Dycon Logo From finite to infinite-dimensional models (FI)

Our team has made some high impact contributions in the description of the limit, as the mesh sizes tend to zero, of numerical schemes for wave equations. These results also provide insight into the link between conservative finite and infinite-dimensional dynamical systems. We have also developed the theoretical control consequences of these facts, which show that, in particular, filtering of high frequency numerical spurious solutions is necessary.

However, the interplay between finite and infinite-dimensional dynamics in control arises in different ways in applications such as collective dynamics and pedestrian flow (see the 2014 SIAM Review paper by S. Motsch and E. Tadmor) or material sciences (see the 2009 lecture Notes by B. L. Pego). Effective models are often described by continuous PDEs involving non-local (in space) terms, modelling interactions between agents.

One of our main goals will be to develop a control theory from finite to infinite-dimensional dynamics with these applications in mind. This will require a deep effort to cope with the non-linear and non-local effects, and the fact that the control often appears in the nonlinear terms and not as a right-hand side source term.

Special attention will be devoted to develop numerical schemes preserving the asymptotic properties of the PDE where this issue was addressed for Kolmogorov equation, providing numerical schemes preserving hypocoercivity and hypoellipticity properties (as in the works of Hérau, Desvillettes, Villani etc.).

This working package will provide new tools (theoretical and computational) when dealing with the important problem of finite versus infinite-dimensional control that so far was mostly developed only in the context of numerical approximation of PDE.