{"title":"On the Pontryagin Maximum Principle under differential constraints of higher order","authors":"F. Cardin, C. Giannotti, A. Spiro","doi":"10.4064/ap220814-15-3","DOIUrl":"https://doi.org/10.4064/ap220814-15-3","url":null,"abstract":"Exploiting our previous results on higher order controlled Lagrangians in [Nonlinear Anal. {bf 207} (2021), 112263], we derive here an analogue of the classical first order Pontryagin Maximum Principle (PMP) for cost minimising problems subjected to higher order differential constraints $frac{d^k x^j}{dt^k} = f^jbig(t, x(t), frac{d x}{dt}(t), ldots, frac{d^{k-1} x}{dt^{k-1}}(t), u(t)big)$, $t in [0,T]$, where $u(t)$ is a control curve in a compact set $K subset mathbb R^m$. This result and its proof can be considered as a detailed illustration of one of the claims of that previous paper, namely that the results of that paper, originally established in a smooth differential geometric framework, yield directly properties holding under much weaker and more common assumptions. In addition, for further clarifying our motivations, in the last section we display a couple of quick indications on how the two-step approach of this paper (i.e., a preliminary easy-to-get differential geometric discussion followed by a refining analysis to weaken the regularity assumptions) might be fruitfully exploited also in the context of control problems governed by partial differential equations or in studies on the dynamics of controlled mechanical systems.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48418599","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}
{"title":"Regular projections and regular covers in\u0000o-minimal structures","authors":"M'hammed Oudrane","doi":"10.4064/ap211206-3-1","DOIUrl":"https://doi.org/10.4064/ap211206-3-1","url":null,"abstract":"In this paper we prove that for any definable subset $Xsubset mathbb{R}^{n}$ in a polynomially bounded o-minimal structure, with $dim(X)<n$, there is a finite set of regular projections (in the sense of Mostowski ). We give also a weak version of this theorem in any o-minimal structure, and we give a counter example in o-minimal structures that are not polynomially bounded. As an application we show that in any o-minimal structure there exist a regular cover in the sense of Parusi'nski.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48892901","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}