{"title":"Optimal Control for Suppression of Singularity in Chemotaxis via Flow Advection","authors":"Weiwei Hu, Ming-Jun Lai, Jinsil Lee","doi":"10.1007/s00245-024-10122-9","DOIUrl":null,"url":null,"abstract":"<div><p>This work focuses on the optimal control design for suppressing the singularity formation in chemotaxis governed by the parabolic-elliptic Patlak–Keller–Segel (PKS) system via flow advection. The main idea of this work lies in utilizing flow advection for enhancing diffusion as to control the nonlinear behavior of the system. The objective is to determine an optimal strategy for adjusting flow strength so that the possible finite time blow-up of the solution can be suppressed. Rigorous proof of the existence of an optimal solution and derivation of first-order optimality conditions for solving such a solution are presented. Spline collocation methods are employed for solving the optimality conditions. Numerical experiments based on 2D cellular flows in a rectangular domain are conducted to demonstrate our ideas and designs.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"89 3","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Optimization","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00245-024-10122-9","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This work focuses on the optimal control design for suppressing the singularity formation in chemotaxis governed by the parabolic-elliptic Patlak–Keller–Segel (PKS) system via flow advection. The main idea of this work lies in utilizing flow advection for enhancing diffusion as to control the nonlinear behavior of the system. The objective is to determine an optimal strategy for adjusting flow strength so that the possible finite time blow-up of the solution can be suppressed. Rigorous proof of the existence of an optimal solution and derivation of first-order optimality conditions for solving such a solution are presented. Spline collocation methods are employed for solving the optimality conditions. Numerical experiments based on 2D cellular flows in a rectangular domain are conducted to demonstrate our ideas and designs.
期刊介绍：
The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.
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