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A uniqueness result for a non-strictly convex problem in the calculus of variations 变分中一个非严格凸问题的唯一性结果
3区 数学 Pub Date : 2023-11-07 DOI: 10.1051/cocv/2023079
Benjamin Lledos
Abstract. We prove the uniqueness of the solution for a non-strictly convex problem in the Calculus of Variations of the form [[EQUATION]] φ (∇ v ) − λv . Here, φ is a convex function not differentiable at the origin and λ is a Lipschitz function. To prove this result we show that under fairly general assumptions, the minimizers are globally Lipschitz continuous.
摘要证明了φ(∇v)−λv形式[[EQUATION]]变分学中非严格凸问题解的唯一性。这里φ是一个在原点不可微的凸函数,λ是一个Lipschitz函数。为了证明这一结果,我们证明了在相当一般的假设下,极小值是全局Lipschitz连续的。
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引用次数: 0
Global exact controllability of the viscous and resistive MHD system in a rectangle thanks to the lateral sides and to distributed phantom forces 由于侧向和分布的幻力,粘阻MHD系统在矩形内具有全局精确的可控性
3区 数学 Pub Date : 2023-11-01 DOI: 10.1051/cocv/2023078
Jiajiang Liao
We consider the 2-D incompressible viscous and resistive magnetohydrodynamics (MHD) system in a rectangle, with controls on the lateral sides. The velocity satisfies Dirichlet boundary conditions, while the magnetic field follows perfectly conducting wall boundary conditions on the remaining, uncontrolled part of the boundary. We extend the small-time global exact null controllability result of Coron et al. in [Ann PDE 5(2):1-49, 2019] from Navier-Stokes equations to MHD equations, with a little help of distributed phantom forces, which can be chosen arbitrarily small in any given Sobolev spaces. Our analysis relies on Coron’s return method, the well-prepared dissipation method, long-time nonlinear Cauchy-Kovalevskaya estimates and Badra’s local exact controllability result.
我们考虑矩形中的二维不可压缩粘阻磁流体动力学(MHD)系统,其两侧有控制。速度满足狄利克雷边界条件,而磁场在边界的其余不受控制部分遵循完全导电壁边界条件。我们将Coron等人在[Ann PDE 5(2):1-49, 2019]中的小时全局精确零可控性结果从Navier-Stokes方程扩展到MHD方程,并借助分布式幻力,可以在任何给定的Sobolev空间中选择任意小的幻力。我们的分析依赖于Coron的回归法、精心准备的耗散法、长时间非线性Cauchy-Kovalevskaya估计和Badra的局部精确可控性结果。
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引用次数: 0
Small-time local controllability of the bilinear Schrödinger equation with a nonlinear competition 具有非线性竞争的双线性Schrödinger方程的小时局部可控性
3区 数学 Pub Date : 2023-10-31 DOI: 10.1051/cocv/2023077
Mégane Bournissou
We consider the local controllability near the ground state of a 1D Schrödinger equation with bilinear control. Specifically, we investigate whether nonlinear terms can restore local controllability when the linearized system is not controllable. In such settings, it is known that the quadratic terms induce drifts in the dynamics which prevent small-time local controllability when the controls are small in very regular spaces. In this paper, using oscillating controls, we prove that the cubic terms can entail the small-time local controllability of the system, despite the presence of such a quadratic drift. This result, which is new for PDEs, is reminiscent of Sussmann's $S(theta)$ sufficient condition of controllability for ODEs. Our proof however relies on a different general strategy involving a new concept of tangent vector, better suited to the infinite-dimensional setting.
考虑了一类双线性控制的一维Schrödinger方程在基态附近的局部可控性。具体来说,我们研究了当线性化系统不可控时,非线性项是否能恢复局部可控性。在这种情况下,众所周知,二次项在动力学中会引起漂移,当控制在非常规则的空间中很小时,这种漂移会阻止小时间局部可控性。在本文中,利用振荡控制,我们证明了三次项可以导致系统的小时局部可控性,尽管存在这样的二次漂移。这个结果对于偏微分方程来说是新的,它让人想起Sussmann关于偏微分方程可控性的$S( θ)$充分条件。然而,我们的证明依赖于一种不同的一般策略,涉及一个新的切向量概念,更适合于无限维的设置。
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引用次数: 0
Dirichlet problem for noncoercive nonlinear elliptic equations with singular drift term in unbounded domains 无界区域上具有奇异漂移项的非强制非线性椭圆方程的Dirichlet问题
3区 数学 Pub Date : 2023-10-24 DOI: 10.1051/cocv/2023076
Patrizia Di Gironimo, Sara Monsurrò, Gabriella Zecca
In this paper we study a Dirichlet problem for noncoercive nonlinear elliptic equations with first order term in an unbounded domain. We obtain Stampacchia type existence, regularity and uniqueness results, when the singular drift term is controlled through a function in a suitable functional space, strictly containing Lebesgue one. The main tools are a weak maximum principle together with some a priori estimates proved by contradiction.
研究了无界区域上非强制一阶非线性椭圆方程的Dirichlet问题。当奇异漂移项在合适的函数空间中被函数控制时,我们得到了Stampacchia型的存在性、正则性和唯一性结果。主要的工具是一个弱极大值原理和一些由矛盾证明的先验估计。
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引用次数: 0
Formation of a nontrivial finite-time stable attractor in a class of polyhedral sweeping processes with periodic input 一类具有周期输入的多面体扫描过程中非平凡有限时间稳定吸引子的形成
3区 数学 Pub Date : 2023-10-16 DOI: 10.1051/cocv/2023074
Ivan Gudoshnikov, Oleg Makarenkov, Dmitrii Rachinskii
We consider a differential inclusion known as a polyhedral sweeping process. The general sweeping process was introduced by J.-J. Moreau as a modeling framework for quasistatic deformations of elastoplastic bodies, and a polyhedral sweeping process is typically used to model stresses in a network of elastoplastic springs. Krejčí’s theorem states that a sweeping process with periodic input has a global attractor which consists of periodic solutions, and all such periodic solutions follow the same trajectory up to a parallel translation. We show that in the case of polyhedral sweeping process with periodic input the attractor has to be a convex polyhedron X of a fixed shape. We provide examples of elasto-plastic spring models leading to structurally stable situations where X is a one or two- dimensional polyhedron. In general, an attractor of a polyhedral sweeping process may be either exponentially stable or finite-time stable and the main result of the paper consists of sufficient conditions for finite-time stability of the attractor, with upper estimates for the settling time. The results have implications for the shakedown theory.
我们考虑一种称为多面体扫描过程的微分包涵体。j - j介绍了一般的清扫过程。作为弹塑性体准静态变形的建模框架和多面体扫描过程通常用于弹塑性弹簧网络中的应力建模。Krejčí的定理指出,具有周期输入的扫掠过程有一个由周期解组成的全局吸引子,并且所有这样的周期解遵循相同的轨迹直至平行平移。我们证明了在具有周期输入的多面体扫描过程中,吸引子必须是一个固定形状的凸多面体X。我们提供的例子弹塑性弹簧模型导致结构稳定的情况下,X是一个一维或二维多面体。一般来说,多面体扫描过程的吸引子可能是指数稳定的,也可能是有限时间稳定的,本文的主要结果包括吸引子有限时间稳定的充分条件和沉降时间的上估计。这些结果对敲诈理论有启示意义。
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引用次数: 0
Necessary conditions for local controllability of a particular class of systems with two scalar controls 一类具有两个标量控制的特定系统的局部可控性的必要条件
3区 数学 Pub Date : 2023-10-16 DOI: 10.1051/cocv/2023073
Laetitia Giraldi, Pierre Lissy, Clément Moreau, Jean-Baptiste Pomet
We consider affine control systems with two scalar controls, such that one control vector field vanishes at an equilibrium state. We state two necessary conditions of local controllability around this equilibrium, involving the iterated Lie brackets of the system vector fields, with controls that are either bounded, small in $mathrm{L}^{infty}$ or small in $mathrm{W}^{1,infty}$. These results are illustrated with several examples.
考虑具有两个标量控制的仿射控制系统,其中一个控制向量场在平衡状态下消失。我们陈述了围绕该平衡的局部可控性的两个必要条件,涉及系统向量场的迭代Lie括号,控制要么是有界的,在$mathrm{L}^{infty}$中小,要么在$mathrm{W}^{1,infty}$中小。用几个例子说明了这些结果。
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引用次数: 2
A discretize-then-optimize approach to PDE-constrained shape optimization pde约束形状优化的离散再优化方法
3区 数学 Pub Date : 2023-10-04 DOI: 10.1051/cocv/2023071
Roland Herzog, Estefanía Loayza-Romero
We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a smooth manifold, termed the manifold of planar triangular meshes. The latter can be endowed with a complete Riemannian metric, which allows large mesh deformations without jeopardizing mesh quality; see [14]. Nonetheless, the discrete shape optimization problem of finding optimal vertex positions does not, in general, possess a globally optimal solution. To overcome this ill-possedness, we propose to add a mesh quality penalization term to the objective function. This allows us to simultaneously render the shape optimization problem solvable, and keep track of the mesh quality. We prove the existence of globally optimal solutions for the penalized problem and establish first-order necessary optimality conditions independently of the chosen Riemannian metric. Because of the independence of the existence results of the choice of the Riemannian metric, we can numerically study the impact of different Riemannian metrics on the steepest descent method. We compare the Euclidean, elasticity, and a novel complete metric, combined with Euclidean and geodesic retractions to perform the mesh deformation.
考虑离散化的二维pde约束形状优化问题,其中形状用三角形网格表示。考虑到连通性,允许顶点位置的空间最近被确定为光滑流形,称为平面三角形网格流形。后者可以被赋予一个完整的黎曼度量,它允许大的网格变形而不损害网格质量;参见[14]。尽管如此,寻找最优顶点位置的离散形状优化问题通常不具有全局最优解。为了克服这种病态,我们建议在目标函数中加入网格质量惩罚项。这使得我们可以同时渲染可解决的形状优化问题,并跟踪网格质量。证明了惩罚问题的全局最优解的存在性,并建立了独立于所选黎曼度量的一阶必要最优性条件。由于黎曼度量选择存在结果的独立性,我们可以数值研究不同黎曼度量对最陡下降法的影响。我们比较了欧几里得、弹性和一种新的完整度量,结合欧几里得和测地线收缩来执行网格变形。
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引用次数: 2
Semi-classical observation suffices for observability: wave and Schr"odinger equations 半经典观测满足可观测性:波方程和薛定谔方程
3区 数学 Pub Date : 2023-09-26 DOI: 10.1051/cocv/2023069
Nicolas Burq, Belhassen Dehman, Jérôme Le Rousseau
For the linear wave and Schr"odinger equations we show how observability can be deduced from the observability of solutions localized in frequency with a dyadic scale.
对于线性波和Schr odinger方程,我们展示了如何从具有二进尺度的频率局域解的可观测性推导出可观测性。
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引用次数: 0
The effect of diffusions and sources on semilinear elliptic problems 扩散和源对半线性椭圆问题的影响
3区 数学 Pub Date : 2023-09-25 DOI: 10.1051/cocv/2023068
Emerson Abreu, Everaldo Medeiros, Marcos Montenegro
This paper deals with properties of non-negative solutions of the boundary value problem in the presence of diffusion a and source f in a bounded domain Ω ⊂ Rn, n ≥ 1, where a and f are non-decreasing continuous functions on [0,L0) and f is positive. Part of the results are new even if we restrict ourselves to the Gelfand type case L0 = ∞, a(t) = t and f is a convex function. We study the behavior of related extremal parameters and solutions with respect to L0 and also to a and f in the C0 topology. The work is carried out in a unified framework for 0 < L0 ≤ ∞ under some interactive conditions between a and f.
本文讨论了有界域上扩散a和源f存在时边值问题的非负解的性质Ω∧Rn, n≥1,其中a和f是[0,L0)上的非递减连续函数,f为正函数。部分结果是新的,即使我们将自己限制在Gelfand类型的情况下,L0 =∞,a(t) = t, f是一个凸函数。我们研究了C0拓扑中L0和a、f的相关极值参数及其解的行为。这项工作是在一个统一的框架下进行的。在a与f之间的某些交互条件下L0≤∞。
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引用次数: 0
Whispering gallery modes for a transmission problem 窃窃私语画廊模式的传输问题
3区 数学 Pub Date : 2023-09-19 DOI: 10.1051/cocv/2023067
Spyridon Filippas
We construct a specific family of eigenfunctions for a Laplace operator with coefficients having a jump across an interface. These eigenfunctions have an exponential concentration arbitrarily close to the interface, and therefore could be considered as whispering gallery modes. The proof is based on an appropriate Agmon estimate. We deduce as a corollary that the quantitative unique continuation result for waves propagating in singular media proved by the author in~cite{fQuantJump} is optimal.
我们为系数在界面上有跳跃的拉普拉斯算子构造了一个特定的特征函数族。这些特征函数在接近界面处具有指数浓度,因此可以被认为是低语走廊模式。证明是基于一个适当的Agmon估计。我们推论出作者在cite{fQuantJump}中所证明的波在奇异介质中传播的定量唯一延拓结果是最优的。
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Esaim-Control Optimisation and Calculus of Variations
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