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Observer-Based Exponential Stability of the Fractional Heat Equation 基于观测器的分数阶热方程的指数稳定性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-05-10 DOI: 10.1007/s00245-025-10266-2
Hugo Parada
{"title":"Observer-Based Exponential Stability of the Fractional Heat Equation","authors":"Hugo Parada","doi":"10.1007/s00245-025-10266-2","DOIUrl":"10.1007/s00245-025-10266-2","url":null,"abstract":"<div><p>In this work, the exponential stability of the nonlocal fractional heat equation is studied. The fractional Laplacian is defined via a singular integral. Using the spectral properties of the fractional Laplacian and a state decomposition, a feedback control is constructed by considering the first <i>N</i> modes and an observer defined via a bounded operator. Different configurations are examined, including interior controller with interior observation, and interior controller with exterior observation. Using the recent result about the simplicity of the eigenvalues (Fall et al. in Calc Var Partial Differ Equ 62(8):233, 2023), some of our stabilization results are valid for <span>(sin (0,1))</span>, in particular for <span>(sin (0,1/2))</span> in which case the fractional heat equation is not null controllable.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10266-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143932322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quantitative Convergence of Quadratically Regularized Linear Programs 二次正则化线性规划的定量收敛性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-05-10 DOI: 10.1007/s00245-025-10267-1
Alberto González-Sanz, Marcel Nutz
{"title":"Quantitative Convergence of Quadratically Regularized Linear Programs","authors":"Alberto González-Sanz,&nbsp;Marcel Nutz","doi":"10.1007/s00245-025-10267-1","DOIUrl":"10.1007/s00245-025-10267-1","url":null,"abstract":"<div><p>Linear programs with quadratic (“ridge”) regularization are of recent interest in optimal transport: unlike entropic regularization, the squared-norm penalty gives rise to sparse approximations of optimal transport couplings. More broadly, quadratic regularization is used in overparametrized learning problems to single out a particular solution. It is well known that the solution of a quadratically regularized linear program over any polytope converges stationarily to the minimal-norm solution of the linear program when the regularization parameter tends to zero. However, that result is merely qualitative. Our main result quantifies the convergence by specifying the exact threshold for the regularization parameter, after which the regularized solution also solves the linear program. Moreover, we bound the suboptimality of the regularized solution before the threshold. These results are complemented by a convergence rate for the regime of large regularization. We apply our general results to the setting of optimal transport, where we shed light on how the threshold and suboptimality depend on the number of data points.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143932323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Finite Element Approximation of Lyapunov Equations Related to Parabolic Stochastic PDEs 抛物型随机偏微分方程Lyapunov方程的有限元逼近
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-05-03 DOI: 10.1007/s00245-025-10260-8
Adam Andersson, Annika Lang, Andreas Petersson, Leander Schroer
{"title":"Finite Element Approximation of Lyapunov Equations Related to Parabolic Stochastic PDEs","authors":"Adam Andersson,&nbsp;Annika Lang,&nbsp;Andreas Petersson,&nbsp;Leander Schroer","doi":"10.1007/s00245-025-10260-8","DOIUrl":"10.1007/s00245-025-10260-8","url":null,"abstract":"<div><p>A numerical analysis for the fully discrete approximation of an operator Lyapunov equation related to linear stochastic partial differential equations (SPDEs) driven by multiplicative noise is considered. The discretization of the Lyapunov equation in space is given by finite elements and in time by a semiimplicit Euler scheme. The main result is the derivation of the rate of convergence in operator norm. Moreover, it is shown that the solution of the equation provides a representation of a quadratic and path dependent functional of the SPDE solution. This fact yields a deterministic numerical method to compute such functionals. As a secondary result, weak error rates are established for a fully discrete finite element approximation of the SPDE with respect to this functional. This is obtained as a consequence of the approximation analysis of the Lyapunov equation. It is the first weak convergence analysis for fully discrete finite element approximations of SPDEs driven by multiplicative noise that obtains double the strong rate of convergence, especially for path dependent functionals and smooth spatial noise. Numerical experiments illustrate the results empirically, and it is demonstrated that the deterministic method has advantages over Monte Carlo sampling in a stability context.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10260-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143900786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The (alpha )-Dependence of the Invariant Measure for the Stochastic Navier–Stokes Equation Driven by (alpha )-Stable Lévy Processes (alpha ) -稳定l<s:1>过程驱动的随机Navier-Stokes方程的(alpha ) -依赖性不变测度
