  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.
