| Abstract: |
| We study boundary optimal control problems governed by the biharmonic equation, motivated by applications in thin plate deformation and structural optimization. The presence of boundary controls introduces analytical and numerical challenges due to the involvement of fractional Sobolev trace spaces. In particular, the control variable is defined on the boundary and naturally belongs to fractional spaces, which complicates both the functional setting and the numerical approximation. Additionally, the fourth-order structure of the biharmonic operator requires higher regularity of the state variable, posing a challenge on classical conforming discretizations. To address these difficulties, we employ a lifting strategy that decomposes the state variable into a homogeneous component and a lifting function encoding the boundary control. This reformulation transforms the original boundary control problem into a distributed one, thereby avoiding the explicit treatment of fractional spaces. The resulting system is discretized using a Hermann--Miyoshi mixed finite element method. This approach reduces the fourth-order equation to a system of lower-order equations and allows the use of standard finite element spaces. We analyze the well-posedness of the continuous optimal control problem and derive a first-order necessary condition. Numerical experiments are presented to demonstrate the effectiveness of the formulation |
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