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-30 DOI: 10.1007/s00245-025-10259-1
Ting Li, Xianming Liu
{"title":"The (alpha )-Dependence of the Invariant Measure for the Stochastic Navier–Stokes Equation Driven by (alpha )-Stable Lévy Processes","authors":"Ting Li,&nbsp;Xianming Liu","doi":"10.1007/s00245-025-10259-1","DOIUrl":"10.1007/s00245-025-10259-1","url":null,"abstract":"<div><p>Our study focuses on the convergence behavior of invariant measures associated with Navier–Stokes equations forced by cylindrical <span>(alpha )</span>-stable noises, which could be either non-degenerate or degenerate. Despite the absence of uniqueness in the invariant measures, we successfully demonstrate the convergence of invariant measures from both non-degenerate and degenerate noise cases to their corresponding Navier–Stokes equations driven by Brownian motions as <span>(alpha )</span> tends to 2, under the Wasserstein metric. Especially, for the non-degenerate case, the convergence of invariant measures is established in the sense of the Wasserstein-1 metric.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10259-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143892671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Weak Closed-Loop Solvability of Linear Quadratic Stochastic Optimal Control Problems with Partial Information 部分信息线性二次型随机最优控制问题的弱闭环可解性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-27 DOI: 10.1007/s00245-025-10262-6
Xun Li, Guangchen Wang, Jie Xiong, Heng Zhang
{"title":"Weak Closed-Loop Solvability of Linear Quadratic Stochastic Optimal Control Problems with Partial Information","authors":"Xun Li,&nbsp;Guangchen Wang,&nbsp;Jie Xiong,&nbsp;Heng Zhang","doi":"10.1007/s00245-025-10262-6","DOIUrl":"10.1007/s00245-025-10262-6","url":null,"abstract":"<div><p>This paper investigates a linear quadratic stochastic optimal control (LQSOC) problem with partial information. Firstly, by introducing two Riccati equations and a backward stochastic differential equation (BSDE), we solve this LQSOC problem under standard positive semidefinite assumptions. Secondly, by means of a perturbation approach, we study open-loop solvability of this problem when the weighting matrices in the cost functional are indefinite. Thirdly, we investigate weak closed-loop solvability of this problem and prove the equivalence between open-loop and weak closed-loop solvabilities. Finally, we give an example to illustrate the way for obtaining a weak closed-loop optimal strategy.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143877687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Non Linear Optimal Control Problem Related to a Road De-icing Device: Analysis and Numerical Experiments 道路除冰装置的非线性最优控制问题：分析与数值实验
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-27 DOI: 10.1007/s00245-025-10261-7
Frédéric Bernardin, Jérôme Lemoine, Arnaud Münch
{"title":"A Non Linear Optimal Control Problem Related to a Road De-icing Device: Analysis and Numerical Experiments","authors":"Frédéric Bernardin,&nbsp;Jérôme Lemoine,&nbsp;Arnaud Münch","doi":"10.1007/s00245-025-10261-7","DOIUrl":"10.1007/s00245-025-10261-7","url":null,"abstract":"<div><p>In order to design a road de-icing device by heating, we consider in a two dimensional setting the optimal control of an advection–diffusion equation with a nonlinear boundary condition of the Stefan-Boltzmann type. The problem models the heating of a road during a winter period to keep positive its surface temperature above a given threshold. The heating device is performed through the circulation of a coolant in a porous layer of the road. We prove the well-posedeness of the nonlinear optimal control problem, subject to unilateral constraints on the control and the state, set up a gradient based algorithm then discuss some numerical results associated with real data obtained from experimental measurements. The study, initially developed in a one dimensional simpler setting in [1], aims to quantify the minimal energy to be provided to keep the road surface without frost or snow.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143879520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Well-Posedness of Infinite Horizon FBSDEs with Non-zero Terminals and LQ Problems with Random Coefficients 非零端点无限视界FBSDEs的适定性与随机系数LQ问题
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-27 DOI: 10.1007/s00245-025-10263-5
Jinghua Li, Zhiyong Yu
