We discuss the existence of solutions to an optimal control problem for the mixed Dirichlet-Neumann boundary value problem for strongly non-linear elliptic equations with an exponential type of nonlinearity. A density of surface traction $u$ acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution $y_d\in L^2(\Omega)$ and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this, we make use of a variant of the classical Tikhonov regularization. We eliminate the differential constraints between control and state and allow such pairs run freely in their respective sets of feasibility by introducing some additional variable which plays the role of ``defect". We show that this special residual function can be determined in a unique way. We introduce a special family of regularized optimization problems and show that each of these problem is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.

中文翻译：

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具有指数型非线性的不适定强非线性椭圆方程的最优Neumann边界控制问题的Tikhonov正则化

我们讨论具有指数类型非线性的强非线性椭圆方程的混合Dirichlet-Neumann边值问题的最优控制问题的解的存在。作用在开放域边界的一部分上的表面牵引力的密度被视为边界控制。最佳控制问题是最小化给定分布Ly2（\ Omega）$中的$ y_d \与当前系统状态之间的差异。当我们不能期望对每个允许的控制都有原始边界值问题的解决方案时，我们将处理这种非线性情况。取而代之的是，我们使用经典Tikhonov正则化的一种变体。我们表明，可以以独特的方式确定此特殊残差函数。我们介绍了一个特殊的正则化优化问题系列，并显示了这些问题中的每一个都是一致的，适定的，并且由于正则化的参数趋于零，因此它们的解决方案可以（在极限内）获得原始问题的最优解。结果，我们为给定的非线性Dirichlet BVP类建立了最优解的充分条件，并提出了逼近它们的方法。我们表明，可以以独特的方式确定此特殊残差函数。我们介绍了一个特殊的正则化优化问题系列，并显示了这些问题中的每一个都是一致的，适定的，并且由于正则化的参数趋于零，因此它们的解决方案可以（在极限内）获得原始问题的最优解。结果，我们为给定的非线性Dirichlet BVP类建立了最优解的充分条件，并提出了逼近它们的方法。