Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects

Medjo, T. Tachim; Témam, Roger; Ziane, Mohammed

doi:10.1115/1.2830523

Cited by 18 publications

(19 citation statements)

References 112 publications

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“…In this case, the gradient can be approximated by using sensitivity methods or methods based on the adjoint equation; see e.g. [1,15,16,51,50,61,84,85]. Efficient (and accurate) solutions can be designed by such methods [1,7,15,29,54,84,85] which may lead however to high-dimensional problems that can turn out to be computationally expensive to solve, especially for fluid flows applications.…”

Section: Introductionmentioning

confidence: 99%

“…[1,15,16,51,50,61,84,85]. Efficient (and accurate) solutions can be designed by such methods [1,7,15,29,54,84,85] which may lead however to high-dimensional problems that can turn out to be computationally expensive to solve, especially for fluid flows applications. The task becomes even more challenging when a dynamic programming approach is adopted, involving typically to solve (infinite-dimensional) Hamilton-Jacobi-Bellman (HJB) equations [8,9,24,34,35,36,37].…”

Section: Introductionmentioning

confidence: 99%

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Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs

Chekroun

Liu

2014

Acta Appl Math

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Abstract. This article proposes a new approach for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. The approach fits into the long tradition of seeking for slaving relationships between the small scales and the large ones (to be controlled) but differ by the introduction of a new type of manifolds to do so, namely the finite-horizon parameterizing manifolds (PMs). Given a finite horizon [0, T ] and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the high modes by the controlled low ones so that the unexplained high-mode energy is reduced -in a mean-square sense over [0, T ] -when this parameterization is applied.Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes introduced in [26]. These formulas allow for an effective derivation of reduced systems of ordinary differential equations (ODEs), aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. The design of low-dimensional suboptimal controllers is then obtained by (indirect) techniques from finite-dimensional optimal control theory, applied to the PM-based reduced ODEs.A priori error estimates between the resulting PM-based low-dimensional suboptimal controller u * R and the optimal controller u * are derived under a second-order sufficient optimality condition. These estimates demonstrate that the closeness of u * R to u * is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with the suboptimal controller u * R and the optimal controller u * ; and (ii) the energy kept in the high modes of the PDE solution either driven by u * R or u * itself. The practical performances of such PM-based suboptimal controllers are numerically assessed for optimal control problems associated with a Burgers-type equation; the locally as well as globally distributed cases being both considered. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs

Chekroun

Liu

2014

Acta Appl Math

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“…The use of alternative metrics beyond L 2 ([0, T ], L 2 ( )) is not new in optimal control of fluids [for a review see Medjo et al (2008)]. In Bewley et al (2001), for example, enstrophy-based metrics and Sobolev-type cost functionals were discussed in the context of turbulence control via boundary forcing for a channel flow.…”

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confidence: 99%

Strong Solvability of a Variational Data Assimilation Problem for the Primitive Equations of Large-Scale Atmosphere and Ocean Dynamics

Korn

2021

J Nonlinear Sci

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For the primitive equations of large-scale atmosphere and ocean dynamics, we study the problem of determining by means of a variational data assimilation algorithm initial conditions that generate strong solutions which minimize the distance to a given set of time-distributed observations. We suggest a modification of the adjoint algorithm whose novel elements is to use norms in the variational cost functional that reflects the $$H^1$$ H 1 -regularity of strong solutions of the primitive equations. For such a cost functional, we prove the existence of minima and a first-order adjoint condition for strong solutions that provides the basis for computing these minima. We prove the local convergence of a gradient-based descent algorithm to optimal initial conditions using the second-order adjoint primitive equations. The algorithmic modifications due to the $$H^1$$ H 1 -norms are straightforwardly to implement into a variational algorithm that employs the standard $$L^2$$ L 2 -metrics.

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“…geometries in zero-flow situations [57,45,41], maximizing mixing [48,24], controlling particular trajectories [5,9], or optimizing fluid mixing across flow barriers [7,4]). Most do not utilize optimal control theory to control particle trajectories but rely on other aspects of optimization, control, or numerical methods (some exceptions: controlling the Navier--Stokes equations [43,31] and multiobjective mixing control [48]). Here, we specifically examine globally controlling trajectories of an existing flow, whose nonautonomous velocities may only be known from observational or experimental data.…”

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confidence: 99%

Controlling Trajectories Globally via Spatiotemporal Finite-Time Optimal Control

Zhang¹,

Balasuriya²

2020

SIAM J. Appl. Dyn. Syst.

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The problems of (i) maximizing or minimizing Lagrangian mixing in fluids via the introduction of a spatiotemporally varying control velocity and (ii) globally controlling the finite-time location of trajectories beginning at all initial conditions in a chaotic system are considered. A particular form of solution to these is designed which uses a new methodology for computing a spatiotemporally dependent optimal control. An L 2 -error norm for trajectory locations over a finite-time horizon is combined with a penalty energy norm for the control velocity in defining the global cost function. A computational algorithm for cost minimization is developed, and theoretical results on global error and cost presented. Numerical simulations (using velocities which are specified, and obtained as data from computational fluid dynamics simulations) are used to demonstrate the efficacy and validity of the approach in determining the required spatiotemporally defined control velocity.

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Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects

Abstract: In this article, we review from the mathematical and numerical points of view some of the recent progresses in the area of flow control.

Cited by 18 publications

References 112 publications

Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs

Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs

Strong Solvability of a Variational Data Assimilation Problem for the Primitive Equations of Large-Scale Atmosphere and Ocean Dynamics

Controlling Trajectories Globally via Spatiotemporal Finite-Time Optimal Control

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