### Abstract

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.

Original language | English (US) |
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Journal | Physica D: Nonlinear Phenomena |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

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### Keywords

- Linear response theory
- Statistical control
- Statistical energy principle

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Condensed Matter Physics

### Cite this

**Using statistical functionals for effective control of inhomogeneous complex turbulent dynamical systems.** / Majda, Andrew; Qi, Di.

Research output: Contribution to journal › Article

}

TY - JOUR

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

AU - Majda, Andrew

AU - Qi, Di

PY - 2019/1/1

Y1 - 2019/1/1

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

UR - http://www.scopus.com/inward/record.url?scp=85059941498&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85059941498&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2018.12.003

DO - 10.1016/j.physd.2018.12.003

M3 - Article

AN - SCOPUS:85059941498

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -