A probabilistic framework for robustness analysis of optimal time-delayed and extended time-delayed feedback controllers for chaos control

Abstract This work presents a probabilistic framework for analyzing the robustness of optimal time-delayed feedback (TDF) and extended time-delayed feedback (ETDF) controllers in stabilizing unstable periodic orbits (UPOs) embedded in the chaotic attractor of a Duffing oscillator. The study integrates established tools, optimal control design, stochastic uncertainty modeling, and probabilistic stability assessment into a unified methodology for chaos control. Parametric and excitation uncertainties are represented through probability density functions derived from the Maximum Entropy Principle. The optimal controller gains are obtained via the cross-entropy optimization method by minimizing an objective function based on Floquet multipliers and Lyapunov exponents. Monte Carlo simulations are employed to estimate the probability distributions of the maximum Lyapunov exponent and the mean distance between the controlled and target UPOs, providing quantitative measures of robustness. The results indicate that both TDF and ETDF controllers maintain high robustness for low-period UPOs, even under significant uncertainty. TDF achieves a slightly higher probability of stability for period-1 and period-2 UPOs, while ETDF remains effective for higher-period orbits where TDF fails. Moreover, for the period-1 UPO, the Lyapunov exponents obtained with ETDF are consistently more negative than those of TDF, implying that the stabilized orbits under ETDF possess stronger stability margins. Although ETDF gradually loses robustness as the orbit period increases, it remains the only viable approach for stabilizing high-period UPOs. Overall, the proposed framework offers a systematic and probabilistic assessment of controller performance, contributing a coherent methodology for the analysis and design of robust chaos control strategies under stochastic perturbations.