theory, dynamical systems theory, complex network theory, computational methods, and probabilistic and statistical methods.