非线性最优控制问题中零解的衰减敏感性
Decaying Sensitivity of the Zero Solution for a Class of Nonlinear Optimal Control Problems
AI总结：
本文研究了非线性最优控制问题中敏感性的空间衰减特性，通过非线性解耦动力学和二次成本问题，证明了单个节点的初始扰动会导致最优轨迹的节点范数随图距离指数衰减，为将已知的线性二次结果扩展到非线性系统提供了第一步。
摘要：
我们研究了具有图结构相互作用拓扑的非线性最优控制问题中敏感性的空间衰减特性。对于具有非线性解耦动力学和二次成本的问题，我们证明了对单个节点的零初始条件的扰动会诱导出一个最优轨迹，其节点范数随该扰动节点与图距离的指数衰减。基于非线性可控性条件的分析，为将已知的线性二次结果扩展到非线性系统提供了第一步。数值示例展示了理论发现。
英文摘要：
We study spatial decay properties of sensitivities in a nonlinear optimal control problem with a graph-structured interaction topology. For a problem with nonlinear decoupled dynamics and quadratic cost, we show that a perturbation of the zero initial condition at a single node induces an optimal trajectory whose node-wise norms decay exponentially with the graph distance from the perturbed node. The analysis, based on a nonlinear null-controllability condition, provides a first step toward extending known spatial decay results from linear-quadratic to nonlinear systems. A numerical example illustrates the theoretical findings.
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