Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems

Vossen, Georg; Volkwein, Stefan

doi:10.3934/naco.2012.2.465

Cited by 13 publications

(13 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This, and its ease of use makes POD very competitive in practical use, despite of a certain heuristic flavor. In this work, we review results for a POD a-posteriori analysis; see, e.g., [73] and [18,35,36,70,71,76,78]. We use a fairly standard perturbation method to deduce how far the suboptimal control, computed on the basis of the POD model, is from the (unknown) exact one.…”

Section: Introductionmentioning

confidence: 99%

Chapter 1: Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control

Gubisch¹,

Volkwein²

2017

Model Reduction and Approximation

Self Cite

139

View full text Add to dashboard Cite

Section: Introductionmentioning

confidence: 99%

Chapter 1: Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control

Gubisch¹,

Volkwein²

2017

Model Reduction and Approximation

Self Cite

139

View full text Add to dashboard Cite

“…During the review process of this paper, the proposed method has been combined with a posteriori error analysis . Further extensions are possible for time‐variant or nonlinear systems.…”

Section: Discussionmentioning

confidence: 99%

“…Note that this requires a certain minimal number of time steps although, for our examples, the number is still moderate and standard tools for nonlinear optimization are applicable. Note that backward time discretization leads to an implicit LTI system where the standard model reduction implementation balreal for Balanced Truncation in MATLAB and IRKA is not applicable, compared with for details for possible extensions. The composite trapezoidal rule is used for the integrals in the cost functional.…”

Section: Numerical Examplesmentioning

confidence: 99%

_2,α‐norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations

Vossen

2013

Optim Control Appl Methods

View full text Add to dashboard Cite

SUMMARYThis paper deals with a numerical solution method for optimal control problems subject to parabolic and hyperbolic evolution equations. Firstly, the problem is semi‐discretized in space with the boundary or distributed controls as input and those parts of the discretized state appearing in the cost functional as output variables. The corresponding transfer function is then approximated optimally with respect to the scriptH2,α‐norm providing an optimally reduced optimal control problem, which is finally solved by a first‐discretize‐then‐optimize approach. To enable the application of this reduction method, a new constrained optimal model reduction problem subject to reduced systems with real system matrices is considered. Necessary optimality conditions and a transformation procedure for the reduced system to a canonical form of real matrices are presented. The method is illustrated with numerical examples where also complicated controls with many bang‐bang arcs are investigated. The approximation quality of the optimal control and its correlation to the decay rate of the Hankel singular values of the system are numerically studied. A comparison to the approach of using Balanced Truncation for model reduction is applied. Copyright © 2013 John Wiley & Sons, Ltd.

show abstract

“…A posteriori error bounds have been proposed previously for POD approximations in [31] to estimate the distance between the computed suboptimal control and the unknown optimal control; also see [30,35] for an application of this approach to other model order reduction techniques. In [4,6], reduced basis approximations and associated a posteriori error estimation procedures have been derived to estimate the error in the optimal value of the cost functional.…”

Section: Introductionmentioning

confidence: 99%

A certified reduced basis method for parametrized elliptic optimal control problems

Kärcher

Grepl

2014

ESAIM: COCV

View full text Add to dashboard Cite

In this paper, we employ the reduced basis method as a surrogate model for the solution of linear-quadratic optimal control problems governed by parametrized elliptic partial differential equations. We present a posteriori error estimation and dual procedures that provide rigorous bounds for the error in several quantities of interest: the optimal control, the cost functional, and general linear output functionals of the control, state, and adjoint variables. We show that, based on the assumption of affine parameter dependence, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. Numerical results are presented to confirm the validity of our approach.Mathematics Subject Classification. 49K20, 49M29, 35J25, 65N15, 65K05, 93C20.

show abstract

Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems

Cited by 13 publications

References 33 publications

Chapter 1: Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control

Chapter 1: Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control

_2,α‐norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations

A certified reduced basis method for parametrized elliptic optimal control problems

Contact Info

Product

Resources

About