- 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.