SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 271-296, February 2024.
Abstract. There are three different forms of adjoint inclusions that appear in the most advanced necessary optimality conditions for optimal control problems involving differential inclusions: Euler–Lagrange inclusion (with partial convexification) [A. D. Ioffe, J. Optim. Theory Appl., 182 (2019), pp. 285–309], fully convexified Hamiltonian inclusion [F. H. Clarke, Mem. Amer. Math. Soc., 173 (2005), 816], and partially convexified Hamiltonian inclusion [P. D. Loewen and R. T. Rockafellar, SIAM J. Control Optim., 34 (1996), pp. 1496–1511], [A. D. Ioffe, Trans. Amer. Math. Soc., 349 (1997), pp. 2871–2900], [R. B. Vinter, SIAM J. Control Optim., 52 (2014), pp. 1237–1250] (for convex-valued differential inclusions in the first two references). This paper addresses all three types of necessary conditions for problems with (in general) nonconvex-valued differential inclusions. The first of the two main theorems, with the Euler–Lagrange inclusion, is equivalent to the main result of [A. D. Ioffe, J. Optim. Theory Appl., 182 (2019), pp. 285–309] but proved in a substantially different and much more direct way. The second theorem contains conditions that guarantee necessity of both types of Hamiltonian conditions. It seems to be the first result of such a sort that covers differential inclusions with possibly unbounded values and contains the most recent results of [F. H. Clarke, Mem. Amer. Math. Soc., 173 (2005), 816] and [R. B. Vinter, SIAM J. Control Optim., 52 (2014), pp. 1237–1250] as particular cases. And again, the proof of the theorem is based on a substantially different approach.

中文翻译：

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微分包含最优控制问题的最大原理

SIAM 控制与优化杂志，第 62 卷，第 1 期，第 271-296 页，2024 年 2 月。
摘要。在涉及微分包含的最优控制问题的最先进的必要最优性条件中，存在三种不同形式的伴随包含： 欧拉-拉格朗日包含（部分凸化）[AD Ioffe，J. Optim。Theory Appl., 182 (2019), pp. 285–309]，完全凸化哈密顿包含 [F. H.克拉克，Mem。阿米尔。数学。Soc., 173 (2005), 816]，以及部分凸化哈密顿包含[P。D. Loewen 和 R. T. Rockafellar，SIAM J. Control Optim.，34 (1996)，第 1496–1511 页]，[A. D.约夫，翻译。阿米尔。数学。社会学会，349 (1997)，第 2871–2900 页]，[R。B. Vinter, SIAM J. Control Optim., 52 (2014), pp. 1237–1250]（对于前两个参考文献中的凸值微分包含）。本文解决了（通常）非凸值微分包含问题的所有三种类型的必要条件。两个主要定理中的第一个，加上欧拉-拉格朗日包含，等价于[A. D. Ioffe，J. Optim。Theory Appl., 182 (2019), pp. 285–309]，但以一种截然不同且更直接的方式证明。第二个定理包含保证两种哈密顿条件的必要性的条件。这似乎是此类的第一个结果，它涵盖了具有可能无限值的微分包含，并且包含 [F. H.克拉克，Mem。阿米尔。数学。Soc.，173（2005），816]和[R。B. Vinter, SIAM J. Control Optim., 52 (2014), pp. 1237–1250] 作为特殊情况。同样，该定理的证明是基于一种截然不同的方法。