This paper focusses on the optimal control problems governed by fourth-order linear elliptic equations with clamped boundary conditions in the framework of the Hessian discretisation method (HDM). The HDM is an abstract framework that enables the convergence analysis of numerical methods through a quadruplet known as a Hessian discretisation (HD) and three core properties of HD. The HDM covers several numerical schemes such as the conforming finite element methods, the Adini and Morley non-conforming finite element methods (ncFEMs), method based on gradient recovery (GR) operators and the finite volume methods (FVMs). Basic error estimates and superconvergence results are established for the state, adjoint and control variables in the HDM framework. The article concludes with numerical results that illustrates the theoretical convergence rates for the GR method, Adini ncFEM and FVM.

翻译：本文着重论述在赫斯离散法(HDM)框架内具有固定边界条件的四阶线性椭圆方程式所制约的最佳控制问题。HDM是一个抽象的框架,它通过称为赫斯离散(HD)的四重曲线和HD的三种核心特性,能够对数字方法进行趋同分析。HDM包含若干数字方法,如符合的有限元素方法、Adini和Morley不兼容的有限元素方法、基于梯度回收操作员的方法和有限体积方法。为HDM框架中的状态、连接变量和控制变量确定了基本误差估计和超级趋同结果。文章最后以数字结果来说明GR方法的理论趋同率、Adini ncFEM和FVM。