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Stability in distribution and stabilization ofswitching jump diffusions 开关跳扩散分布的稳定性和稳定性
IF 1.4 3区 数学 Pub Date : 2022-09-28 DOI: 10.1051/cocv/2022062
K. Tran, D. Nguyen, G. Yin
This paper aims to study stability in distribution of Markovian switching jump diffusions. The main motivation stems from stability andstabilizinghybrid systems in which there is no trivial solution.An explicit criterion for stability in distribution is derived. The stabilizing effects of Markov chains, Brownian motions, and Poisson jumps are revealed. Based on these criteria, stabilization problems of stochastic differential equations with Markovian switching and Poisson jumps are developed.
本文旨在研究马尔可夫切换跳变扩散分布的稳定性。主要动机源于稳定性和稳定混合系统,其中没有简单的解决方案。导出了分布稳定性的一个显式判据。揭示了马尔可夫链、布朗运动和泊松跳跃的稳定效应。基于这些准则,研究了具有马尔可夫切换和泊松跳变的随机微分方程的镇定问题。
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引用次数: 0
Optimal control of hyperbolic type discrete and differential inclusions described by the Laplace 用拉普拉斯描述的双曲型离散和微分内含物的最优控制
IF 1.4 3区 数学 Pub Date : 2022-09-20 DOI: 10.1051/cocv/2022061
E. Mahmudov
The paper is devoted to the optimization of a first mixed initial-boundary value problem for hyperbolic differential inclusions (DFIs) with Laplace operator. For this, an auxiliary problem with a hyperbolic discrete inclusion is defined and, using locally conjugate mappings, necessary and sufficient optimality conditions for hyperbolic discrete inclusions are proved. Then, using the method of discretization of hyperbolic DFIs and the already obtained optimality conditions for discrete inclusions, the optimality conditions for the discrete approximate problem are formulated in the form of the Euler-Lagrange type inclusion. Thus, using specially proved equivalence theorems, which are the only tool for constructing Euler-Lagrangian inclusions, we establish sufficient optimality conditions for hyperbolic DFIs. Further, the way of extending the obtained results to the multidimensional case is indicated. To demonstrate the above approach, some linear problems and polyhedral optimization with hyperbolic DFIs are investigated.
利用拉普拉斯算子研究了一类双曲型微分包含的第一混合初边值问题的优化问题。为此,定义了一个具有双曲型离散包含的辅助问题,并利用局部共轭映射证明了双曲型离散包含的充分最优性必要条件。然后,利用双曲DFIs的离散化方法,结合已经得到的离散包含的最优性条件,将离散近似问题的最优性条件表述为欧拉-拉格朗日型包含的形式。因此,利用特殊证明的等价定理(这是构造欧拉-拉格朗日包含的唯一工具),我们建立了双曲型DFIs的充分最优性条件。此外,还指出了将所得结果推广到多维情况的方法。为了证明上述方法,研究了一些具有双曲DFIs的线性问题和多面体优化问题。
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引用次数: 3
Purely time-dependent optimal control of quasilinear parabolic PDEs with sparsity enforcing penalization 具有稀疏性惩罚的拟线性抛物型偏微分方程的纯时变最优控制
IF 1.4 3区 数学 Pub Date : 2022-09-14 DOI: 10.1051/cocv/2022058
Fabian Hoppe, Ira Neitzel
We prove first- and second-order optimality conditions for sparse, purely time-dependent optimal control problems governed by a quasilinear parabolic PDE. In particular, we analyze sparsity patterns of the optimal controls induced by different sparsity enforcing functionals in the purely timedependent control case and illustrate them by numerical examples. Our findings are based on results obtained by abstraction of well known techniques from the literature.
我们证明了拟线性抛物型PDE控制下的稀疏纯时变最优控制问题的一阶和二阶最优性条件。特别地,我们分析了在纯时变控制情况下由不同的稀疏增强函数引起的最优控制的稀疏模式,并通过数值例子加以说明。我们的发现是基于从文献中抽象出的著名技术得到的结果。
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引用次数: 1
Controllability of a Stokes system with a diffusive boundary condition 具有扩散边界条件的Stokes系统的可控性
IF 1.4 3区 数学 Pub Date : 2022-09-12 DOI: 10.1051/cocv/2022057
Takéo Takahashi, R'emi Buffe
We are interested by the controllability of a fluid-structure interaction system where the fluid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the fluid's domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable.The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estimates corresponds to the case of Ventcel boundary conditions.This work can be seen as a first step to handle the real system where the structure is modeled by the wave or the plate equation.
