The paper is devoted to the duality of the Mayer problem for $\kappa$-th order viable differential inclusions with endpoint constraints, where $\kappa$ is an arbitrary natural number. Thus, this paper for constructing the dual problems to viable differential inclusions of any order with endpoint constraints can make a great contribution to the modern development of optimal control theory. For this, using locally conjugate mappings in the form of Euler–Lagrange type inclusions and transversality conditions, sufficient optimality conditions are obtained. It is noteworthy that the Euler–Lagrange type inclusions for both primary and dual problems are ‘duality relations’. To demonstrate this approach, some semilinear problems and polyhedral optimization with fourth order differential inclusions are considered. These problems show that sufficient conditions and dual problems can be easily established for problems of a reasonable order.

本文专门研究 Mayer 问题的对偶性$\kappa$- 具有端点约束的可行差分包含，其中$\kappa$是任意自然数。因此，本文针对具有端点约束的任意阶可行微分包含构造对偶问题，可以对最优控制理论的现代发展做出重大贡献。为此，使用欧拉-拉格朗日类型包含和横向条件形式的局部共轭映射，可以获得足够的最优性条件。值得注意的是，初级问题和对偶问题的欧拉-拉格朗日类型包含是“对偶关系”。为了证明这种方法，考虑了一些半线性问题和四阶微分包含的多面体优化。这些问题表明，对于合理阶的问题，可以很容易地建立充分条件和对偶问题。