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Non-Autonomous Degenerate Parabolic Equations with Robin Boundary Conditions: Carleman Estimates and Null-Controllability
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-15 DOI: 10.1007/s00245-025-10227-9
Mohammad Akil, Genni Fragnelli, Sarah Ismail
{"title":"Non-Autonomous Degenerate Parabolic Equations with Robin Boundary Conditions: Carleman Estimates and Null-Controllability","authors":"Mohammad Akil,&nbsp;Genni Fragnelli,&nbsp;Sarah Ismail","doi":"10.1007/s00245-025-10227-9","DOIUrl":"10.1007/s00245-025-10227-9","url":null,"abstract":"<div><p>The Earth’s climate system naturally adjusts to maintain a balance between the energy received from the Sun and the energy reflected back into space, a concept known as the “Earth’s radiation budget”. However, this balance has been disrupted by human activities, leading to global warming. Starting from the energy balance model proposed by Budyko and Sellers, and considering a time-dependent diffusion coefficient, we prove the null-controllability of non-autonomous degenerate parabolic problems, in the sense that the Earth achieves a desired temperature, by finding new Carleman estimates for the non-homogeneous adjoint problems. At the degeneracy point, we impose Robin boundary condition which is appropriate for modeling heat transfer at the Earth’s surface. Moreover, we provide the equivalence between null-controllability and observability inequality for the non-autonomous case. At the end, we present some extensions and open problems.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10227-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143423369","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
Global Weak Solutions in a Haptotactic Cross-Diffusion System Modeling Oncolytic Virotherapy with Nonlinear Diffusion
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-14 DOI: 10.1007/s00245-025-10232-y
Yue Zhou, Changchun Liu
{"title":"Global Weak Solutions in a Haptotactic Cross-Diffusion System Modeling Oncolytic Virotherapy with Nonlinear Diffusion","authors":"Yue Zhou,&nbsp;Changchun Liu","doi":"10.1007/s00245-025-10232-y","DOIUrl":"10.1007/s00245-025-10232-y","url":null,"abstract":"<div><p>This paper discusses an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy </p><div><div><span>$$begin{aligned} {left{ begin{array}{ll} u_t=Delta u^m-nabla cdot (unabla v) +mu u(1-u)-uz,;;&amp; ;xin Omega ,~t&gt;0, v_t=-(u+w)v,;;&amp; ;xin Omega ,~t&gt;0, w_t=Delta w-nabla cdot (wnabla v)-w+uz,;;&amp; ;xin Omega ,~t&gt;0, z_t=DDelta z-z-uz+beta w,;;&amp; ;xin Omega ,~t&gt;0, frac{partial u^m}{partial nu }-ufrac{partial v}{partial nu }=frac{partial w}{partial nu }-wfrac{partial v}{partial nu }=frac{partial z}{partial nu }=0,;;&amp; ;xin partial Omega ,~t&gt;0, u(x,0)=u_{0},~v(x,0)=v_{0},~w(x,0)=w_{0}, z(x,0)=z_{0},;;&amp; ;xin Omega , end{array}right. } end{aligned}$$</span></div></div><p>in a smooth bounded domain <span>( Omega subset {mathbb {R}}^{N}(N=1,2) )</span> with <span>( m&gt;1, beta&gt;0, mu &gt;0 )</span>, and <span>( D&gt;0)</span>. We prove that for any large initial datum, the problem admits a global ‘very’ weak solution for any <span>(m&gt;1)</span>.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143404145","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
Convergence of the Stochastic Navier–Stokes-(alpha ) Solutions Toward the Stochastic Navier–Stokes Solutions 随机纳维-斯托克斯（Navier-Stokes）-（alpha ）解向随机纳维-斯托克斯解的收敛性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-13 DOI: 10.1007/s00245-025-10228-8
Jad Doghman, Ludovic Goudenège
