{"title":"The discretized backstepping method: An application to a general system of $ 2times 2 $ linear balance laws","authors":"Mathias Dus","doi":"10.3934/mcrf.2022006","DOIUrl":"https://doi.org/10.3934/mcrf.2022006","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we introduce the numerical backstepping method by applying it to a problem of finite-time stabilization for a system of <inline-formula><tex-math id=\"M2\">begin{document}$ 2 times 2 $end{document}</tex-math></inline-formula> balance laws discretized thanks to the upwind scheme. On the one hand, we illustrate on an example that the scheme used to compute the feedback control cannot be chosen arbitrarily. On the other hand, an algorithm is given to construct this control properly and an approached finite-time stabilization result is proven.</p>","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"129 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123810965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A differential game control problem with state constraints","authors":"Nidhal Gammoudi, H. Zidani","doi":"10.3934/mcrf.2022008","DOIUrl":"https://doi.org/10.3934/mcrf.2022008","url":null,"abstract":"We study the Hamilton-Jacobi (HJ) approach for a two-person zero-sum differential game with state constraints and where controls of the two players are coupled within the dynamics, the state constraints and the cost functions. It is known for such problems that the value function may be discontinuous and its characterization by means of an HJ equation requires some controllability assumptions involving the dynamics and the set of state constraints. In this work, we characterize this value function through an auxiliary differential game free of state constraints. Furthermore, we establish a link between the optimal strategies of the constrained problem and those of the auxiliary problem and we present a general approach allowing to construct approximated optimal feedbacks to the constrained differential game for both players. Finally, an aircraft landing problem in the presence of wind disturbances is given as an illustrative numerical example.","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"68 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132861430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A general maximum principle for partially observed mean-field stochastic system with random jumps in progressive structure","authors":"Tian Chen, Zhen Wu","doi":"10.3934/mcrf.2022012","DOIUrl":"https://doi.org/10.3934/mcrf.2022012","url":null,"abstract":"We study the progressive optimal control for partially observed stochastic system of mean-field type with random jumps. The cost function and the observation are also of mean-field type. The control is allowed to enter the diffusion, jump coefficient and the observation. The control domain need not be convex. We obtain the maximum principle for the partially observable progressive optimal control by a special spike variation. The maximum principle in the progressive structure is different from the classical case.","PeriodicalId":418020,"journal":{"name":"Mathematical Control & Related Fields","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132231703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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