TY - JOUR

T1 - Using statistical functionals for effective control of inhomogeneous complex turbulent dynamical systems

AU - Majda, Andrew J.

AU - Qi, Di

N1 - Funding Information:
The research of A. J. M. is partially supported by the Office of Naval Research through MURI N00014-16-1-2161 and DARPA through W911NF-15-1-0636 . D. Q. is supported as a postdoctoral fellow on the second grant. The authors are grateful for the comments from the reviewers that have helped to improve this manuscript.
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2019/5

Y1 - 2019/5

N2 - Efficient statistical control strategies are developed for general complex turbulent systems with energy conserving nonlinearity. Instead of direct control on the high-dimensional turbulent equations concerning a large number of instabilities, a statistical functional that characterizes the total statistical structure of the complex system is adopted here as the control object. First the statistical energy equation reduces the control of the complex nonlinear system to a linear statistical control problem; then the explicit form of the forcing control is recovered through nonlocal inversion of the optimal control functional using approximate statistical linear response theory for attribution of the feedback. Through this control strategy with statistical energy conservation, the explicit form of the control forcing is determined offline only requiring the initial configuration of total statistical energy change and the autocorrelation functions in the most sensitive modes of the target statistical equilibrium, with no need of knowing the explicit forcing history and running the complex system. The general framework of the statistical control method can be applied directly on various scenarios with both homogeneous and inhomogeneous perturbations. The effectiveness of the statistical control strategy is demonstrated using the Lorenz ’96 system and a turbulent barotropic system with topography and a large number of instabilities.

AB - Efficient statistical control strategies are developed for general complex turbulent systems with energy conserving nonlinearity. Instead of direct control on the high-dimensional turbulent equations concerning a large number of instabilities, a statistical functional that characterizes the total statistical structure of the complex system is adopted here as the control object. First the statistical energy equation reduces the control of the complex nonlinear system to a linear statistical control problem; then the explicit form of the forcing control is recovered through nonlocal inversion of the optimal control functional using approximate statistical linear response theory for attribution of the feedback. Through this control strategy with statistical energy conservation, the explicit form of the control forcing is determined offline only requiring the initial configuration of total statistical energy change and the autocorrelation functions in the most sensitive modes of the target statistical equilibrium, with no need of knowing the explicit forcing history and running the complex system. The general framework of the statistical control method can be applied directly on various scenarios with both homogeneous and inhomogeneous perturbations. The effectiveness of the statistical control strategy is demonstrated using the Lorenz ’96 system and a turbulent barotropic system with topography and a large number of instabilities.

KW - Linear response theory

KW - Statistical control

KW - Statistical energy principle

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U2 - 10.1016/j.physd.2018.12.003

DO - 10.1016/j.physd.2018.12.003

M3 - Article

AN - SCOPUS:85059941498

SN - 0167-2789

VL - 392

SP - 34

EP - 56

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

ER -