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.
利用我们在[非线性分析.｛bf 207｝（2021），112263]中关于高阶受控拉格朗日方程的先前结果，我们在这里推导了在高阶微分约束$frac｛d^k x ^j｝｛dt^k｝=f^jbig（t，x（t）， frac｛d x｝｛dt｝（t），$tin[0，t]$，其中$u（t）$是紧集$Ksubetmathbb R^m$中的控制曲线。这一结果及其证明可以被视为前一篇论文的一项权利要求的详细说明，即该论文的结果最初建立在光滑的微分几何框架中，直接产生在更弱、更常见的假设下成立的性质。此外，为了进一步阐明我们的动机，在最后一节中，我们展示了本文的两步方法（即，初步的易于获得的微分几何讨论，然后进行精炼分析，以削弱正则性假设）如何在偏微分方程控制问题或受控动力学研究中得到有效利用机械系统。
{"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":null,"pages":null},"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":null,"pages":null},"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}
We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form Φ(z1, z2) = (φ(z1, z2), z2). If φ has degree 1 in the first variable, the dynamics on each horizontal fiber can be described in terms of Möbius transformations but the global dynamics on the 2-torus exhibit some complexity, encoded in terms of certain T-symmetric polynomials. We describe the dynamical behavior of such mappings Φ and give criteria for different configurations of fixed point curves and rotation belts in terms of zeros of a related one-variable polynomial.
{"title":"Dynamics of low-degree rational inner skew-products on $mathbb{T}^2$","authors":"A. Sola, R. Tully-Doyle","doi":"10.4064/ap211108-28-2","DOIUrl":"https://doi.org/10.4064/ap211108-28-2","url":null,"abstract":"We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form Φ(z1, z2) = (φ(z1, z2), z2). If φ has degree 1 in the first variable, the dynamics on each horizontal fiber can be described in terms of Möbius transformations but the global dynamics on the 2-torus exhibit some complexity, encoded in terms of certain T-symmetric polynomials. We describe the dynamical behavior of such mappings Φ and give criteria for different configurations of fixed point curves and rotation belts in terms of zeros of a related one-variable polynomial.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49617796","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}
. Let G : R → R be a continuous function. Under some assumptions on G , s, α, p and q we prove that { : f ∈ p,q implies G is a linear function. Here A sp,q ( R n , | · | α ) stands for either the Besov space B s p,q ( R n , | · | α ) or the Triebel-Lizorkin space F s p,q ( R n , | · | α ). These spaces unify and generalize many classical function spaces such as Sobolev spaces of power weights. One of the main difficulties to study this problem is that the norm of the A s p,q ( R n , |·| α ) spaces with α 6 = 0 is not translation invariant, so some new techniques must be developed.
{"title":"On the composition operators on Besov and Triebel–Lizorkin spaces with power weights","authors":"D. Drihem","doi":"10.4064/ap220314-23-9","DOIUrl":"https://doi.org/10.4064/ap220314-23-9","url":null,"abstract":". Let G : R → R be a continuous function. Under some assumptions on G , s, α, p and q we prove that { : f ∈ p,q implies G is a linear function. Here A sp,q ( R n , | · | α ) stands for either the Besov space B s p,q ( R n , | · | α ) or the Triebel-Lizorkin space F s p,q ( R n , | · | α ). These spaces unify and generalize many classical function spaces such as Sobolev spaces of power weights. One of the main difficulties to study this problem is that the norm of the A s p,q ( R n , |·| α ) spaces with α 6 = 0 is not translation invariant, so some new techniques must be developed.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44480292","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":"Hardy’s inequalities, Hilbert inequalities and fractional integrals on function spaces of $q$-integral $p$-variation","authors":"K. Ho","doi":"10.4064/AP200224-11-1","DOIUrl":"https://doi.org/10.4064/AP200224-11-1","url":null,"abstract":"","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45683154","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}
Abstract In this paper, the authors consider an initial boundary value problem for the heat flow of equation of surfaces with constant mean curvatures, which was investigated in [On the heat flow of equation of surfaces of constant mean curvatures, Manuscripta Mathematica, 2011, 134: 259-271] by Huang et al., where global well-posedness and finite time blow-up of regular solutions were obtained. Their results are complemented in this paper in the sense that some new conditions on the initial data are provided for the solutions to develop finite time singularity.
