Optimal controllability in space of continuum adhesive systems: an a posteriori approach

Within the framework of Continuum Mechanics, and building upon Geometric Control Theory, we develop a novel approach to analyze the spatial controllability of deformable media. Here, controllability in space refers to the capability of a continuum body to respond to external stimuli and, in conjunction with internal forces, to modify its shape in order to attain a prescribed target configuration. In detail, we derive the field equations governing the equilibrium of an adhesive system and, following a variational-based procedure, we single out the effects arising from three distinct contributions: (i) internal elastic forces, (ii) internal non-elastic forces, and (iii) external forces. The latter can be treated from two complementary perspectives. One is called a priori controllability and relies on the fact that the external force is assigned in advance, by prescribing its functional expression in terms of the system’s state and/or to external (mechanical or non-mechanical) factors. On the other hand, we speak of a posteriori controllability when the external force is not given at the beginning of the problem but computed in order for the adhesive complex to reach a prescribed deformation. Finally, we demonstrate how fundamental results of Geometric Control Theory can be applied to Continuum Mechanics in a clear and rigorous manner, by interpreting controls as spatial functions that drive a given system adhesive toward configurations characterized by prescribed values of displacement and stress.