A Forward Reachability Perspective on Robust Control Invariance and Discount Factors in Reachability Analysis

Control invariant sets are crucial for various methods that aim to design safe control policies for systems whose state constraints must be satisfied over an indefinite time horizon. In this article, we explore the connections among reachability, control invariance, and Control Barrier Functions (CBFs). Unlike prior formulations based on backward reachability concepts, by examining a forward reachability problem, we are able to establish a strong link between these three concepts. First, our findings show that the inevitable Forward Reachable Tube (FRT), which is the set of states such that every trajectory reaching the FRT must have passed through a given initial set of states, is precisely this initial set of states itself if it is a robust control invariant set with a differentiable boundary. We highlight that this statement may not hold if the boundary is not differentiable. Next, we formulate a differential game between the control and disturbance, where the inevitable FRT is characterized by the zero-superlevel set of the value function. By incorporating a discount factor in the cost function of the game, the barrier constraint of the CBF naturally arises as the constraint that is imposed on the optimal control policy. Combining these results, the value function of our FRT formulation serves as a CBF-like function, and conversely, any valid CBF is also a forward reachability value function inside the control invariant set, thereby revealing the inverse optimality of the CBF. This strong link we establish between the reachability problem and the barrier constraint, while guaranteeing the continuity of the value function, is not achievable by previous backward reachability-based formulations. As such, our work fills a crucial gap in the existing literature that is vital for constructing valid CBFs to ensure safety.