{"title":"Convergence of the Stochastic Navier–Stokes-(alpha ) Solutions Toward the Stochastic Navier–Stokes Solutions","authors":"Jad Doghman,&nbsp;Ludovic Goudenège","doi":"10.1007/s00245-025-10228-8","DOIUrl":"10.1007/s00245-025-10228-8","url":null,"abstract":"<div><p>Loosely speaking, the Navier–Stokes-<span>(alpha )</span> model and the Navier–Stokes equations differ by a spatial filtration parametrized by a scale denoted <span>(alpha )</span>. Starting from a strong two-dimensional solution to the Navier–Stokes-<span>(alpha )</span> model driven by a multiplicative noise, we demonstrate that it generates a strong solution to the stochastic Navier–Stokes equations under the condition <span>(alpha rightarrow 0)</span>. The initially introduced probability space and the Wiener process are maintained throughout the investigation, thanks to a local monotonicity property that abolishes the use of Skorokhod’s theorem. High spatial regularity a priori estimates for the fluid velocity vector field are carried out within periodic boundary conditions.\u0000</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10228-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396661","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
Before and After Default: Information and Optimal Portfolio via Anticipating Calculus 违约前后：通过预期微积分获得信息和最佳投资组合
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-12 DOI: 10.1007/s00245-024-10210-w
Bernardo D’Auria, Giulia Di Nunno, Jose A. Salmeron
{"title":"Before and After Default: Information and Optimal Portfolio via Anticipating Calculus","authors":"Bernardo D’Auria,&nbsp;Giulia Di Nunno,&nbsp;Jose A. Salmeron","doi":"10.1007/s00245-024-10210-w","DOIUrl":"10.1007/s00245-024-10210-w","url":null,"abstract":"<div><p>Default risk calculus plays a crucial role in portfolio optimization when the risky asset is under threat of bankruptcy. However, traditional stochastic control techniques are not applicable in this scenario, and additional assumptions are required to obtain the optimal solution in a before-and-after default context. We propose an alternative approach using forward integration, which allows to avoid one of the restrictive assumptions, the <i>Jacod density hypothesis</i>. We demonstrate that, in the case of logarithmic utility, the weaker <i>intensity hypothesis</i> is the appropriate condition for optimality. Furthermore, we establish the semimartingale decomposition of the risky asset in the filtration that is progressively enlarged to accommodate the default process, under the assumption of the existence of the optimal portfolio. This work aims to provide valuable insights for developing effective risk management strategies when facing default risk.\u0000</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-024-10210-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396678","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
Null Controllability for Stochastic Parabolic Equations Coupled by First and Zero Order Terms
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-12 DOI: 10.1007/s00245-025-10231-z
Mahmoud Baroun, Said Boulite, Abdellatif Elgrou, Lahcen Maniar
{"title":"Null Controllability for Stochastic Parabolic Equations Coupled by First and Zero Order Terms","authors":"Mahmoud Baroun,&nbsp;Said Boulite,&nbsp;Abdellatif Elgrou,&nbsp;Lahcen Maniar","doi":"10.1007/s00245-025-10231-z","DOIUrl":"10.1007/s00245-025-10231-z","url":null,"abstract":"<div><p>We prove the null controllability of forward and backward linear stochastic parabolic equations with first and zero order coupling terms, and with bounded coefficients. The null controllability results rely on novel Carleman estimates for the backward and forward adjoint equations, established through the application of the duality technique. In doing so, we resolve an open question (see Remark 2.2 in [Tang and Zhang in SIAM J Control Optim 48:2191–2216, 2009]). Moreover, we provide more accurate estimates for the null-control costs.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396679","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
Global Well-posedness of the Nonhomogeneous Initial Boundary Value Problem for the Hirota Equation Posed in a Finite Domain 在有限域内求解广田方程的非均质初始边界值问题的全局良好求解性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-12 DOI: 10.1007/s00245-025-10226-w
Mengtao Xu, Chunxiao Guo, Boling Guo, Xin-guang Yang
