In this work, we explore the application of the Virtual Element Methods for Neumann boundary Optimal Control Problems in saddle point formulation. The method is proposed for arbitrarily polynomial order of accuracy and general polygonal meshes. Our contribution includes a rigorous a priori error estimate that holds for general polynomial degree. On the numerical side, we present (i) an initial convergence test that reflects our theoretical findings, and (ii) a second test case based on a more application-oriented experiment. For the latter test, we focus on the role of VEM stabilization, conducting a detailed experimental analysis, and proposing an alternative structure-preserving strategy to circumvent issues related to the choice of the stabilization parameter.

翻译：本文探讨了虚拟元方法在鞍点公式化诺伊曼边界最优控制问题中的应用。该方法适用于任意多项式精度阶次及一般多边形网格。我们的贡献包括一套适用于一般多项式阶次的严格先验误差估计。在数值实验方面，我们展示了：(i) 符合理论预测的初始收敛性测试；(ii) 基于更具应用导向实验的第二测试案例。针对后者，我们重点研究了VEM稳定化机制的作用，开展了详细的实验分析，并提出了一种替代性的结构保持策略以规避稳定化参数选择相关的问题。