Qin Zhou, Binjie Li
This paper analyzes the discretization of a Neumann boundary control problem with a stochastic parabolic equation, where an additive noise occurs in the Neumann boundary condition. The convergence is established for general filtrations, and the convergence rate $ O(τ^{1/4-ε} + h^{1/2-ε}) $ is derived for the natural filtration of the Q-Wiener process.
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New time domain decomposition methods for parabolic optimal control problems II: Neumann-Neumann algorithms
针对无约束抛物型最优控制问题，提出基于时域分解的9种Neumann-Neumann算法变体，分析其收敛性与最优松弛参数，发现更优的高效求解器选择，并用数值实验验证。
New time domain decomposition methods for parabolic control problems I: Dirichlet-Neumann and Neumann-Dirichlet algorithms
针对无约束抛物型最优控制问题，提出带时域分解的新型Dirichlet-Neumann与Neumann-Dirichlet算法，分析其收敛性与最优松弛参数，发现更优的高效求解器选择并通过数值实验验证。
Non-overlapping Schwarz methods in time for parabolic optimal control problems
本文提出一种基于时间区域分解的经典Schwarz方法，用于求解无约束抛物型最优控制问题，发现基于最优系统前后向结构的不同性质，并指出仅使用Dirichlet和Neumann传输条件的变体中，部分仅为良好平滑器，部分则可成为高效求解器。
In this paper, we establish a globally quantitative estimate of unique continuation at one time point for solutions of parabolic equations with Neumann boundary conditions in bounded domains. Our proof is mainly based on Carleman commutator estimates and a global frequency function argument, which is motivated from a recent work [5]. As an application, we obtain an observability inequality from measurable sets in time for all solutions of the above equations.
We prove the existence of a unique viscosity solution to certain systems of fully nonlinear parabolic partial differential equations with interconnected obstacles in the setting of Neumann boundary conditions. The method of proof builds on the classical viscosity solution technique adapted to the setting of interconnected obstacles and construction of explicit viscosity sub- and supersolutions as bounds for Perron's method. Our motivation stems from so called optimal switching problems on bounded domains.
We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter which extend those proposed by Kohn and Serfaty (2010). These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution of the PDE as the parameter tends to zero. Moreover, our construction allows us to treat both the …
A discretization of an optimal control problem of a stochastic parabolic equation driven by multiplicative noise is analyzed. The state equation is discretized by the continuous piecewise linear element method in space and by the backward Euler scheme in time. The convergence rate $ O(τ^{1/2} + h^2) $ is rigorously derived.
We consider a heat conduction problem $S$ with mixed boundary conditions in a $n$-dimensional domain $Ω$ with regular boundary and a family of problems $S_α$ with also mixed boundary conditions in $Ω$, where $α>0$ is the heat transfer coefficient on the portion of the boundary $Γ_{1}$. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux $q$ which is definite on the complementary portion $Γ_{2}$ of the boundary of $Ω$. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint …
We establish the null controllability for linear stochastic fourth order parabolic equations. Utilizing the duality argument, the null controllability is reduced to the observability for backward fourth order stochastic parabolic equations, and the desired observability estimate is obtained by a new global Carleman estimate. Our Carleman estimate is based on a new fundamental identity for a stochastic fourth order parabolic operator.
In this paper, we establish two Carleman estimates for a stochastic degenerate parabolic equation. The first one is for the backward stochastic degenerate parabolic equation with singular weight function. Combining this Carleman estimate and an approximate argument, we prove the null controllability of the forward stochastic degenerate parabolic equation with the gradient term. The second one is for the forward stochastic degenerate parabolic equation with regular weighted function, based on which we obtain the Lipschitz stability for an inverse problem of determining a random source depending…
Null controllability for stochastic semi-discrete parabolic equations
本文研究随机半离散抛物型方程的零能控性，通过建立依赖离散化参数的显式可观测性估计和新的Carleman估计，证明了一类随机半离散抛物型方程的零能控性。
Carleman Estimate for Stochastic Parabolic Equations and Inverse Stochastic Parabolic Problems
本文建立随机抛物方程的全局Carleman估计，并基于此解决两个反问题：通过最终时刻观测确定随机热过程历史（条件稳定性）和侧边界观测反源问题（唯一性）。
Null controllability for stochastic coupled systems of fourth order parabolic equations
本文研究两个倒向四阶随机抛物方程耦合系统的零能控问题，提出新的全局Carleman估计推导伴随系统能观性不等式，实现了单漂移项控制量对双方程的零能控。
Null Controllability for Backward Stochastic Parabolic Convection-Diffusion Equations with Dynamic Boundary Conditions
本文研究带动态边界条件和对流项的线性倒向随机抛物型方程零能控性，通过对偶论证结合新的全局Carleman估计证明能观性不等式，得到零能控性并给出控制成本估计。
Carleman estimate for semi-discrete stochastic parabolic operators in arbitrary dimension and applications to controllability
该文研究任意维半离散随机抛物算子，证明空间半离散的零能控随机抛物系统不具零能控性，同时通过推导半离散倒向随机伴随系统的新Carleman估计得到松弛能观不等式，将其用于导数φ-零能控性分析。
This paper is devoted to a study of the unique continuation property for stochastic parabolic equations. Due to the adapted nature of solutions in the stochastic situation, classical approaches to treat the the unique continuation problem for deterministic equations do not work. Our method is based on a suitable partial Holmgren coordinate transform and a stochastic version of Carleman-type estimate.
