  We present some existence results for three-dimensional quasistatic
morphoelasticity. The state of the growing body is described by its deformation
and the underlying growth tensor and is ruled by the interplay of hyperelastic
energy minimization and growth dynamics. By introducing a regularization in the
model, we prove that solutions can be obtained as limits of time-discrete
solutions, built by means of an exponential-update scheme. By further allowing
the dependence of growth dynamics on an additional scalar field, to be
interpreted as a nutrient or inhibitor, we formulate an optimal control problem
and prove existence of optimal controls and states. Eventually, we tackle the
existence of coupled morphoelastic and nutrient solutions, when the latter is
allowed to diffuse and interact with the growing body.
