## ANR Project ANR-20-CE40-0009: TRECOS

New Trends in Control and Stabilization: Constraints and non-local terms

Project founded by the Agence Nationale de la Recherche for 2021-2024, extended up to 2025.
**Abstract:**
The goal of this project is to address new directions of research in control theory for partial differential equations,
triggered by models from ecology and biology. In particular, our projet will deal with the development of new methods
which will be applicable in many applications, from the treatment of cancer cells to the analysis of the thermic
efficiency of buildings, and from control issues for the biological control of pests to cardiovascular fluid flows.
To achieve these objectives, we will have to solve several theoretical issues in order to design efficient control methods.

We have thus targeted four main challenges:

- The first axis of study concerns the control of parabolic systems, in particular when the control does not act on all the components of the system. In this case, the main difficulty comes from the fact that the components of the system in which the control does not act have to be controlled indirectly through the coupling with the other components of the system. Our goal is to better understand the control properties of such systems, and to develop new robust techniques, which can be applied in particular in nonlinear settings.
- The second axis will focus on the analysis of the control properties of partial differential equations containing nonlocal terms. Such nonlocal terms appear in many applications, as soon as the dynamics of the system depends on global quantities, or locally averaged quantities. This is for instance the case in population dynamics, where the total number of birth depends on the whole population. New tools should be developed in this context, where the classical techniques developed so far in control theory do not apply.
- The third point of study aims at addressing in depth major applications involving fluid mechanics. We will in particular study the modeling of the thermic efficiency of buildings and its related control issues, the modeling of cardiovascular flows and stabilization issues around periodic trajectories in models of interactions of viscous or viscoelastic flow (modeling the blood) and an elastic structure (modeling the blood vessels), and the design of waves energy converter of optimal efficiency.
- The fourth axis of study concerns the control of partial differential equations when the trajectories are required to satisfy some constraints. The study of such question has started very recently and should be developed further and extended to many models, in particular in order to take into account positivity constraints on controlled trajectories, or on some components of it, which are required for the feasibility of the control strategies. This question appears very naturally when some quantities are intrinsically positive, which is the case for instance when they model a population density or a concentration of a chemical reactant.