{"title":"Well-Posedness of Infinite Horizon FBSDEs with Non-zero Terminals and LQ Problems with Random Coefficients","authors":"Jinghua Li,&nbsp;Zhiyong Yu","doi":"10.1007/s00245-025-10263-5","DOIUrl":"10.1007/s00245-025-10263-5","url":null,"abstract":"<div><p>This paper is concerned with fully coupled forward–backward stochastic differential equations (FBSDEs, for short) with non-zero terminals in infinite horizon. By introducing stochastic Lipschitz conditions and constructing infinite horizon domination–monotonicity conditions, the well-posedness of this kind of infinite horizon FBSDEs including the existence, uniqueness and a pair of estimates is proved. Moreover, the theoretical results are applied to solve four kinds of linear-quadratic (LQ, for short) stochastic optimal control problems with random coefficients in infinite horizon. Due to the unboundedness and randomness of coefficients, the results of the FBSDEs and LQ problems obtained in this paper, even if they are degenerated to finite horizon, contain more situations than the results in the literature.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143879519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lipschitz Multivalued Perturbations of Integro-differential Prox-Regular Sweeping Processes 积分-微分拟正则扫描过程的Lipschitz多值摄动
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-19 DOI: 10.1007/s00245-025-10258-2
Tahar Haddad, Sarra Gaouir, Lionel Thibault
{"title":"Lipschitz Multivalued Perturbations of Integro-differential Prox-Regular Sweeping Processes","authors":"Tahar Haddad,&nbsp;Sarra Gaouir,&nbsp;Lionel Thibault","doi":"10.1007/s00245-025-10258-2","DOIUrl":"10.1007/s00245-025-10258-2","url":null,"abstract":"<div><p>Integro-differential sweeping processes with prox-regular sets in Hilbert spaces have been the subject of various recent studies. Diverse applications of such differential inclusions to complementarity problems, electrical circuits, frictionless contact, can be found in the literature. Here we provide a general theorem of existence of solution for such processes perturbed by a Lipschitz multimapping with nonconvex values.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143850899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Determination of Singular Control in the Optimal Management of Natural Resources 自然资源最优管理中奇异控制的确定
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-12 DOI: 10.1007/s00245-025-10257-3
Chris Guiver, Mark R. Opmeer
{"title":"Determination of Singular Control in the Optimal Management of Natural Resources","authors":"Chris Guiver,&nbsp;Mark R. Opmeer","doi":"10.1007/s00245-025-10257-3","DOIUrl":"10.1007/s00245-025-10257-3","url":null,"abstract":"<div><p>A method is presented to simplify the determination of solutions of certain optimal control problems which commonly arise in natural resource management and bioeconomic contexts. The method, termed the resource-value balance method, essentially leverages an equivalent formulation of the original optimal control problem and, as described, in certain cases the method obviates the need for classical tools from optimal control theory, such as the Pontryagin Principle. Indeed, in these cases the method reduces the original problem to one solvable with elementary calculus techniques. Further, the solution provided by the resource-value balance method is shown to equal the singular solution of an associated (and more commonly considered) input-constrained optimal control problem, providing insight into the nature of singular control in this context. The theory is illustrated with examples from bioeconomics.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10257-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143821770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Arched Beams of Bresse Type: Thermoelastic Modeling and Stability Analysis Bresse型拱梁：热弹性建模与稳定性分析
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-04-12 DOI: 10.1007/s00245-025-10255-5
G. E. Bittencourt Moraes, M. A. Jorge Silva
{"title":"Arched Beams of Bresse Type: Thermoelastic Modeling and Stability Analysis","authors":"G. E. Bittencourt Moraes,&nbsp;M. A. Jorge Silva","doi":"10.1007/s00245-025-10255-5","DOIUrl":"10.1007/s00245-025-10255-5","url":null,"abstract":"<div><p>This is the third and final work in a series dedicated to thermoelastic arched beams of Bresse type under Fourier’s law. Herein, our first main goal is to provide a detailed modeling of the thermoelastic Bresse–Fourier systems, addressing thermal couplings and their effects on axial, shear, and bending forces. Then, the stability results are rigorously analyzed, by proving that stability patterns remain consistent under different boundary conditions and thermal couplings. Theoretical contributions include semi-uniform algebraic and uniform exponential decay rates, achieved using semigroup theory. This paper concludes the trilogy by unifying the stability analysis for all remaining systems with new thermal couplings.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143821774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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