受半线性椭圆偏微分方程约束的仿射最优控制问题的稳定性
Stability in affine optimal control problems constrained by semilinear elliptic partial differential equations
AI总结：
本文针对受半线性椭圆偏微分方程约束的仿射最优控制问题，通过分析相关集值映射的度量次正则性研究其稳定性，在更弱假设下覆盖广义非线性扰动，并用Tikhonov正则化误差估计验证结果适用性。
摘要：
本文研究受半线性椭圆偏微分方程约束的仿射最优控制问题的稳定性性质，研究路径为分析与一阶最优性必要条件系统相关的集值映射的所谓度量次正则性。本文首先建立了涉及函数（尤其是所谓的切换函数）可微性的初步结果，基于该思路，本文涵盖了更广义的非线性扰动，且所采用的假设弱于此前控制约束椭圆问题相关文献中使用的假设。最后，本文通过吉洪诺夫正则化的若干误差估计验证了所得结果的适用性。
英文摘要：
This paper investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping associated with the system of first order necessary optimality conditions. Preliminary results concerning the differentiability of the functions involved are established, especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed, and under weaker assumptions, than the ones previously considered in the literature on control constrained elliptic problems. Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization.
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