Nagumo-Type Characterization of Forward Invariance for Constrained Systems
Résumé
We present a solution-independent condition certifying forward invariance of closed sets for constrained systems. The proposed condition is only necessary in the general case, we establish its sufficiency under three assumptions. These assumptions, related to critical boundary points, constrain the dynamics and enforce mild transversality between the set constraining the dynamics and the set to render forward invariant. When the boundary of the latter set is entirely within the interior of the former, our result reduces to the well-known Nagumo characterization of forward invariance. The importance of the proposed assumptions is illustrated via counterexamples. We further show how the framework simplifies when the considered sets are regular; namely, we consider practical, resp. polytopic, sets—defined as the intersection of sublevel sets of smooth, resp. linear, functions—yielding conditions directly checkable from the defining functions. The results carry important implications for safety-critical applications, particularly in the context of hybrid systems.
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