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Control invariant sets are crucial for various methods that aim to designsafe control policies for systems whose state constraints must be satisfiedover an indefinite time horizon. In this article, we explore the connectionsamong reachability, control invariance, and Control Barrier Functions (CBFs) byexamining the forward reachability problem associated with control invariantsets. We present the notion of an "inevitable Forward Reachable Tube" (FRT) asa tool for analyzing control invariant sets. Our findings show that theinevitable FRT of a robust control invariant set with a differentiable boundaryis the set itself. We highlight the role of the differentiability of theboundary in shaping the FRTs of the sets through numerical examples. We alsoformulate a zero-sum differential game between the control and disturbance,where the inevitable FRT is characterized by the zero-superlevel set of thevalue function. By incorporating a discount factor in the cost function of thegame, the barrier constraint of the CBF naturally arises as the constraint thatis imposed on the optimal control policy. As a result, the value function ofour FRT formulation serves as a CBF-like function, which has not beenpreviously realized in reachability studies. Conversely, any valid CBF is alsoa forward reachability value function inside the control invariant set, therebyrevealing the inverse optimality of the CBF. As such, our work establishes astrong link between reachability, control invariance, and CBFs, filling a gapthat prior formulations based on backward reachability were unable to bridge.

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