In this article, based on a multidimensional control problem, in short (MCP), we introduce a modified multidimensional variational control problem involving first‐order partial differential equations and inequality‐type constraints. As well, we formulate and prove optimality conditions for this new variational control problem. Furthermore, we establish (under some generalized convexity assumptions) an equivalence between an optimal solution of (MCP) and a saddle‐point associated with the Lagrange functional (Lagrangian) corresponding to the modified multidimensional control problem. Also, in order to illustrate our main characterization results and their effectiveness, we present several applications.

中文翻译：

---

偏微分方程约束的变分控制问题的鞍点最优性准则

在本文中，基于多维控制问题，简而言之（MCP），我们介绍了一种涉及一阶偏微分方程和不等式类型约束的改进的多维变分控制问题。同样，我们为这个新的变分控制问题制定并证明了最优条件。此外，我们（在一些广义凸假设下）建立了（MCP）的最佳解和与修正的多维控制问题相对应的拉格朗日函数（拉格朗日）相关的鞍点之间的等价关系。另外，为了说明我们的主要表征结果及其有效性，我们提出了几种应用。