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.