Flow, Turbulence and Combustion

, Volume 65, Issue 3–4, pp 299–320 | Cite as

Numerical Approaches to Optimal Control of a Model Equation for Shear Flow Instabilities

  • Markus Högberg
  • Martin Berggren


We investigate two different discretization approaches of a model optimal-control problem, chosen to be relevant for control of instabilities in shear flows. In the first method, a fully discrete approach has been used, together with a finite-element spatial discretization, to obtain the objective function gradient in terms of a discretely-derived adjoint equation. In the second method, Chebyshev collocation is used for spatial discretization, and the gradient is approximated by discretizing the continuously-derived adjoint equation. The discrete approach always results in a faster convergence of the conjugate-gradient optimization algorithm. Due to the shear in the convective velocity, a low diffusivity in the problem complicates the structure of the computed optimal control, resulting in particularly noticeable differences in convergence rate between the methods. When the diffusivity is higher, the control becomes less complicated, and the difference in convergence rate reduces. The use of approximate gradients results in a higher sensitivity to the degrees of freedom in time. When the system contains a strong instability, it only takes a few iteration to obtain an effective control for both methods,even if there are differences in the formal convergence rate. This indicates that it is possible to use the approximative gradients of the objective function in cases where the control problem mainly consists of controlling strong instabilities.

shear flow optimal control adjoint equations discrete adjoint exact gradient 


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Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Markus Högberg
    • 1
  • Martin Berggren
    • 2
  1. 1.Department of MechanicsRoyal Institute of TechnologyStockholmSweden
  2. 2.FFA, the Aeronautical Institute of Sweden, and Department of Scientific ComputingUppsala UniversityUppsalaSweden

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