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

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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 languageEnglish (US)
JournalPhysica D: Nonlinear Phenomena
DOIs
StateAccepted/In press - Jan 1 2019

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functionals
dynamical systems
complex systems
energy conservation
optimal control
nonlinear systems
autocorrelation
energy
topography
nonlinearity
histories
inversions
perturbation
configurations

Keywords

  • Linear response theory
  • Statistical control
  • Statistical energy principle

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Condensed Matter Physics

Cite this

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title = "Using statistical functionals for effective control of inhomogeneous complex turbulent dynamical systems",
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.",
keywords = "Linear response theory, Statistical control, Statistical energy principle",
author = "Andrew Majda and Di Qi",
year = "2019",
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language = "English (US)",
journal = "Physica D: Nonlinear Phenomena",
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AU - Qi, Di

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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.

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