SIAM Journal on Control and Optimization, Volume 58, Issue 6, Page 3262-3292, January 2020.
We study a class of discounted autonomous infinite horizon optimal control problems where the state dynamics may undergo discontinuous changes when the state crosses from one region of the state space to another. Assuming the state and control space are one-dimensional, we provide structural results of the candidate optimal solutions. In particular, we show that candidate optimal state trajectories are monotonic. Further, we prove a theorem on absence of limit cycles, generalizing the Bendixson criterion to a hybrid context. Using these results, we derive necessary and sufficient conditions for the existence of tipping points for this class of problems. A tipping point is an initial condition starting from which there exist multiple candidate optimal solutions with different qualitative behaviors. Next, we specialize these results for piecewise linear quadratic models to obtain an analytic characterization of tipping points in terms of the model parameters.

中文翻译：

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一状态变量混合系统的最优控制问题分析

SIAM控制与优化杂志，第58卷，第6期，第3262-3292页，2020年1月。
我们研究了一类打折的自治无限地平线最优控制问题，当状态从状态空间的一个区域穿越到另一个区域时，状态动态可能会经历不连续的变化。假设状态和控制空间是一维的，我们提供候选最优解的结构结果。特别地，我们表明候选最佳状态轨迹是单调的。此外，我们证明了不存在极限环的一个定理，将Bendixson准则推广到混合环境。使用这些结果，我们得出此类问题的临界点存在的必要和充分条件。临界点是一个初始条件，从该条件开始，存在具有不同定性行为的多个候选最优解。下一个，