In this paper, we consider the infinite-horizon reach-avoid zero-sum game problem, where the goal is to find a set in the state space, referred to as the reach-avoid set, such that the system starting at a state therein could be controlled to reach a given target set without violating constraints under the worst-case disturbance. We address this problem by designing a new value function with a contracting Bellman backup, where the super-zero level set, i.e., the set of states where the value function is evaluated to be non-negative, recovers the reach-avoid set. Building upon this, we prove that the proposed method can be adapted to compute the viability kernel, or the set of states which could be controlled to satisfy given constraints, and the backward reachable set, or the set of states that could be driven towards a given target set. Finally, we propose to alleviate the curse of dimensionality issue in high-dimensional problems by extending Conservative Q-Learning, a deep reinforcement learning technique, to learn a value function such that the super-zero level set of the learned value function serves as a (conservative) approximation to the reach-avoid set. Our theoretical and empirical results suggest that the proposed method could learn reliably the reach-avoid set and the optimal control policy even with neural network approximation.

本文主要研究了无限时间视角下的reach-avoid zero-sum博弈问题，提出了基于收缩Bellman backup的价值函数和Conservative Q-Learning方法，求解约束条件下的到达目标集和控制策略的可行解。

利用深度强化学习进行无限时域达到-避免零和博弈