使用时变障碍函数的路径不连通性代数证明
Algebraic Proofs of Path Disconnectedness using Time-Dependent Barrier Functions
AI总结：
该研究将路径不连通性转化为单积分器控制不可行问题，提出用时变障碍函数给出充要性证明，并基于矩—平方和半定规划层级数值构造多项式障碍函数。
摘要：
给定集合的两个子集若位于该较大集合的不同连通分量中，则它们是路径不连通的。路径不连通性的验证对于证明运动规划和轨迹优化算法的不可行性至关重要。我们将路径不连通性表述为在足够长的时间范围内于初始集合和目标集合之间移动的单积分器控制任务的不可行性。这种控制不可行任务通过生成一个分隔初始集合和最终集合的时变障碍函数来证明。在紧致性条件下，时变障碍函数的存在是路径不连通性的充要条件。在数值方面，对多项式障碍函数的搜索利用半定规划的矩—平方和层级来表述。该障碍函数在足够大的多项式次数下证明路径不连通性。这些半定规划的计算复杂度可通过消去控制变量来降低。文中为若干示例系统综合了不连通性证明。
英文摘要：
Two subsets of a given set are path-disconnected if they lie in different connected components of the larger set. Verification of path-disconnectedness is essential in proving the infeasibility of motion planning and trajectory optimization algorithms. We formulate path-disconnectedness as the infeasibility of a single-integrator control task to move between an initial set and a target set in a sufficiently long time horizon. This control-infeasibility task is certified through the generation of a time-dependent barrier function that separates the initial and final sets. The existence of a time-dependent barrier function is a necessary and sufficient condition for path-disconnectedness under compactness conditions. Numerically, the search for a polynomial barrier function is formulated using the moment-sum-of-squares hierarchy of semidefinite programs. The barrier function proves path-disconnectedness at a sufficiently large polynomial degree. The computational complexity of these semidefinite programs can be reduced by elimination of the control variables. Disconnectedness proofs are synthesized for example systems.
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