一类阻尼线性分数型双曲型问题的度量图最优控制

Q3 Mathematics
Pasquini Fotsing Soh
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引用次数: 0

摘要

分数阶偏微分方程的最优控制在标准域中得到了广泛的研究,但度量图中最优控制的存在性和唯一性,特别是双曲方程的最优控制,仍然很少被探索。大多数研究集中在经典阻尼(如粘性阻尼)或双曲型问题中的整数阶阻尼上,而分数阶阻尼对度量图控制和优化的影响却受到有限的关注。考虑到这些结果在现实问题中的潜在应用,如河流网络中的污染运输、交通流量控制和分支结构中的热传播,这提出了一个重要而有前途的研究空白。本文研究了一个二阶控制问题,该问题涉及一个具有Dirichlet和Neumann边界条件的阻尼线性分数型双曲方程。所考虑的分数阶导数是右卡普托分数阶导数和左黎曼-刘维尔分数阶导数的组合。首先给出了开有界实区间上的存在唯一性结果,证明了一类二次型最优控制问题解的存在性,并给出了最优性系统的刻画。然后,我们研究了具有Dirichlet和Neumann混合边界控制的度量图上的分数阶阻尼双曲问题的类似问题。本文的动机可能源于推进数学理论和控制理论的愿望,特别是在度量图表示的复杂系统的背景下。这些结果的潜在影响或应用跨越了广泛的领域,从工程和网络控制到医学成像和环境科学,在这些领域,理解和优化阻尼双曲系统是必不可少的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Optimal control on a metric graph for a damped linear fractional hyperbolic problem
The optimal control of fractional PDEs has been extensively studied in standard domains, but the existence and uniqueness of optimal controls in metric graphs, particularly for hyperbolic equations, remain less explored. Most studies focus on classical damping (e.g., viscous damping) or integer-order damping in hyperbolic problems, whereas the impact of fractional-order damping on control and optimization in metric graphs has received limited attention. Given the potential applications of these results to real-world problems such as pollution transport in river networks, traffic flow control, and heat propagation in branched structures, this presents a significant and promising research gap. This paper addresses a quadratic control problem involving a damped linear fractional hyperbolic equation subject to Dirichlet and Neumann boundary conditions. The considered fractional derivative is a composition of the right Caputo fractional derivative and the left Riemann–Liouville fractional derivative. We first give some existence and uniqueness results on an open bounded real interval, prove the existence of solutions to a quadratic optimal control problem and provide a characterization via optimality systems. We then investigate the analogous problems for a fractional Damped hyperbolic problem on a metric graph with mixed Dirichlet and Neumann boundary controls. The paper’s motivation likely arises from the desire to advance mathematical theory and control theory, especially in the context of complex systems represented by metric graphs. The potential impact or applications of these results span a wide range of fields, from engineering and network control to medical imaging and environmental science, where understanding and optimizing damped hyperbolic systems are essential.
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来源期刊
Results in Control and Optimization
Results in Control and Optimization Mathematics-Control and Optimization
CiteScore
3.00
自引率
0.00%
发文量
51
审稿时长
91 days
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