Foundational theory is established for nonlinear differential equations with embedded nonlinear optimization problems exhibiting active set changes. Existence, uniqueness, and continuation of solutions are shown, followed by lexicographically smooth (implying Lipschitzian) parametric dependence. The sensitivity theory found here accurately characterizes sensitivity jumps resulting from active set changes via an auxiliary nonsmooth sensitivity system obtained by lexicographic directional differentiation. The results in this article hold under easily verifiable regularity conditions (linear independence of constraints and strong second-order sufficiency), which are shown to imply generalized differentiation index one of a nonsmooth differential-algebraic equation system obtained by replacing the optimization problem with its optimality conditions and recasting the complementarity conditions as nonsmooth algebraic equations. The theory in this article is computationally relevant, allowing for implementation of dynamic optimization strategies (i.e., open-loop optimal control), and recovers (and rigorously formalizes) classical results in the absence of active set changes. Along the way, contributions are made to the theory of piecewise differentiable functions.

中文翻译：

---

具有活动集变化的优化约束微分方程

为非线性微分方程建立了基础理论，其中嵌入的非线性优化问题表现出活动集变化。显示了解的存在性、唯一性和连续性，然后是按字典顺序排列的平滑（暗示 Lipschitzian）参数依赖。这里发现的敏感性理论通过字典方向微分获得的辅助非平滑敏感性系统准确地描述了由主动集合变化引起的敏感性跳跃。本文的结果在易于验证的正则性条件下（约束的线性独立性和强二阶充分性）成立，表明其隐含广义微分指数之一，是通过将优化问题替换为其最优条件并将互补条件重铸为非光滑代数方程而获得的非光滑微分代数方程组。本文中的理论与计算相关，允许实施动态优化策略（即开环最优控制），并在没有活动集更改的情况下恢复（并严格形式化）经典结果。在此过程中，对分段可微函数理论做出了贡献。并在没有活动集变化的情况下恢复（并严格形式化）经典结果。在此过程中，对分段可微函数理论做出了贡献。并在没有活动集变化的情况下恢复（并严格形式化）经典结果。在此过程中，对分段可微函数理论做出了贡献。