Optimization‐constrained differential equations (OCDE) are a class of mathematical problems where differential equations are constrained by an embedded algebraic optimization problem. We analyze the well‐posedness of the local solutions of OCDE based on local optimality. By assuming linear independence constraint qualification and applying the Karush‐Kuhn‐Tucker optimality conditions, an OCDE is transformed into a complementarity system (CS). Under second‐order sufficient condition we show that (a) if strict complementary condition (SCC) holds, the local solution of OCDE is well‐posed, which corresponds to a mode of the derived CS; (b) at points where SCC is violated, a local solution of OCDE exists by sequentially connecting the local solutions of two selected modes of the derived CS. We propose an event‐based algorithm to numerically solve OCDE. We illustrate the approach and algorithm for microbial cultivation, single flash unit and contrived numerical examples.

中文翻译：

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用局部最优条件分析优化约束微分方程的局部适定性

优化约束微分方程（OCDE）是一类数学问题，其中微分方程受嵌入式代数优化问题的约束。我们基于局部最优性分析了OCDE局部解的适定性。通过假设线性独立性约束条件并应用Karush-Kuhn-Tucker最优性条件，OCDE转化为互补系统（CS）。在二阶充分条件下，我们证明（a）如果严格补充条件（SCC）成立，则OCDE的局部解是适当的，这对应于导出的CS的模式；（b）在违反SCC的点上，通过顺序连接派生CS的两个选定模式的局部解来存在OCDE的局部解。我们提出了一种基于事件的算法来对OCDE进行数值求解。