The article presents a framework for certifying path disconnectedness between two sets X0 and X1 within a larger set X. The key insights are:
Path disconnectedness can be formulated as the infeasibility of a single-integrator optimal control problem to move between X0 and X1 within a sufficiently long time horizon T.
This infeasibility can be certified through the existence of a time-dependent barrier function v(t,x) that satisfies:
The existence of such a time-dependent barrier function is a necessary and sufficient condition for path disconnectedness under compactness assumptions on X0, X1 and X.
Numerically, the search for a polynomial barrier function is formulated as a hierarchy of Moment-Sum-of-Squares (SOS) Semidefinite Programs (SDPs). As the degree of the polynomial increases, the barrier function eventually proves path disconnectedness.
The computational complexity of these SDPs can be reduced by eliminating the control variables u, leading to an SDP with fewer variables.
The article demonstrates the application of this framework to synthesize disconnectedness proofs for various example systems.
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by Didier Henri... at arxiv.org 04-11-2024
https://arxiv.org/pdf/2404.06985.pdfDeeper Inquiries