For the first time in this paper, the dual problem is constructed for the problem with generalized first order partial differential inclusions, the duality theorem is proved. For discrete problems, necessary and sufficient optimality conditions are derived in the form of the Euler–Lagrange type inclusion. Thus, it is possible to construct dual problems for problems with partial differential inclusions on the basis of dual operations of addition and infimal convolution of convex functions. To pass from the dual problem to the discrete-approximation problem, important equivalence theorems are proved, without which it is unlikely that certain success can be achieved along this path. Hence, we believe that this method of constructing dual problems can serve as the only possible method for studying duality for a wide class of problems with partial/ordinary differential inclusions. The results obtained are demonstrated on some linear problem and on a problem with first-order polyhedral partial differential inclusions.

本文首次对具有广义一阶偏微分包含的问题构造对偶问题，证明了对偶性定理。对于离散问题，必要和充分的最优性条件以欧拉-拉格朗日类型包含的形式导出。因此，可以在凸函数的加法和最后卷积的对偶运算的基础上构造具有偏微分包含问题的对偶问题。为了从对偶问题过渡到离散逼近问题，证明了重要的等价定理，没有这些定理，沿着这条路不可能取得一定的成功。因此，我们相信，这种构造对偶问题的方法可以作为研究具有偏/常微分包含的广泛问题的对偶性的唯一可能方法。所获得的结果在一些线性问题和一阶多面体偏微分包含问题上得到证明。