Optimal control of fractional semilinear PDEs
Optimal control of fractional semilinear PDEs
复制标题
DOI：
10.1051/cocv/2019003
复制
发表时间：
2017-12
期刊：
影响因子：
--
通讯作者：
Harbir Antil;M. Warma
中科院分区：
文献类型：
--
作者：
Harbir Antil;M. Warma
在本文中，我们考虑使用 2s 阶谱和积分分数扩散算子（其中 s ∈ (0, 1)）对半线性分数偏微分方程进行最优控制。我们首先证明在域和数据的最小正则性假设下两个半线性分数偏微分方程解的有界性。接下来，我们引入非线性的最佳增长条件，以显示半线性椭圆方程解图相对于数据的 Lipschitz 连续性。我们进一步应用我们的想法来证明以半线性分数方程作为约束的最优控制问题的解的存在性。在非线性（两次连续可导）的标准假设下，我们推导出一阶和二阶最优性条件。
In this paper, we consider the optimal control of semilinear fractional PDEs with both spectral and integral fractional diffusion operators of order 2s with s ∈ (0, 1). We first prove the boundedness of solutions to both semilinear fractional PDEs under minimal regularity assumptions on domain and data. We next introduce an optimal growth condition on the nonlinearity to show the Lipschitz continuity of the solution map for the semilinear elliptic equations with respect to the data. We further apply our ideas to show existence of solutions to optimal control problems with semilinear fractional equations as constraints. Under the standard assumptions on the nonlinearity (twice continuously differentiable) we derive the first and second order optimality conditions.