{"title":"Finite time blow-up for the heat flow of\u0000$H$-surface with constant mean curvature","authors":"Haixia Li","doi":"10.4064/ap210403-26-7","DOIUrl":"https://doi.org/10.4064/ap210403-26-7","url":null,"abstract":"Abstract In this paper, the authors consider an initial boundary value problem for the heat flow of equation of surfaces with constant mean curvatures, which was investigated in [On the heat flow of equation of surfaces of constant mean curvatures, Manuscripta Mathematica, 2011, 134: 259-271] by Huang et al., where global well-posedness and finite time blow-up of regular solutions were obtained. Their results are complemented in this paper in the sense that some new conditions on the initial data are provided for the solutions to develop finite time singularity.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47042204","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}
Given a negative plurisubharmonic function φ in a bounded pseudoconvex domain of C, we introduce and study its residual function gφ determined by the asymptotic behavior of φ near its singularity points, both inside the domain and on its boundary. For certain choices of φ, the function gφ coincides with different versions of pluricomplex Green functions. The considerations are motivated by a problem on when two given plurisubharmonic functions can be connected by a plurisubharmonic geodesic. Mathematic Subject Classification: 32U05, 32U15, 32U35, 32W20
{"title":"Rooftop envelopes and residual plurisubharmonic functions","authors":"A. Rashkovskii","doi":"10.4064/ap210624-12-11","DOIUrl":"https://doi.org/10.4064/ap210624-12-11","url":null,"abstract":"Given a negative plurisubharmonic function φ in a bounded pseudoconvex domain of C, we introduce and study its residual function gφ determined by the asymptotic behavior of φ near its singularity points, both inside the domain and on its boundary. For certain choices of φ, the function gφ coincides with different versions of pluricomplex Green functions. The considerations are motivated by a problem on when two given plurisubharmonic functions can be connected by a plurisubharmonic geodesic. Mathematic Subject Classification: 32U05, 32U15, 32U35, 32W20","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41302357","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}
S. Brzostowski, Tadeusz Krasi'nski, Grzegorz Oleksik
We prove the constancy of the Lojasiewicz exponent in non-degenerate μ-constant deformations of surface singularities. This is a positive answer to a question posed by B. Teissier.
{"title":"The Łojasiewicz exponent in non-degenerate deformations of surface singularities","authors":"S. Brzostowski, Tadeusz Krasi'nski, Grzegorz Oleksik","doi":"10.4064/ap210209-1-6","DOIUrl":"https://doi.org/10.4064/ap210209-1-6","url":null,"abstract":"We prove the constancy of the Lojasiewicz exponent in non-degenerate μ-constant deformations of surface singularities. This is a positive answer to a question posed by B. Teissier.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42099747","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}
We show several weighted regularity criteria for the axisymmetric solutions to the three-dimensional magnetohydrodynamic equations, involving u, u; u, b, ∂ru , ∂rb ; u, b, ∂zu, ∂zb; u, b, ∂ru, ∂rb; or ω, where u, u, u are the angular, swirl and axial components of the velocity respectively and ω denotes the swirl component of the vorticity.
{"title":"Remarks on the regularity criteria for the axisymmetric MHD system","authors":"Zujin Zhang, Yali Zhang","doi":"10.4064/ap200221-15-9","DOIUrl":"https://doi.org/10.4064/ap200221-15-9","url":null,"abstract":"We show several weighted regularity criteria for the axisymmetric solutions to the three-dimensional magnetohydrodynamic equations, involving u, u; u, b, ∂ru , ∂rb ; u, b, ∂zu, ∂zb; u, b, ∂ru, ∂rb; or ω, where u, u, u are the angular, swirl and axial components of the velocity respectively and ω denotes the swirl component of the vorticity.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70581093","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":"Characterization of hypercyclic weighted\u0000composition operators on the space of holomorphic functions","authors":"M. Golinski, A. Przestacki","doi":"10.4064/ap210215-8-9","DOIUrl":"https://doi.org/10.4064/ap210215-8-9","url":null,"abstract":"","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70582771","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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