Optimal control and approximation for elliptic bilateral obstacle problems
Optimal control and approximation for elliptic bilateral obstacle problems
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椭圆双边障碍问题的最优控制与逼近
DOI：
10.1016/j.cnsns.2021.105938
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发表时间：
2021
影响因子：
3.9
通讯作者：
Zeng Shengda
中科院分区：
文献类型：
--
作者：
Liu Jinjie;Yang Xinmin;Zeng Shengda
研究了一类椭圆型双边障碍系统（EBOS），该系统包含一个非线性非齐次偏微分算子和一个由Clarke广义梯度描述的多值项.首先，我们得到了变分-半变分不等式（EBOS）的弱形式，并证明了双边障碍问题的唯一可解性。其次，我们考虑了由（EBOS）控制的非线性最优控制问题，其中控制变量为双边障碍物，并在温和的条件下建立了障碍物控制问题最优解的存在性。然后，采用正则化技术和惩罚方法，我们引入了一个家庭的近似问题所考虑的最优控制问题。最后给出了逼近问题的任意解序列收敛于原最优控制问题的最优解的收敛性结果。
The aim of this paper is to study an elliptic bilateral obstacle system (EBOS, for short) involving a nonlinear and nonhomogeneous partial differential operator and a multivalued term which is described by Clarke’s generalized gradient. First, we obtain the weak formulation of (EBOS) which is a variational-hemivariational inequality, and prove the unique solvability of the bilateral obstacle problem. Second, we consider a nonlinear optimal control problem governed by (EBOS) in which the control variable is the bilateral obstacle, and establish the existence of an optimal solution to the obstacle control problem under mild conditions. Then, employing the regularization technique and penalty approach, we introduce a family of approximating problems corresponding to the optimal control problem under consideration. Finally, a convergence result that any sequence of solutions for the approximating problems converges to an optimal solution of the original optimal control problem is delivered.