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

TL;DR

The paper addresses how a localized initial perturbation in a graph-structured network of decoupled nonlinear subsystems affects the global nonlinear OCP with a quadratic cost. It proves a dimension-free exponential decay of the perturbation with graph distance under a nonlinear asymptotic controllability assumption, using a perturbation propagation framework and reduced OCP arguments. The main contribution is a nonlinear extension of spatial decay results known for linear-quadratic problems, providing explicit bounds of the form $\|x_\mathcal{W}^*\|_{L_2} \le S \rho^{\mathrm{d_G}(i^*,\mathcal{W})} |x_{0,i^*}|$ with $\rho<1$, plus a numerical demonstration on a chain of vehicles. This work supports scalable decentralized control by showing localization of disturbances, enabling localized solution methods and separable value-function representations in nonlinear settings.

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 localized perturbation of the zero reference leads to an optimal trajectory that decays exponentially with the graph distance. The analysis, based on a nonlinear 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.

Figures (2)

• Figure 1: Graph for $s=5$ vehicles.
• Figure 2: Decay of the ${L}_2$-norms of the optimal state trajectories $x_i$ for $i^*\in\{1,12,25\}$.