Decaying Sensitivity of the Zero Solution for a Class of Nonlinear Optimal Control Problems

TL;DR

This paper investigates how perturbations at a single node in a nonlinear control system on a graph decay exponentially with distance, extending linear results to nonlinear systems.

Contribution

It establishes exponential decay of sensitivities in nonlinear optimal control problems with graph-structured interactions, based on a nonlinear null-controllability condition.

Findings

Perturbations at a node induce exponentially decaying trajectories with graph distance.

The analysis extends spatial decay results from linear to nonlinear systems.

Numerical example confirms theoretical decay behavior.

Abstract

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.