Robust optimal Robin boundary control for the transient heat equation with random input data

Martínez-Frutos, Jesús; Kessler, Mathieu; Münch, Arnaud; Periago, Francisco

doi:10.1002/nme.5210

Cited by 18 publications

(7 citation statements)

References 24 publications

(46 reference statements)

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“…We will be able to compare our numerical results with experimental measurements collected on the Egletons demonstrator described in Section 1. Following [13], it is also interesting to analyze in our nonlinear setting, the influence on the optimal controls of the perturbation of the data f 1 , f 2 and ε, due notably to measurements.…”

Section: Resultsmentioning

confidence: 99%

Modeling and optimizing a road de-icing device by a nonlinear heating

Bernardin

Münch

2019

ESAIM: M2AN

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In order to design a road de-icing device by heating, we consider in the one dimensional setting the optimal control of a parabolic equation with a nonlinear boundary condition of the Stefan–Boltzmann type. Both the punctual control and the corresponding state are subjected to a unilateral constraint. This control problem models the heating of a road during a winter period to keep the road surface temperature above a given threshold. The one-dimensional modeling used in this work is a first step of the modeling of a road heating device through the circulation of a coolant in a porous layer of the road. We first prove, under realistic physical assumptions, the well-posedness of the direct problem and the optimal control problem. We then perform some numerical experiments using real data obtained from experimental measurements. This model and the corresponding numerical results allow to quantify the minimal energy to be provided to keep the road surface without frost or snow.

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Section: Resultsmentioning

confidence: 99%

Modeling and optimizing a road de-icing device by a nonlinear heating

Bernardin

Münch

2019

ESAIM: M2AN

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“…This is a standard computation so it is omitted here. See the works of Rosseel and Wells and Martínez‐Frutos et al for similar situations.…”

Section: Numerical Resolution Of (P)mentioning

confidence: 99%

“…Although controlling the average of the state variable provides a first idea of robustness, it may be useless whether the dispersion of the system response is large. In the context of optimal control theory, the problem of controlling both the average and the variance of the state variable has been studied in other works at the theoretical and numerical points of view. Significant differences between optimal controls, minimizing only the average of the state variable and controls minimizing both the mean and the variance of the state variable, were numerically observed in those references.…”

Section: Introductionmentioning

confidence: 99%

A polynomial chaos‐based approach to risk‐averse piezoelectric control of random vibrations of beams

Marín

Martínez-Frutos

Periago

2018

Numerical Meth Engineering

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Summary This paper proposes a risk‐averse formulation for the problem of piezoelectric control of random vibrations of elastic structures. The proposed formulation, inspired by the notion of risk aversion in economy, is applied to the piezoelectric control of a Bernoulli‐Euler beam subjected to uncertainties in its input data. To address the high computational burden associated to the presence of random fields in the model and the discontinuities involved in the cost functional and its gradient, a combination of a nonintrusive anisotropic polynomial chaos approach for uncertainty propagation with a Monte Carlo sampling method is proposed. In the first part, the well‐posedness of the control problem is established by proving the existence of optimal controls. In the second part, an adaptive gradient‐based method is proposed for the numerical resolution of the problem. Several experiments illustrate the performance of the proposed approach and the significant differences that may occur between the classical deterministic formulation of the problem and its stochastic risk‐averse counterpart.

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“…In [59] a parabolic PDE system concerning stochastic inverse heat conduction problems was investigated, where the thermal conductivity and heat capacity coefficients were assumed to be random fields. In [48] the authors studied optimization of thermal processes described as parabolic PDEs by considering uncertainties in the thermal conductivity, convective heat transfer coefficients, initial data of the process as well as control variables. The work in [4,49] investigated optimal shape design problems with elliptic PDE models, where uncertainties arise from material, geometrical and loading properties.…”

Section: Introductionmentioning

confidence: 99%

Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation

et al. 2020

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In this paper, we consider chance constrained optimization of elliptic partial differential equation (CCPDE) systems with random parameters and constrained state variables. We demonstrate that, under standard assumptions, CCPDE is a convex optimization problem. Since chance constrained optimization problem are generally nonsmooth and difficult to solve directly, we propose a smoothing inner-outer approximation method to generate a sequence of smooth approximate problems for the CCPDE. Thus, the optimal solution of the convex CCPDE is approximable through optimal solutions of the inner-outer approximation problems. A numerical example demonstrates the viability of the proposed approach.

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Robust optimal Robin boundary control for the transient heat equation with random input data

Cited by 18 publications

References 24 publications

Modeling and optimizing a road de-icing device by a nonlinear heating

Modeling and optimizing a road de-icing device by a nonlinear heating

A polynomial chaos‐based approach to risk‐averse piezoelectric control of random vibrations of beams

Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation

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