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Shape turnpike, numerical control and optimal design for PDEs

Gontran Lance 1, 2
2 CaGE - Control And GEometry
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : This thesis, which is in the field of optimal control of PDEs and shape optimization, addresses both theoretical and numerical issues. The first step of the thesis is the demonstration of the turnpike property for shape optimization problems or "shape turnpike". Indeed, for a quadratic cost under the constraints of a parabolic PDE controlled by a time-dependent domain acting as a source term, we manage to show by a relaxation method that the time-optimal shape is most of the time close to a stationary shape solution of a stationary optimization problem. Numerical simulations illustrate this result and have shown the need for an adapted numerical formulation. In the second part, we thus develop a general methodology for the numerical solution of optimal control problems with the FreeFEM and IpOpt softwares. Through several examples, from the simple linear quadratic case to the more complicated case of the optimal design of a micro-swimmer, using adjoint methods or automatic differentiation, we illustrate the effectiveness of the combination of FreeFEM and IpOpt as an optimization tool. Finally, we conclude with a description of the numerical methods used for the problem of the first part and give several propositions for future research.
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Submitted on : Friday, December 17, 2021 - 2:33:07 PM
Last modification on : Thursday, April 7, 2022 - 1:58:33 PM
Long-term archiving on: : Friday, March 18, 2022 - 7:13:39 PM

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  • HAL Id : tel-03486151, version 1

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Gontran Lance. Shape turnpike, numerical control and optimal design for PDEs. Optimization and Control [math.OC]. Sorbonne Université, 2021. English. ⟨NNT : 2021SORUS238⟩. ⟨tel-03486151⟩

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