A priori and a posteriori controllability of volumetric growth: a theoretical approach

We propose a theoretical setting in which volumetric growth, grounded in the field of Continuum Mechanics, is addressed within the framework of Geometric Control Theory. To the best of our knowledge, this approach has never been explored before and represents a novelty in both disciplines. Specifically, we integrate the modeling of volumetric growth in continuous systems with the main tools of Geometric Control Theory to study how the interactions exchanged with the external environment must be coordinated to reach a desired grown configuration. In this regard, we generalize some results available in the literature and investigate two different perspectives: the a priori and the a posteriori controllability. In the first case, the contribution of external interactions is given in terms of a non-conventional force, whose functional dependence on the system’s state is prescribed. Within the second case, non-conventional forces are not assigned, but determined as a control function which, obeying certain controllability criteria, is able to drive the system toward preferred configurations. We thus apply Geometric Control Theory and, by focusing on the geometrical properties of the system of differential equations describing growth, we specialize the notions of controllability/non-controllability to growth dynamics and infer analytical results in terms of conditions on the material parameters of the system. This way, we ensure the capability of controlling growth and/or elastic generalized trajectories. To test the results of our work, a benchmark example is finally considered.