Shape turnpike for linear parabolic PDE models
Lance G, Trélat E, Zuazua E (2020)
Publication Type: Journal article
Publication year: 2020
Journal
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)
Involved external institutions
France (FR)How to cite
APA:
Lance, G., Trélat, E., & Zuazua, E. (2020). Shape turnpike for linear parabolic PDE models. Systems & Control Letters, 142. https://dx.doi.org/10.1016/j.sysconle.2020.104733
MLA:
Lance, Gontran, Emmanuel Trélat, and Enrique Zuazua. "Shape turnpike for linear parabolic PDE models." Systems & Control Letters 142 (2020).
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