Shape turnpike for linear parabolic PDE models

Lance G, Trélat E, Zuazua Iriondo E (2020)

Publication Type: Journal article

Publication year: 2020

Journal

Systems & Control Letters Elsevier

Book Volume: 142

Article Number: 104733

DOI: 10.1016/j.sysconle.2020.104733

Abstract

We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in terms of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations.

Authors with CRIS profile

Enrique Zuazua Iriondo Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Additional Organisation(s)

Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

Sorbonne Université FR France (FR)

How to cite

APA:

Lance, G., Trélat, E., & Zuazua Iriondo, E. (2020). Shape turnpike for linear parabolic PDE models. Systems & Control Letters, 142. https://doi.org/10.1016/j.sysconle.2020.104733

MLA:

Lance, Gontran, Emmanuel Trélat, and Enrique Zuazua Iriondo. "Shape turnpike for linear parabolic PDE models." Systems & Control Letters 142 (2020).

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