In this paper, we present a numerical scheme to solve the initial-boundary value problem for backward stochastic partial differential equations of parabolic type. Based on the Galerkin method, we approximate the original equation by a family of backward stochastic differential equations (BSDEs, for short), and then solve these BSDEs by the time discretization. Combining the truncation with respect to the spatial variable and the backward Euler method on time variable, we obtain the global $L^2$ error estimate.
Optimal error estimates of the stochastic parabolic optimal control problem with integral state constraint
针对带加性噪声与积分状态约束的随机抛物型最优控制问题，本文结合时间隐式有限元离散推导控制、状态、伴随状态和乘子的最优先验误差估计，并提出具有收敛速率保证的梯度投影算法。
In this paper, we study the Hölder-type interpolation inequality and observability inequality from measurable sets in time for parabolic equations either with L^p unbounded potentials or with electric potentials. The parabolic equations under consideration evolve in bounded C^{1,1} domains of R^N (N\geq3) with homogeneous Neumann boundary conditions. The approach for the interpolation inequality is based on a modified reduction method and some stability estimates for the corresponding elliptic operator.
Null controllability for stochastic fourth order semi-discrete parabolic equations
本文针对空间有限差分离散、时间连续的随机四阶半离散抛物型方程，通过建立依赖网格尺寸的全局Carleman估计和松弛可观测性估计，并利用对偶技巧证明了其$ϕ$-零能控性。
Null controllability for semi-discrete stochastic semilinear parabolic equations
本文研究空间有限差分离散的随机半线性抛物型方程零能控性，通过显式给出Carleman参数与离散化参数并结合Banach不动点方法证明其成立。
Null Controllability for Cascade systems of Coupled Backward Stochastic Parabolic Equations with One Distributed Control
本文证明了含反应、对流项及一般二阶抛物算子的n个耦合后向随机抛物方程级联系统的零能控性，通过施加单一分布控制，开发了新的全局Carleman估计推导能观性不等式，并给出零控制成本估计。
Null Controllability for Stochastic Parabolic Equations with convection terms
本文通过在漂移和扩散部分引入有界对流项，利用对偶技巧建立新颖的 Carleman 估计，研究了正向与反向线性随机抛物型方程的零能控性，解决了先前未解的问题并给出了更精确的零控制成本估计。
In this paper, we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy-Neumann problems. First, we will obtain embedding results for weighted Sobolev spaces, that have proved decisive in reaching well-posedness for nonlinear degenerate problems. Then, we show that the above systems can be steered in $L^2$ from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear piecewise static controls. Moreover, we extend the above result relaxing the sign constraint on the initial date.
Stability Estimate for an Inverse Stochastic Parabolic Problem of Determining Unknown Time-varying Boundary
本文研究随机抛物型方程中未知时变边界的反问题，利用新的Carleman估计证明未知边界对数依赖于内部测量，从而可从任意内部区域观测跟踪边界行为。
In this paper, we consider forward stochastic nonlinear parabolic equations, with a control localized in the drift term. Under suitable assumptions, we prove the small-time global null-controllability, with a truncated nonlinearity. We also prove the statistical local null-controllability of the true system. The proof relies on a precise estimation of the cost of null-controllability of the stochastic heat equation and on an adaptation of the source term method to the stochastic setting. The main difficulty comes from the estimation of the nonlinearity in the fixed point argument due to the la…
Quantitative uniqueness estimates for stochastic parabolic equations on the whole Euclidean space
本文将随机热方程唯一延拓的定量估计从有界区域推广到全欧几里得空间，利用局部抛物型频率函数方法证明，并进一步得到时间可测集上的可观测性估计。
The Energy-Dissipation Principle provides a variational tool for the analysis of parabolic evolution problems: solutions are characterized as so-called null-minimizers of a global functional on entire trajectories. This variational technique allows for applying the general results of the calculus of variations to the underlying differential problem and has been successfully applied in a variety of deterministic cases, ranging from doubly nonlinear flows to curves of maximal slope in metric spaces. The aim of this note is to extend the Energy-Dissipation Principle to stochastic parabolic evolut…
Controllability for forward stochastic parabolic equations with dynamic boundary conditions without extra forces
本文研究动态边界条件下前向随机抛物方程的可控性，通过谱不等式和迭代策略建立可观测性，进而证明零可控性和近似可控性。
Null controllability for stochastic parabolic equations with Robin boundary conditions
本文研究带线性 Robin 边界条件且含零阶、一阶有界系数项的正反向线性随机抛物型方程，通过对偶方法为负 Sobolev 空间中的伴随方程弱解建立两个新的全局 Carleman 估计，从而证明其零能控性。
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