离散时间线性系统的最大熵最优密度控制与薛定谔桥
Maximum entropy optimal density control of discrete-time linear systems and Schrödinger bridges
AI总结：
本文针对离散时间线性系统提出带高斯密度约束的最大熵最优密度控制方法，推导其显式形式，并将相关状态过程刻画为pinned过程，揭示其与薛定谔桥的等价关系。
摘要：
我们考虑确定性离散时间线性系统的最优密度控制的熵正则化版本。熵正则化，或称最优控制的最大熵（MaxEnt）方法，由于具有自然探索策略等许多优点，尤其在强化学习中引起了广泛关注。尽管有这些优点，正则化所诱导的高熵控制策略会给系统引入概率不确定性，这严重限制了MaxEnt最优控制在安全关键系统中的适用性。为了弥补这一不足，我们在指定时刻对MaxEnt最优控制施加高斯密度约束，以直接控制状态不确定性。具体而言，我们推导了MaxEnt最优密度控制的显式形式。此外，我们还考虑了密度约束被固定点约束替代的情形。然后，我们将相关的状态过程刻画为一个pinned过程，这是布朗桥向线性系统的推广。最后，我们揭示了MaxEnt最优密度控制给出了与离散时间线性系统相关联的所谓薛定谔桥。
英文摘要：
We consider an entropy-regularized version of optimal density control of deterministic discrete-time linear systems. Entropy regularization, or a maximum entropy (MaxEnt) method for optimal control has attracted much attention especially in reinforcement learning due to its many advantages such as a natural exploration strategy. Despite the merits, high-entropy control policies induced by the regularization introduce probabilistic uncertainty into systems, which severely limits the applicability of MaxEnt optimal control to safety-critical systems. To remedy this situation, we impose a Gaussian density constraint at a specified time on the MaxEnt optimal control to directly control state uncertainty. Specifically, we derive the explicit form of the MaxEnt optimal density control. In addition, we also consider the case where density constraints are replaced by fixed point constraints. Then, we characterize the associated state process as a pinned process, which is a generalization of the Brownian bridge to linear systems. Finally, we reveal that the MaxEnt optimal density control gives the so-called Schrödinger bridge associated to a discrete-time linear system.
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