AI 中文总结
研究非自治鞍结分岔的最优控制问题，利用分岔与速率诱导过冲的联系将其转化为无约束最小化问题，关键是速率函数，还给出其近似以获近似控制封闭形式表达式，适用于需重复求解控制问题的情况。
AI 中文摘要
非自治鞍结分岔常见于数学模型中，但针对此类系统的最优控制理论尚未完善。我们给出了非自治鞍结范式的概念性工作。通过利用分岔与速率诱导过冲之间的联系，开发了一种将最优控制问题重新表述为无约束最小化问题的方法。关键部分是一个速率函数，它为每个非自治强迫分配临界速率。在鞍结范式中，临界速率是唯一的，速率函数定义良好。不过速率函数通常未知，为此我们还给出了速率函数的近似，得到近似控制的封闭形式表达式，这使得该方法在需要重复求解控制问题的情况下很有吸引力。
英文摘要
Nonautonomous saddle node bifurcations commonly appear in mathematical models, however, an optimal control theory tailored to such systems is not yet well established. We present conceptual work on the nonautonomous saddle node normal form. We develop an approach that reformulates the optimal control problem into an unconstrained minimization problem by exploiting a connection between bifurcation and rate-induced overshoots. The key component is a rate function that assigns the critical rate to every nonautonomous forcing. We show that in the saddle node normal form, the critical rate is unique, and hence the rate function is well defined. While this reformulation is valid in any case, the rate function is often unknown. To resolve this, we additionally present an approximation of the rate function that leads to a closed form expression of the approximated control. This makes the approach appealing in situations, in which we need repeated solves of the control problem.
