涉及李括号的 HJ 不等式与带成本调节的反馈可镇定性
HJ inequalities involving Lie brackets and feedback stabilizability with cost regulation
AI总结：
针对状态趋近闭目标集且支付非负代价的最优控制问题，将定义控制 Lyapunov 函数的经典耗散关系推广为涉及代价和迭代李括号的较弱微分不等式，提出度数 k 最小约束函数，并证明其存在性可构建基于李括号的反馈，在调节代价的同时实现系统采样镇定。
摘要：
针对一个状态需渐近趋近闭目标集同时支付非负积分代价的最优控制问题，我们将定义控制 Lyapunov 函数的经典耗散关系推广至一个较弱的微分不等式。后者同时涉及代价以及动力学中向量场直至大于等于1的某度数 k 的迭代李括号，我们称其任何（经适当定义的）解为度数 k 最小约束函数。我们证明，度数 k 最小约束函数的存在性使我们能够构建一个基于李括号的反馈，该反馈在调节（即一致有界）代价的同时，将系统采样镇定至目标集。
英文摘要：
With reference to an optimal control problem where the state has to approach asymptotically a closed target while paying a non-negative integral cost, we propose a generalization of the classical dissipative relation that defines a Control Lyapunov Function to a weaker differential inequality. The latter involves both the cost and the iterated Lie brackets of the vector fields in the dynamics up to a certain degree k greater than or equal to 1, and we call any of its (suitably defined) solutions a degree-k Minimum Restraint Function. We prove that the existence of a degree-k Minimum Restraint Function allows us to build a Lie-bracket-based feedback which sample stabilizes the system to the target while regulating (i.e., uniformly bounding) the cost.
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