## Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Dirichlet Boundary Control of Navier-Stokes Flow (1997)

Citations: | 15 - 1 self |

### BibTeX

@MISC{Heinkenschloss97formulationand,

author = {Matthias Heinkenschloss},

title = {Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Dirichlet Boundary Control of Navier-Stokes Flow},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

The optimal boundary control of Navier--Stokes flow is formulated as a constrained optimization problem and a sequential quadratic programming (SQP) approach is studied for its solution. Since SQP methods treat states and controls as independent variables and do not insist on satisfying the constraints during the iterations, care must be taken to avoid a possible incompatibility of Dirichlet boundary conditions and incompressibility constraint. In this paper, compatibility is enforced by choosing appropriate function spaces. The resulting optimization problem is analyzed. Differentiability of the constraints and surjectivity of linearized constraints are verified and adjoints are computed. An SQP method is applied to the optimization problem and compared with other approaches.

### Citations

1498 |
Sobolev spaces
- ADAMS
- 1975
(Show Context)
Citation Context ..., we consider the closed subspace G = ae g 2 H 1 0 (\Gamma c ) j Z \Gamma c g \Delta n dx = 0 oe ; of H 1 0 (\Gamma c ). For details on these and other function spaces used in this paper, we refer to =-=[3]-=-, [12], [26]. Let fl : H 1 (\Omega\Gamma ! H 1=2 (\Gamma) be the continuous trace operator. If we multiply (2), (3) by sufficiently smooth test functions v and q, respectively, then we are lead to the... |

766 |
Finite element methods for Navier-Stokes equations
- Girault, Raviart
- 1986
(Show Context)
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217 |
Updating quasi-newton matrices with limited storage
- Nocedal
- 1980
(Show Context)
Citation Context ...l optimization) often cannot be done easily. The computation of H k h in the infinite dimensional context can proceed similar to the implementation of limited memory quasi-- Newton updates, see e.g., =-=[25]-=-. Only the Euclidean scalar product has to be replaced by the infinite dimensional one. The system in step 5 is equivalent to computing s u 2 V withsa(s u ; v) + b(s u ; u b k ; v) + b(u b k ; s u ; v... |

115 | Sequential quadratic programming
- Boggs, Tolle
- 1995
(Show Context)
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- VICENTE
- 1996
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38 |
On some control problems in fluid mechanics
- Abergel, Temam
(Show Context)
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33 | A reduced Hessian method for large-scale constrained optimization
- Biegler, Nocedal, et al.
- 1995
(Show Context)
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29 |
Swobodny: Analysis and finite element approximation of optimal control problems for the stationary Navier–Stokes equations with Dirichlet controls
- Gunzburger, Hou, et al.
- 1991
(Show Context)
Citation Context ...ga\Gamma + ff 2 kgk 2 G ; (6) J(u; g) = 1 2 Z D (u 2 ) 2 dx + ff 2 kgk 2 G ; (7) and J(u; g) = 1 2 Z D / @u 2 @x 1 \Gamma @u 1 @x 2 ! 2 dx + ff 2 kgk 2 G : (8) The objective (6) with p = 4 is used in =-=[14]-=-, [16] for flow matching. Flow separation along a horizontal line D in two dimensions (N = 2) leads to (7), see [10], and minimization of the vorticity in a subdomain D ae\Omega for N = 2 leads to the... |

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Augmented Lagrangian–SQP methods for nonlinear optimal control problems of tracking type
- ITO, KUNISCH
- 1996
(Show Context)
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An analysis of reduced Hessian methods for constrained optimization
- Byrd, Nocedal
- 1991
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Citation Context ...n update used for H(x; ). It is beyond the scope of the paper to give an overview over the possible choices of H and d and their local convergence analysis. We refer to the literature, e.g. [6], [7], =-=[8]-=-, [22], for a detailed discussion. To be specific, we state one version of a reduced SQP method below. It uses BFGS updates to approximate the reduced Hessian and d = 0. Algorithm 2.1 (Reduced SQP--BF... |

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- Alt, Malanowski
- 1993
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Citation Context ... in [10]. While we focus on the SQP method, the study in this paper is also applicable to other methods, e.g., the augmented Lagrangian--SQP method in [20], and in particular Lagrange--Newton methods =-=[4]-=-, [27]. We comment more on this at the end of Section 2. The paper is organized as follows. In x 2 we review the SQP method to establish notation and to provide the background for the presentation in ... |

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- GUNZBURGER, HOU, et al.
- 1993
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Citation Context ...ma + ff 2 kgk 2 G ; (6) J(u; g) = 1 2 Z D (u 2 ) 2 dx + ff 2 kgk 2 G ; (7) and J(u; g) = 1 2 Z D / @u 2 @x 1 \Gamma @u 1 @x 2 ! 2 dx + ff 2 kgk 2 G : (8) The objective (6) with p = 4 is used in [14], =-=[16]-=- for flow matching. Flow separation along a horizontal line D in two dimensions (N = 2) leads to (7), see [10], and minimization of the vorticity in a subdomain D ae\Omega for N = 2 leads to the funct... |

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- Desai, Ito
- 1994
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Citation Context ...Gamma @u 1 @x 2 ! 2 dx + ff 2 kgk 2 G : (8) The objective (6) with p = 4 is used in [14], [16] for flow matching. Flow separation along a horizontal line D in two dimensions (N = 2) leads to (7), see =-=[10]-=-, and minimization of the vorticity in a subdomain D ae\Omega for N = 2 leads to the functional (8). For a discussion of other objective functions see, e.g., [16]. Optimal control of Navier--Stokes fl... |

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Citation Context ... then xsis a solution of (13). We now give a brief derivation of the SQP method. This presentation is rather standard and follows the ones given in [9], [18], [22], [23]. See also the presentation in =-=[11]-=-. It is included to establish some notation and to motivate the use of the SQP method. In the following we assume that C u (x) is continuously invertible for all x under consideration. This implies th... |

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- Heinkenschloss
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Citation Context ... 2 8h 2 fwjC x (xs)w = 0g (16) for some oe ? 0, then xsis a solution of (13). We now give a brief derivation of the SQP method. This presentation is rather standard and follows the ones given in [9], =-=[18]-=-, [22], [23]. See also the presentation in [11]. It is included to establish some notation and to motivate the use of the SQP method. In the following we assume that C u (x) is continuously invertible... |

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- Temam
- 1977
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- 1994
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Citation Context ...10]. While we focus on the SQP method, the study in this paper is also applicable to other methods, e.g., the augmented Lagrangian--SQP method in [20], and in particular Lagrange--Newton methods [4], =-=[27]-=-. We comment more on this at the end of Section 2. The paper is organized as follows. In x 2 we review the SQP method to establish notation and to provide the background for the presentation in the su... |

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A prospective look at SQP methods for semilinear parabolic control problems, in Optimal control of partial differential equations
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- 1991
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Citation Context ... x (xs)w = 0g (16) for some oe ? 0, then xsis a solution of (13). We now give a brief derivation of the SQP method. This presentation is rather standard and follows the ones given in [9], [18], [22], =-=[23]-=-. See also the presentation in [11]. It is included to establish some notation and to motivate the use of the SQP method. In the following we assume that C u (x) is continuously invertible for all x u... |

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Citation Context ...quations. SQP methods find a solution of the constrained nonlinear minimization problem by solving a sequence of quadratic minimization problems. Although they have been successfully applied in [11], =-=[17]-=- to the solution of optimal control of Navier--Stokes flow, a rigorous theoretical justification is missing. This paper is a contribution towards that goal. We formulate and analyze an SQP method for ... |

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