{"title":"Global Well-posedness of the Nonhomogeneous Initial Boundary Value Problem for the Hirota Equation Posed in a Finite Domain","authors":"Mengtao Xu,&nbsp;Chunxiao Guo,&nbsp;Boling Guo,&nbsp;Xin-guang Yang","doi":"10.1007/s00245-025-10226-w","DOIUrl":"10.1007/s00245-025-10226-w","url":null,"abstract":"<div><p>We study a system described by a type of initial and boundary value problem of the Hirota equation with nonhomogeneous boundary conditions posed on a bounded interval. Firstly, we prove the local well-posedness of the system in the space <span>(H^s(0,1))</span> by using an explicit solution formula and contraction mapping principle for any <span>(sge 1)</span>. Secondly, we obtain the global well-posedness in <span>(H^1(0,1))</span> and <span>(H^2(0,1))</span> by the norm estimation. Especially, the main difficulty is that the characteristic equation corresponding to Hirota equation needs to be solved by construction and that nonlinear terms are taken into consideration. In addition, the norm estimate for global well-posedness of solution in <span>(H^1(0,1))</span> and <span>(H^2(0,1))</span> are complicated.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396680","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
Linear-Quadratic Mean-Field Backward Stackelberg Game with Mixed Terminal Perturbation
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-08 DOI: 10.1007/s00245-025-10225-x
Tian Chen, Xinwei Feng, Yunxiao Jia
{"title":"Linear-Quadratic Mean-Field Backward Stackelberg Game with Mixed Terminal Perturbation","authors":"Tian Chen,&nbsp;Xinwei Feng,&nbsp;Yunxiao Jia","doi":"10.1007/s00245-025-10225-x","DOIUrl":"10.1007/s00245-025-10225-x","url":null,"abstract":"<div><p>This paper investigates a linear-quadratic (LQ) Stackelberg game for mean-field type backward stochastic system, in which the cost functional is also mean-field type. In our model, the leader first announces the terminal goal satisfying pointwise and affine constraints and open-loop dynamic decisions at the initial time which takes into account the best response of the follower. Then two interrelated optimization problems are sequentially solved by the follower (a backward LQ problem) and the leader (a backward-forward LQ problem). The open-loop Stackelberg equilibrium is obtained by virtue of duality theory and represented by some fully coupled mean-field backward-forward stochastic differential equations with mixed initial-terminal conditions, whose global solvability is discussed in some nontrivial cases by Riccati decoupling method and discounting method. As an application, we address a product pricing problem.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143361670","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
The Sticky Particle Dynamics of the 1D Pressureless Euler-Alignment System as a Gradient Flow
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-08 DOI: 10.1007/s00245-025-10223-z
Sondre Tesdal Galtung
{"title":"The Sticky Particle Dynamics of the 1D Pressureless Euler-Alignment System as a Gradient Flow","authors":"Sondre Tesdal Galtung","doi":"10.1007/s00245-025-10223-z","DOIUrl":"10.1007/s00245-025-10223-z","url":null,"abstract":"<div><p>We show how the sticky dynamics for the one-dimensional pressureless Euler-alignment system can be obtained as an <span>(L^2)</span>-gradient flow of a convex functional. This is analogous to the Lagrangian evolution introduced by Natile and Savaré for the pressureless Euler system, and by Brenier et al. for the corresponding system with a self-interacting force field. Our Lagrangian evolution can be seen as the limit of sticky particle Cucker–Smale dynamics, similar to the solutions obtained by Leslie and Tan from a corresponding scalar balance law, and provides us with a uniquely determined distributional solution of the original system in the space of probability measures with quadratic moments and corresponding square-integrable velocities. Moreover, we show that the gradient flow also provides an entropy solution to the balance law of Leslie and Tan, and how their results on cluster formation follow naturally from (non-)monotonicity properties of the so-called natural velocity of the flow.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10223-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143361671","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