我们感兴趣的是流固相互作用系统的可控性,其中流体是粘性和不可压缩的,而结构是弹性的,位于流体域的一部分边界上。在本文中,我们通过考虑线性化和用热方程代替结构的波/板方程来简化该系统。我们证明了Stokes方程与热方程在其边界处耦合的对应系统是零可控的。证明是基于Carleman估计和插值不等式。其中一个Carleman估计对应于Ventcel边界条件的情况。这项工作可以看作是处理实际系统的第一步,其中结构是由波或板方程模拟的。
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引用次数: 2
Stabilization of the wave equation through nonlinear Dirichlet actuation 非线性狄利克雷驱动下波动方程的镇定
IF 1.4 3区 数学 Pub Date : 2022-08-29 DOI: 10.1051/cocv/2022077
Nicolas Vanspranghe, Francesco Ferrante, C. Prieur
In this paper, we consider the problem of nonlinear (in particular, saturated) stabilization of the high-dimensional wave equation with Dirichlet boundary conditions. The wave dynamics are subject to a dissipative nonlinear velocity feedback and generate a strongly continuous semigroup of contractions on the optimal energy space L 2 (Ω) × H −1 (Ω). It is first proved that any solution to the closed-loop equations converges to zero in the aforementioned topology. Secondly, under the condition that the feedback nonlinearity has linear growth around zero, polynomial energy decay rates are established for solutions with smooth initial data. This constitutes new Dirichlet counterparts to well-known results pertaining to nonlinear stabilization in H 1 (Ω) × L 2 (Ω) of the wave equation with Neumann boundary conditions.
本文研究了具有Dirichlet边界条件的高维波动方程的非线性(特别是饱和)镇定问题。波浪动力学受耗散非线性速度反馈影响,在最优能量空间l2 (Ω) × H−1 (Ω)上产生强连续收缩半群。首先证明了在上述拓扑结构下闭环方程的任意解收敛于零。其次,在反馈非线性在零附近线性增长的条件下,建立了初始数据光滑解的多项式能量衰减率。这构成了与具有诺伊曼边界条件的波动方程h1 (Ω) × l2 (Ω)的非线性稳定有关的著名结果的新的狄利克雷对立物。
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引用次数: 0
A General Maximum Principle for Progressive Optimal Stochastic Control Problems with Markov Regime-Switching 马尔可夫状态切换渐进最优随机控制问题的一般极大值原理
IF 1.4 3区 数学 Pub Date : 2022-08-25 DOI: 10.1051/cocv/2022054
Yuanzhuo Song, Zhanghua Wu
In this paper, we give a general maximum principle for optimal controls of stochastic systems driven by Markov chains. The control is allowed to enter both diffusion and jump terms and the control domain is not necessarily convex. We apply a new spike variation and the stochastic integral of progressive processes to obtain the main result.
本文给出了马尔可夫链驱动的随机系统最优控制的一般极大值原理。控制允许同时进入扩散项和跳跃项,并且控制域不一定是凸的。我们应用了一个新的尖峰变化和渐进过程的随机积分来得到主要结果。
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引用次数: 2
Qualitative analysis of solutions to mixed-order positive linear coupled systems with bounded or unbounded delays 具有有界或无界时滞的混合阶正线性耦合系统解的定性分析
IF 1.4 3区 数学 Pub Date : 2022-08-23 DOI: 10.1051/cocv/2023057
Tuan Hoang The, Thinh Van La
This paper addresses the qualitative theory of mixed-order delay positive linear coupled systems. First, we introduce a general result on the existence and uniqueness of solutions to mixed-order delay linear systems. Next, we obtain necessary and sufficient criteria which characterize the positivity of mixed-order delay linear coupled systems. Our main contributions are in Section 5. More precisely, by using a smoothness property of solutions to fractional differential equations and developing a new appropriated comparison principle for solutions to mixed-order delay positive systems, we prove the attractivity of mixed-order non-homogeneous linear positive coupled systems under the impact of bounded or unbounded delays. We also establish a necessary and sufficient condition to ensure the stability of homogeneous systems. As a consequence of these results, we show the smallest asymptotic bound of solutions to mixed-order delay non-homogeneous linear positive coupled systems where disturbances are continuous and bounded. Finally, we provide numerical simulations to illustrate the proposed theoretical results.
本文研究了混合阶时滞正线性耦合系统的定性理论。首先,我们给出了混合阶延迟线性系统解的存在唯一性的一般结果。其次,我们得到了表征混合阶延迟线性耦合系统正性的充分必要判据。我们的主要贡献在第5部分。更准确地说,我们利用分数阶微分方程解的光滑性,建立了一种新的适合于混合阶时滞正系统解的比较原理,证明了混合阶非齐次线性正耦合系统在有界或无界时滞作用下的吸引性。建立了齐次系统稳定性的充分必要条件。作为这些结果的结果,我们给出了扰动连续有界的混合阶延迟非齐次线性正耦合系统解的最小渐近界。最后,我们提供了数值模拟来说明所提出的理论结果。
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引用次数: 0
On the reconstruction of cavities in a nonlinear model arising from cardiac electrophysiology 由心脏电生理引起的非线性模型中腔体的重建
IF 1.4 3区 数学 Pub Date : 2022-08-08 DOI: 10.1051/cocv/2023026
E. Beretta, M. Cerutti, D. Pierotti, L. Ratti
In this paper we deal with the problem of determining perfectly insulating regions (cavities) from one boundary measurement in a nonlinear elliptic equation arising from cardiac electrophysiology. Based on the results obtained in [9] we propose a new reconstruction algorithm based on Γ-convergence. The relevance and applicability of this approach is then shown through several numerical experiments.
本文讨论了由心脏电生理引起的非线性椭圆方程的一次边界测量确定完全绝缘区域(空腔)的问题。基于文献[9]的结果,我们提出了一种新的基于Γ-convergence的重构算法。然后通过几个数值实验证明了该方法的相关性和适用性。
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引用次数: 0
Singular perturbations in stochastic optimal control with unbounded data 无界数据随机最优控制中的奇异扰动
IF 1.4 3区 数学 Pub Date : 2022-08-01 DOI: 10.1051/cocv/2023020
M. Bardi, Hicham Kouhkouh
Abstract. We study singular perturbations of a class of two-scale stochastic control systems with unbounded data. The assumptions are designed to cover some relaxation problems for deep neural networks. We construct effective Hamiltonian and initial data and prove the convergence of the value function to the solution of a limit (effective) Cauchy problem for a parabolic equation of HJB type. We use methods of probability, viscosity solutions and homogenization.
摘要研究了一类具有无界数据的双尺度随机控制系统的奇异摄动。这些假设旨在涵盖深度神经网络的一些松弛问题。构造了HJB型抛物方程的有效哈密顿量和初始数据,并证明了值函数对极限(有效)Cauchy问题解的收敛性。我们使用概率法、粘度解法和均质法。
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引用次数: 3
Phase field topology optimisation for 4D printing 面向4D打印的相场拓扑优化
IF 1.4 3区 数学 Pub Date : 2022-07-08 DOI: 10.1051/cocv/2023012
H. Garcke, K. F. Lam, Robert Nurnberg, A. Signori
This work concerns a structural topology optimisation problem for 4D printing based on the phase field approach. The concept of 4D printing as a targeted evolution of 3D printed structures can be realised in a two-step process. One first fabricates a 3D object with multi-material active composites and apply external loads in the programming stage. Then, a change in an environmental stimulus and the removal of loads cause the object deform in the programmed stage. The dynamic transition between the original and deformed shapes is achieved with appropriate applications of the stimulus. The mathematical interest is to find an optimal distribution for the materials such that the 3D printed object achieves a targeted configuration in the programmed stage as best as possible.Casting the problem as a PDE-constrained minimisation problem, we consider a vector-valued order parameter representing the volume fractions of the different materials in the composite as a control variable. We prove the existence of optimal designs and formulate first order necessary conditions for minimisers. Moreover, by suitable asymptotic techniques, we relate our approach to a sharp interface description. Finally, the theoretical results are validated by several numerical simulations both in two and three space dimensions.
这项工作涉及基于相场方法的4D打印结构拓扑优化问题。4D打印的概念作为3D打印结构的有针对性的演变可以在两个步骤的过程中实现。首先用多材料活性复合材料制造三维物体,并在编程阶段施加外部载荷。然后,环境刺激的变化和负载的去除导致物体在程序阶段变形。通过适当的刺激,实现了原始形状和变形形状之间的动态转换。数学上的兴趣是找到材料的最佳分布,使3D打印对象在编程阶段尽可能达到目标配置。将该问题转换为pde约束最小化问题,我们将表示复合材料中不同材料的体积分数的向量值顺序参数作为控制变量。我们证明了最优设计的存在性,并给出了最小化的一阶必要条件。此外,通过适当的渐近技术,我们将我们的方法与尖锐的界面描述联系起来。最后,通过二维和三维的数值模拟验证了理论结果。
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引用次数: 3
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Esaim-Control Optimisation and Calculus of Variations
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