  In this paper, we introduce the concept of Developmental Partial Differential
Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a
time-varying manifold with complete coupling between the PDE and the manifold's
evolution. In other words, the manifold's evolution depends on the solution to
the PDE, and vice versa the differential operator of the PDE depends on the
manifold's geometry. DPDE is used to study a diffusion equation with source on
a growing surface whose growth depends on the intensity of the diffused
quantity. The surface may, for instance, represent the membrane of an egg
chamber and the diffused quantity a protein activating a signaling pathway
leading to growth. Our main objective is to show controllability of the surface
shape using a fixed source with variable intensity for the diffusion. More
specifically, we look for a control driving a symmetric manifold shape to any
other symmetric shape in a given time interval. For the diffusion we take
directly the Laplace-Beltrami operator of the surface, while the surface growth
is assumed to be equal to the value of the diffused quantity. We introduce a
theoretical framework, provide approximate controllability and show numerical
results. Future applications include a specific model for the oogenesis of
Drosophila melanogaster.
