In this paper, we consider a bilinear optimal control with final observation for the heat equation. We first give the existence and uniqueness results of weak solutions under an integral constraints on the control and the state. We prove the existence and uniqueness of the solution to a quadratic boundary optimal control problem by means of compactness Sobolev space embedding. Then we provide a characterization of the optimal control via the Euler-Lagrange first-order optimality conditions. The unique optimal control is characterized in terms of the solution of the optimality system, which consists of the state system coupled with an adjoint system.

中文翻译：

---

双线性边界优化控制与热方程的最终观察

在本文中，我们考虑对热方程进行最终观测的双线性最优控制。我们首先给出了在控制和状态的积分约束下弱解的存在唯一性结果。我们通过紧致Sobolev空间嵌入证明了二次边界最优控制问题解的存在性和唯一性。然后我们通过 Euler-Lagrange 一阶最优性条件提供最优控制的特征。独特的最优控制的特征在于最优系统的解，它由状态系统和伴随系统组成。