In this paper we study an optimal control problem for the mixed Dirichlet–Neumann boundary value problem for the strongly non-linear elliptic equation with p-Laplace operator and \(L^1\)-nonlinearity in their right-hand side. 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 \(y_d\in L^2(\varOmega )\) and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for any admissible control. After defining a suitable functional class in which we look for solutions and assuming that this problem admits at least one feasible solution, we prove the existence of optimal pairs. In order to handle the strong non-linearity in the right-hand side of elliptic equation, we involve a special two-parametric fictitious optimization problem. We derive existence of optimal solutions to the regularized optimization problems at each \(({\varepsilon },k)\)-level of approximation and discuss the asymptotic behaviour of the optimal solutions to regularized problems as the parameters \({\varepsilon }\) and k tend to zero and infinity, respectively.

中文翻译：

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具有p -Laplace算子和$$ L ^ 1 $$ L1-非线性类型的不适定强非线性椭圆方程的Neumann边界控制问题

在本文中，我们研究了带有p -Laplace算子和\（L ^ 1 \）-非线性在其右侧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。作用在开放域边界的一部分上的表面牵引力的密度u被作为边界控制。最优控制问题是最小化给定分布\（y_d \ in L ^ 2（\ varOmega）\）之间的差异和当前系统状态。当我们不能期望对任何允许的控制都具有状态方程的解时，我们将处理这种非线性情况。在定义了一个合适的功能类并在其中寻找解决方案并假设此问题至少接受了一个可行的解决方案之后，我们证明了最优对的存在。为了处理椭圆方程右侧的强非线性，我们涉及一个特殊的两参数虚拟优化问题。我们在每个\（（{{varepsilon}，k）\）逼近水平上得出正则化优化问题的最优解的存在，并讨论正则化问题的最优解的渐近行为作为参数\（{\ varepsilon} \）和k 分别趋于零和无穷大。