The paper is devoted to the Lagrange problem in the bounded region for first-order partial differential inclusions (PDIs). For this, using discretisation method and locally adjoint mappings (LAMs), in the form of Euler–Lagrange type inclusions and conjugate boundary conditions, sufficient optimality conditions are obtained. The transition to a continuous problem with PDIs is possible using a specially proved equivalence theorem. To demonstrate this approach, some semilinear problems and polyhedral optimisation with first-order partial differential inclusions are considered. Furthermore, the numerical results also are provided.

中文翻译：

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有界区域一阶偏微分夹杂物的最优控制

本文致力于研究一阶偏微分包含 (PDI) 有界区域中的拉格朗日问题。为此，使用离散化方法和局部伴随映射（LAM），以欧拉-拉格朗日型夹杂物和共轭边界条件的形式，获得了充分的最优性条件。使用经过特殊证明的等价定理可以转换到 PDI 的连续问题。为了证明这种方法，考虑了一些半线性问题和具有一阶偏微分包含的多面体优化。此外，还提供了数值结果。