Nonlinear Monte Carlo Methods with Polynomial Runtime for Bellman Equations of Discrete Time High-Dimensional Stochastic Optimal Control Problems
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-04 DOI: 10.1007/s00245-024-10213-7
Christian Beck, Arnulf Jentzen, Konrad Kleinberg, Thomas Kruse
{"title":"Nonlinear Monte Carlo Methods with Polynomial Runtime for Bellman Equations of Discrete Time High-Dimensional Stochastic Optimal Control Problems","authors":"Christian Beck,&nbsp;Arnulf Jentzen,&nbsp;Konrad Kleinberg,&nbsp;Thomas Kruse","doi":"10.1007/s00245-024-10213-7","DOIUrl":"10.1007/s00245-024-10213-7","url":null,"abstract":"<div><p>Discrete time <i>stochastic optimal control</i> problems and <i>Markov decision processes</i> (MDPs), respectively, serve as fundamental models for problems that involve sequential decision making under uncertainty and as such constitute the theoretical foundation of <i>reinforcement learning</i>. In this article we study the numerical approximation of MDPs with infinite time horizon, finite control set, and general state spaces. Our set-up in particular covers infinite-horizon optimal stopping problems of discrete time Markov processes. A key tool to solve MDPs are <i>Bellman equations</i> which characterize the value functions of the MDPs and determine the optimal control strategies. By combining ideas from the <i>full-history recursive multilevel Picard approximation method</i>, which was recently introduced to solve certain nonlinear partial differential equations, and ideas from <i>Q</i><i>-learning</i> we introduce a class of suitable <i>nonlinear Monte Carlo methods</i> and prove that the proposed methods do not suffer from the <i>curse of dimensionality</i> in the numerical approximation of the solutions of Bellman equations and the associated discrete time stochastic optimal control problems.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-024-10213-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108327","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
Feedback Stabilization of Convective Brinkman-Forchheimer Extended Darcy Equations
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-01-31 DOI: 10.1007/s00245-024-10217-3
Sagar Gautam, Kush Kinra, Manil T. Mohan
{"title":"Feedback Stabilization of Convective Brinkman-Forchheimer Extended Darcy Equations","authors":"Sagar Gautam,&nbsp;Kush Kinra,&nbsp;Manil T. Mohan","doi":"10.1007/s00245-024-10217-3","DOIUrl":"10.1007/s00245-024-10217-3","url":null,"abstract":"<div><p>In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a <i>d</i>-dimensional torus: </p><div><div><span>$$begin{aligned} frac{partial {varvec{y}}}{partial t}-mu Delta {varvec{y}}+({varvec{y}}cdot nabla ){varvec{y}}+alpha {varvec{y}}+beta vert {varvec{y}}vert ^{r-1}{varvec{y}}+gamma vert {varvec{y}}vert ^{q-1}{varvec{y}}+nabla p={varvec{g}}+{varvec{u}}, nabla cdot {varvec{y}}=0, end{aligned}$$</span></div></div><p>where <span>(din {2,3})</span>, <span>(mu ,alpha ,beta &gt;0)</span>, <span>(gamma in {mathbb {R}})</span>, <span>(r,qin [1,infty ))</span> with <span>(r&gt;qge 1)</span>. We prove the exponential stabilization of CBFeD system by finite- and infinite-dimensional feedback controllers. The solvability of the controlled problem is achieved by using the abstract theory of <i>m</i>-accretive operators and density arguments. As an application of the above solvability result, by using infinite-dimensional feedback controllers, we demonstrate exponential stability results such that the solution preserves an invariance condition for a given closed and convex set. By utilizing the unique continuation property of controllability for finite-dimensional systems, we construct a finite-dimensional feedback controller which exponentially stabilizes CBFeD system locally, where the control is localized in a smaller subdomain. Furthermore, we establish the local exponential stability of CBFeD system via proportional controllers.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143110089","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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