  Heterogeneous growth plays an important role in the shape and pattern
formation of thin elastic structures ranging from the petals of blooming lilies
to the cell walls of growing bacteria. Here we address the stability and
regulation of such growth, which we modeled as a quasi-static time evolution of
a metric, with fast elastic relaxation of the shape. We consider regulation via
coupling of the growth law, defined by the time derivative of the target
metric, to purely local properties of the shape, such as the local curvature
and stress. For cylindrical shells, motivated by rod-like E. \textit{coli}, we
show that coupling to curvature alone is generically linearly unstable and that
additionally coupling to stress can lead to stably elongating structures. Our
approach can readily be extended to gain insights into the general classes of
stable growth laws for different target geometries.
