Abstract

We consider problems of optimal control of processes described by the Dirichlet boundary value problem for elliptic equations with mixed derivatives and unbounded nonlinearity. The controls are contained in the coefficients multiplying the highest derivatives of the state equation. Finite-difference approximations to nonlinear optimization models are constructed and investigated, and to find an approximate solution of a nonlinear boundary value problem for the state, an iteration process implementing the problem is constructed. The convergence of the iteration process used to prove the existence and uniqueness of the solution of a nonlinear difference scheme approximating the original boundary value problem for the state is studied rigorously. We establish the mesh \(W_{2,0}^2(\omega )\) -norm estimates consistent with the smoothness of the desired solution for the rate of convergence of difference schemes that approximate the nonlinear state equation. The convergence of the approximations to the optimal control problems with respect to state, functional, and control is investigated; the regularization of the approximations is carried out.

中文翻译：

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系数无界非线性的非发散型椭圆方程的前导系数最优控制问题的近似

摘要

我们考虑具有混合导数和无界非线性的椭圆方程的狄利克雷边值问题所描述的过程的最优控制问题。控制包含在乘以状态方程的最高导数的系数中。构造和研究非线性优化模型的有限差分近似，并为找到状态的非线性边值问题的近似解，构造了实现该问题的迭代过程。严格研究了迭代过程的收敛性，该过程用于证明近似于状态的原始边值问题的非线性差分格式的解的存在性和唯一性。我们建立网格\(W_{2,0}^2(\omega )\) - 范数估计与近似非线性状态方程的差分方案的收敛速度的期望解的平滑度一致。研究了关于状态、功能和控制的最优控制问题的近似收敛；执行近似的正则化。