Shape turnpike for linear parabolic PDE models

TL;DR

This paper investigates the turnpike property for optimal control of time-varying shapes governed by linear parabolic PDEs, demonstrating that optimal solutions stay close to static solutions over time.

Contribution

It establishes the existence, necessary conditions, and exponential turnpike property for optimal shapes in linear parabolic PDE control problems, with numerical illustrations.

Findings

Optimal solutions exhibit measure-turnpike behavior.

Exponential turnpike property holds in Hausdorff distance.

Numerical simulations confirm the theoretical results.

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 term of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations.