{"title":"Numerical Method for a Controlled Sweeping Process with Nonsmooth Sweeping Set","authors":"Chadi Nour, Vera Zeidan","doi":"10.1007/s10957-024-02470-x","DOIUrl":null,"url":null,"abstract":"<p>The numerical method developed in Nour and Zeidan (IEEE Control Syst. Lett. 6:1190-1195, 2022) via the exponential penalization technique for optimal control problems involving sweeping processes with sweeping set <i>C</i> generated by <i>one</i> <i>smooth</i> function, is generalized in this paper to the case where <i>C</i> is <i>nonsmooth</i>. That is, <i>C</i> is the intersection of a <i>finite number</i> (greater than one) of sublevel sets of smooth functions. The change from one to greater than one generating smooth functions is quite challenging. Indeed, while in the latter case <i>C</i> could be reformulated as being generated by one function, however, this function is only <i>Lipschitz</i>, and hence, the method established for one generating smooth function is not applicable to this framework. Therefore, this general setting requires a new approach, which represents the novelty of this paper.</p>","PeriodicalId":50100,"journal":{"name":"Journal of Optimization Theory and Applications","volume":"40 1","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Optimization Theory and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10957-024-02470-x","RegionNum":3,"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
The numerical method developed in Nour and Zeidan (IEEE Control Syst. Lett. 6:1190-1195, 2022) via the exponential penalization technique for optimal control problems involving sweeping processes with sweeping set C generated by onesmooth function, is generalized in this paper to the case where C is nonsmooth. That is, C is the intersection of a finite number (greater than one) of sublevel sets of smooth functions. The change from one to greater than one generating smooth functions is quite challenging. Indeed, while in the latter case C could be reformulated as being generated by one function, however, this function is only Lipschitz, and hence, the method established for one generating smooth function is not applicable to this framework. Therefore, this general setting requires a new approach, which represents the novelty of this paper.
期刊介绍：
The Journal of Optimization Theory and Applications is devoted to the publication of carefully selected regular papers, invited papers, survey papers, technical notes, book notices, and forums that cover mathematical optimization techniques and their applications to science and engineering. Typical theoretical areas include linear, nonlinear, mathematical, and dynamic programming. Among the areas of application covered are mathematical economics, mathematical physics and biology, and aerospace, chemical, civil, electrical, and mechanical engineering.
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