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Algebraic Proofs of Path Disconnectedness using Time-Dependent Barrier Functions


Core Concepts
The core message of this article is that path disconnectedness between two sets X0 and X1 within a larger set X can be certified through the existence of a time-dependent barrier function that separates the sets. This barrier function is a necessary and sufficient condition for path disconnectedness under compactness assumptions.
Abstract
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: v(0,x) > 0 for all x in X0 v(T,x) ≤ 0 for all x in X1 The Lie derivative Luv(t,x) ≥ 0 for all (t,x,u) in [0,T] x X x U, where U is the control set. 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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Deeper Inquiries

How could this framework be extended to handle unbounded domains or non-compact sets

To extend this framework to handle unbounded domains or non-compact sets, one approach could be to introduce appropriate bounding functions or constraints that limit the behavior of the system within a bounded region. For unbounded domains, one could define a large but finite bounding box or sphere that encapsulates the unbounded domain. This bounding region would act as a constraint to ensure that the system remains within a finite space. Additionally, one could introduce penalty functions or regularization terms in the optimization problem to discourage trajectories from venturing too far into the unbounded regions. For non-compact sets, one could consider approximating the non-compact set with a sequence of compact sets that cover the entire space. By formulating the problem over these compact approximations, one can still apply the same methodology for finding time-dependent barrier functions and proving path-disconnectedness. The key challenge in handling non-compact sets lies in ensuring that the approximations capture the essential characteristics of the non-compact set without introducing significant errors or inaccuracies in the analysis.

What are the limitations of the Moment-SOS hierarchy approach, and are there alternative numerical methods that could be used to find time-dependent barrier functions

The Moment-SOS hierarchy approach, while powerful for solving polynomial optimization problems, has limitations in terms of scalability and computational complexity. As the degree of the polynomials increases, the size of the semidefinite programming (SDP) problems grows rapidly, leading to challenges in solving them efficiently for high-dimensional systems or high-degree polynomials. Additionally, the numerical conditioning of the SDPs can deteriorate as the degree increases, making it harder to find accurate solutions. Alternative numerical methods that could be used to find time-dependent barrier functions include optimization techniques such as interior-point methods, genetic algorithms, particle swarm optimization, or reinforcement learning approaches. These methods may offer advantages in terms of scalability, convergence speed, and robustness to numerical issues. For example, reinforcement learning algorithms could learn the barrier function by interacting with the system dynamics, potentially providing a more adaptive and data-driven approach to finding path-disconnectedness certificates.

What are the connections between this work on path disconnectedness and recent results on non-intersection certificates between semialgebraic sets

The work on path disconnectedness and recent results on non-intersection certificates between semialgebraic sets are closely related in the context of geometric constraint satisfaction and optimization. Both areas focus on certifying the infeasibility of certain geometric configurations or trajectories within a given space. While path disconnectedness deals with proving the impossibility of connecting two sets through trajectories, non-intersection certificates focus on demonstrating that two sets do not overlap or intersect under certain constraints. The connection between these two areas lies in their common goal of providing rigorous mathematical proofs of geometric relationships or constraints. By leveraging techniques from polynomial optimization, semidefinite programming, and geometric analysis, researchers can develop robust methods for certifying the feasibility or infeasibility of geometric configurations. The insights and methodologies from one area can often be applied or adapted to the other, leading to a cross-pollination of ideas and approaches in the field of geometric optimization and constraint verification.
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