Nonuniqueness of minimizers for semilinear optimal control problems

Infinite Horizon Optimal Control Problems for a Class of Semilinear Parabolic Equations

Infinite horizon open loop optimal control problems for semilinear parabolic equations are investigated. The controls are subject to a cost-functional which promotes sparsity in time. The focus is put on deriving first order optimality conditions. This is achieved without relying on a well-defined control-to-state mapping in a neighborhood of minimizers. The technique of proof is based on the approximation of the original problem by a family of finite horizon problems. The optimality conditions allow to deduce sparsity properties of the optimal controls in time.

具有混合逐点控制-状态约束的分布和边界半线性椭圆最优控制问题解及其相应乘子的Lipschitz连续性

Lipschitz continuity of solutions and corresponding multipliers to distributed and boundary semilinear elliptic optimal control problems with mixed pointwise control-state constraints

本文研究了由半线性椭圆方程支配、具有混合逐点控制-状态约束的最优控制问题，证明了极小化元及相应乘子的存在性，并通过bootstrapping技术和分数阶Sobolev空间工具证明了最优解与Lagrange乘子的Lipschitz连续性。

含非利普希茨非线性项的半线性椭圆型偏微分方程的最优控制

Optimal Control of Semilinear Elliptic Partial Differential Equations with Non-Lipschitzian Nonlinearities

针对含非利普希茨非线性项的半线性椭圆型偏微分方程最优控制问题，证明某类方程解映射的弗雷歇可微性，建立一阶必要最优性条件（耦合偏微分方程组形式），并提出最优控制正则性等相关问题。

半线性椭圆偏微分方程中非光滑叠加算子的辨识与优化

On the Identification and Optimization of Nonsmooth Superposition Operators in Semilinear Elliptic PDEs

针对半线性椭圆PDE中非光滑叠加算子的低正则性辨识优化问题，推导了局部极小值的平稳性系统，提出梯度投影法并证明其函数空间收敛性，发现最优算子具有类激活函数特征，且通过数值实验验证了理论结果。

Critical cones for sufficient second order conditions in PDE constrained optimization

In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for d…

仿射最优控制问题的有限元误差分析

Finite element error analysis of affine optimal control problems

本文研究受半线性椭圆型PDEs约束的仿射最优控制问题数值逼近的误差估计，在较弱的局部增长条件下，通过有限元或变分离散格式显著改进了最优控制与状态的误差估计。