The SQP method for control constrained optimal control of the Burgers equation

Tröltzsch, Fredi; Volkwein, Stefan

doi:10.1051/cocv:2001127

Cited by 27 publications

(32 citation statements)

References 15 publications

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“…Following the concepts in [1,12,30], for a proof of Theorem 6.3 we first investigate the strong regularity of (6.1) at d * = (y * , u * , p * ). This is the contents of the next lemma.…”

Section: Convergence Analysis Of the Sqp-iterationmentioning

confidence: 99%

“…We point out that the approach taken in the proof of Lemma 6.4 is related to the technique utilized in [30,.…”

Section: Convergence Analysis Of the Sqp-iterationmentioning

confidence: 99%

“…Once we have established the strong regularity of (6.1) at d * , the proof of Theorem 6.3, i.e., the locally quadratic convergence rate for (6.2), follows from [1,12]; see also [30,Theorem 3.3]. Since the Newton process (6.2) is equivalent to the SQPiteration in Algorithm 4.1 we readily obtain the same locally quadratic convergence rate for {(x n , p n )} produced by Algorithm 4.1.…”

Section: Convergence Analysis Of the Sqp-iterationmentioning

confidence: 99%

“…For the analysis of the SQP-method (outer iteration) a generalized equations approach is utilized. In a different context, the potential of generalized equations for the analysis of algorithms for constrained optimal control problems was exploited previously in, e.g., [1,12,30] and the references therein. Under a strong regularity property (see [27]) we prove that the SQP-iteration converges at a locally quadratic rate.…”

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confidence: 99%

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A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints

Hintermüller¹,

Hinze²

2006

SIAM J. Optim.

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Abstract. Sequential quadratic programming (SQP) methods for the optimal control of the instationary Navier-Stokes equations with pointwise constraints on the control are considered. Due to the presence of the constraints the quadratic subproblems (QP) of SQP require a more sophisticated solver compared to the unconstrained case. In this paper, a semismooth Newton method is proposed for efficiently solving the QPs. The convergence analysis, which is performed in an appropriate function space setting, relies on the concept of slant differentiability for proving locally superlinear convergence of the QP-solver. For the analysis of the outer SQP-iteration a generalized equations approach is utilized. Sufficient conditions for guaranteeing strong regularity of the generalized equation are established which, in turn, allows to argue a quadratic rate of convergence of the SQP-method. The paper ends with a report on numerical results supporting the theoretical findings.

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“…Following the concepts in [1,12,30], for a proof of Theorem 6.3 we first investigate the strong regularity of (6.1) at d * = (y * , u * , p * ). This is the contents of the next lemma.…”

Section: Convergence Analysis Of the Sqp-iterationmentioning

confidence: 99%

“…We point out that the approach taken in the proof of Lemma 6.4 is related to the technique utilized in [30,.…”

Section: Convergence Analysis Of the Sqp-iterationmentioning

confidence: 99%

Section: Convergence Analysis Of the Sqp-iterationmentioning

confidence: 99%

mentioning

confidence: 99%

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A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints

Hintermüller¹,

Hinze²

2006

SIAM J. Optim.

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“…The Lagrange-Newton technique will be the first approach which we consider in this paper. We will give a rather compact description referring to [19,24,26] for results on convergence.…”

Section: Introductionmentioning

confidence: 99%

Adjoint concepts for the optimal control of Burgers equation

Noack

Walther

2006

Comput Optim Applic

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Adjoint techniques are a common tool in the numerical treatment of optimal control problems. They are used for efficient evaluations of the gradient of the objective in gradientbased optimization algorithms. Different adjoint techniques for the optimal control of Burgers equation with Neumann boundary control are studied. The methods differ in the point in the numerical algorithm at which the adjoints are incorporated. Discretization methods for the continuous adjoint are discussed and compared with methods applying the application of the discrete adjoint. At the example of the implicit Euler method and the Crank Nicolson method it is shown that a discretization for the adjoint problem that is adjoint to the discretized optimal control problem avoids additional errors in gradient-based optimization algorithms. The approach of discrete adjoints coincides with that of automatic differentiation tools (AD) which provide exact gradient calculations on the discrete level.

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Distributed optimal control of the viscous Burgers equation via a Legendre pseudo‐spectral approach

Sabeh

Shamsi

Dehghan

2015

Math Methods in App Sciences

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Communicated by Y. S. XuThis paper presents a computational technique based on the pseudo-spectral method for the solution of distributed optimal control problem for the viscous Burgers equation. By using pseudo-spectral method, the problem is converted to a classical optimal control problem governed by a system of ordinary differential equations, which can be solved by welldeveloped direct or indirect methods. For solving the resulting optimal control problem, we present an indirect method by deriving and numerically solving the first-order optimality conditions. Numerical tests involving both unconstrained and constrained control problems are considered.

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The SQP method for control constrained optimal control of the Burgers equation

Cited by 27 publications

References 15 publications

A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints

A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints

Adjoint concepts for the optimal control of Burgers equation

Distributed optimal control of the viscous Burgers equation via a Legendre pseudo‐spectral approach

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