We study a diffusion process on a finite interval under the influence of a controllable drift where the particle resets to the left-hand side upon reaching the right-hand side. Assigning a pay-off for being nearer the right-hand side, but a penalty for reaching it, induces an inherent trade-off. We seek the drift feedback that maximises the long-term reward. By reducing the problem to a constrained variational problem we deduce that, for a wide class of problems, the optimal feedback law is remarkably straightforward: below a threshold state exert maximum drift; beyond the threshold exert minimum drift.

中文翻译：

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首段复位中的最优反馈控制

我们研究了在可控漂移的影响下有限间隔上的扩散过程，其中粒子在到达右侧时重置到左侧。为靠近右手边分配回报，但对到达右手边进行惩罚，会导致内在的权衡。我们寻求最大化长期回报的漂移反馈。通过将问题简化为一个受约束的变分问题，我们推断，对于一大类问题，最优反馈定律非常简单：低于阈值状态产生最大漂移；超过阈值施加最小漂移。