The present paper studies the duality theory for the Mayer problem with second-order evolution differential inclusions with delay and state constraints. Although all the proofs in the paper relating to dual problems are carried out in the case of delay, these results are new for problems without delay, too. To this end, first we use an auxiliary problem with second-order discrete- and discrete-approximate inclusions. Second, applying infimal convolution concept of convex functions, step-by-step, we construct the dual problems for discrete, discrete–approximate, and differential inclusions, and prove duality results, where the Euler–Lagrange type inclusions are “duality relations” for both primary and dual problems and that the dual problem for discrete–approximate problem make a bridge between the dual problems of discrete and continuous problems. Thus, proceeding to the limit procedure, we establish the dual problem for the continuous problem. In addition, semilinear problems with discrete and differential inclusions of second order with delay are also considered. These problems show that supremum in the dual problems are realized over the set of solutions of the Euler–Lagrange-type discrete/differential inclusions, respectively.

中文翻译：

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具有时滞的二阶离散和微分包含的对偶性

本文研究了具有时滞和状态约束的二阶演化微分包含的Mayer问题的对偶理论。尽管本文中有关双重问题的所有证明都是在延迟情况下进行的，但这些结果对于没有延迟的问题也是新的。为此，首先我们使用带有二阶离散和离散近似包含的辅助问题。其次，逐步应用凸函数的极小卷积概念，我们构造了离散，离散-近似和微分包含的对偶问题，并证明了对偶结果，其中Euler-Lagrange类型包含为“对偶关系”无论是主要问题还是双重问题，离散-近似问题的双重问题都在离散问题和连续问题的双重问题之间架起了一座桥梁。因此，进行极限程序，我们建立了连续问题的对偶问题。另外，还考虑了具有二阶延迟的离散和微分包含的半线性问题。这些问题表明对偶问题中的极值分别通过Euler–Lagrange型离散/微分包含解的集合实现。