Our team has made several contributions in the description of the limit behaviour, as the mesh sizes tend to zero, of **numerical schemes for wave and Schrödinger equations** from a control theoretical perspective. These results show that, in particular, filtering the high frequency numerical spurious solutions is necessary (and a good remedy) to assure the convergence of numerical schemes from a control perspective.

These results also provide insight into the link between **conservative finite** and **infinite-dimensional dynamical systems** and their asymptotic behaviour.

However, the interplay between finite and infinite-dimensional dynamics in control arises in other contexts as well, such as in applications to **collective dynamics** and **pedestrian flow** or **in material sciences**. The corresponding effective mean-field models are often described by continuous PDEs involving non-local (in space) terms, modelling interactions between agents, and this adds significant novelties to the qualitative behaviour of these systems and raises new interesting problems from a control theoretical perspective.

One of our main is further developing the needed control theory to link finite to infinite-dimensional dynamics, with these applications in mind. This requires a significant effort to cope with the non-linear and non-local effects and the fact that the control often appears in a bilinear manner and not as a right-hand side source term.

Special attention will also be devoted to developing numerical schemes preserving the asymptotic properties of the PDE. This issue was addressed for the **Kolmogorov equation**, providing numerical schemes preserving **hypocoercivity** and **hypoellipticity** properties of the continuous PDE.