  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.
