The present paper investigates systems exhibiting two attractors, and we discuss the problem of steering the state from one attractor to the other attractor by our idea of associating with the stabilization problem an optimal control problem. We first formulate the steering problem and give partial answers for the problem in a two-dimensional case by using the ordinary differential equation based on the infinite horizon optimal control model with negative discounts. Furthermore, under some conditions, we verify that the phase space can be separated into some openly connected components depending on the asymptotic behavior of the orbit starting from initial points in their components. This classification of initial points suggests that the system enables robust stabilizable control. Moreover, we illustrate some numerical results for the control obtained by applying our focused system for the Bonhoeffer–Van der Pol model.
本文研究了具有两个吸引子的系统，并将镇定问题与最优控制问题联系起来，讨论了状态从一个吸引子转向另一个吸引子的问题。首先利用基于负折扣无限视界最优控制模型的常微分方程，给出了二维情况下的转向问题的部分解。此外，在某些条件下，我们验证了相位空间可以被分割成一些开放连接的分量，这取决于轨道从它们的分量中的初始点开始的渐近行为。这种初始点的分类表明，该系统能够实现鲁棒稳定控制。此外，我们还举例说明了将我们的聚焦系统应用于Bonhoeffer-Van der Pol模型所得到的一些数值结果。
{"title":"Accessibility and stabilization by infinite horizon optimal control with negative discounting","authors":"Fumihiko NAKAMURA","doi":"10.14492/hokmj/2021-551","DOIUrl":"https://doi.org/10.14492/hokmj/2021-551","url":null,"abstract":"The present paper investigates systems exhibiting two attractors, and we discuss the problem of steering the state from one attractor to the other attractor by our idea of associating with the stabilization problem an optimal control problem. We first formulate the steering problem and give partial answers for the problem in a two-dimensional case by using the ordinary differential equation based on the infinite horizon optimal control model with negative discounts. Furthermore, under some conditions, we verify that the phase space can be separated into some openly connected components depending on the asymptotic behavior of the orbit starting from initial points in their components. This classification of initial points suggests that the system enables robust stabilizable control. Moreover, we illustrate some numerical results for the control obtained by applying our focused system for the Bonhoeffer–Van der Pol model.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136204781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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