Optimal control of elliptic surface PDEs with pointwise bounds on the state

Optimal control of PDEs in a complex space setting; application to the Schrödinger equation

In this paper we discuss optimality conditions for abstract optimization problems over complex spaces. We then apply these results to optimal control problems with a semigroup structure. As an application we detail the case when the state equation is the Schrödinger one, with pointwise constraints on the "bilinear" control. We derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly.

带点态控制约束的点跟踪椭圆型分布最优控制问题的新误差分析

A New Error Analysis for a Point Tracking Elliptic Distributed Optimal Control Problem with Pointwise Control Constraints

本文针对带点跟踪和点态控制约束的线性二次椭圆型分布最优控制问题，在统一框架下对标准有限元离散与变分离散开展误差分析，其自包含性依托有限元分析标准技术与椭圆型偏微分方程经典结果。

带逐点控制约束的椭圆型Neumann边界控制问题有限元方法的一种新误差分析

A New Error Analysis for Finite Element Methods for Elliptic Neumann Boundary Control Problems with Pointwise Control Constraints

该研究针对带Neumann边界控制与逐点控制约束的线性二次椭圆最优控制问题，提出一种适用于光滑系数下标准有限元、粗糙系数下多尺度有限元的新误差分析方法。

Error estimates for a pointwise tracking optimal control problem of a semilinear elliptic equation

We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. We devise two strategies of discretization to approximate a solution of the optimal control problem: a semidiscrete scheme where the control variable is not discretized -- the so-called variational discretization approach -- and a fully discrete scheme where the control variable is discretized with piecewise constant functions. For both solution techniqu…

A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics

In insurance mathematics optimal control problems over an infinite time horizon arise when computing risk measures. Their solutions correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In this paper we propose a deep neural network algorithm for solving such partial differential equations in high dimensions. The algorithm is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with random terminal time.

关于半线性椭圆PDE的双线性点跟踪最优控制问题

A bilinear pointwise tracking optimal control problem for a semilinear elliptic PDE

本文研究了半线性椭圆PDE的双线性点跟踪最优控制问题，推导了最优解的存在性，并证明了局部最优控制的正则性性质。

Finite Element Methods for One Dimensional Elliptic Distributed Optimal Control Problems with Pointwise Constraints on the Derivative of the State

We investigate $C^1$ finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state formulated as fourth order variational inequalities for the state variable. For the problem with Dirichlet boundary conditions, we use an existing $H^{\frac52-ε}$ regularity result for the optimal state to derive $O(h^{\frac12-ε})$ convergence for the approximation of the optimal state in the $H^2$ norm. For the problem with mixed Dirichlet and Neumann boundary conditions, we show that the optimal state belongs to $H^3$ under ap…

Nitsche-XFEM用于具有界面的椭圆PDE最优控制问题

Nitsche-XFEM for optimal control problems governed by elliptic PDEs with interfaces

本文提出基于Hansbo的Nitsche-XFEM方法，用于求解具有界面的椭圆PDE最优控制问题，推导了状态、共轭状态和控制的最优误差估计，并验证了非均质界面条件的适用性。

带随机输入的线性椭圆型偏微分方程最优控制的可靠误差估计

Reliable Error Estimates for Optimal Control of Linear Elliptic PDEs with Random Inputs

针对带随机输入的线性椭圆型偏微分方程风险中性最优控制问题，提出蒙特卡洛采样结合有限元的离散化方法，建立了解的指数尾界，证明有限维解可可靠估计原问题解，并通过数值模拟验证理论。

分数椭圆PDEs最优控制与分数阶Sobolev空间的对偶空间刻画

Optimal Control of Fractional Elliptic PDEs with State Constraints and Characterization of the dual of Fractional Order Sobolev Spaces

本文研究了由分数阶椭圆PDEs支配的最优控制问题中的状态约束，开发了多种数学工具，如分数阶Sobolev空间的对偶空间刻画和分数PDEs的测度值数据的适定性，推导了最优控制问题的适定性并得到一阶最优性条件，强调了经典情况（s=1）已被E.Casas研究，但现有结果难以直接应用于分数阶情况。

不确定性下的最优控制与联合机会状态约束：几乎处处界、方差缩减及其在（双）线性椭圆型偏微分方程中的应用

Optimal control under uncertainty with joint chance state constraints: almost-everywhere bounds, variance reduction, and application to (bi-)linear elliptic PDEs

本文研究不确定性下带联合机会状态约束的PDE最优控制，证明几乎处处界存在唯一性，并利用球面-径向分解显著降低蒙特卡洛估计方差，数值验证于线性和双线性椭圆PDE。