This project aims at developing theoretical and applied results in the fields of Control Theory and study of Physical Networks. More generally, the project develops and coordinates research in this domain in applied mathematics and in engineering sciences. Indeed, it combines the research activities of researchers from mathematics departments (Eduardo Cerpa and Nicolás Carreño) and from Systems Theory and Automatic Control research centers (Christophe Prieur and Federico Bribiesca Argomedo). The project contributes towards the consolidation of the French and Chilean mathematical and control theory community and to its integration within the international community.

This project focuses on the control of dynamical systems and its applications to physical networks.

On the theoretical plan we will focus on two main research lines: stabilization and observation of systems modeled by partial differential equations (PDE) and stability of networked control systems under constraints on either the control or on the concurrent communication capacity. The interest on the study of control properties of infinite dimensional systems under constraints is rather recent but very active. Among the different classes of constraints that we have in mind, let us cite

• magnitude limitations on the inputs (in the form of saturated controls);
• the consideration of nonlinearities in the dynamics when solving the control problem. Such nonlinearities may be either distributed (e.g. a damping along the space domain), or part of a boundary condition (e.g. a destabilizing nonlinear ordinary differential equation coupled with a wave equation);
• the effect of communication constraints when considering networks. It is known that for many networks, it is not possible to control at all times, but only at some sampling times, and through suitable information quantization. These constraints require special care, due to the coupling of discrete-time events with the continuous-time dynamics of the plant.

As for the applications to physical systems, we will consider several potential applications, in particular the drilling problem in oil production systems in the hyperbolic case. The physical length of the considered systems (several kilometers) implies the existence of mechanical wave propagation, which has to be explicitly taken into account in the modeling and in the control. The design of suitable control strategies asks to deal with limit structural vibrations. Among other possibilities, considered solutions include: Backstepping method controller for uncertain variables; Observer design using only measurements available at the surface of the well; Adaptive controller or observer to reduce the impact of modeling uncertainties.

Another potential application is the boundary control of information networks. It is known that controlling networks and servers is crucial to optimize the service-on-demand of information over the Internet. The natural model for information transmission through networks system of conservation laws in interaction with finite-dimension models to take into account the state of the buffers in the network. Also, these results can have implications in much larger classes of systems, such as power grids and fuel distribution networks.

As for coupled parabolic PDEs, we will focus on the applications to observer design and parameter identification for electrochemical battery models. The main challenge in this application is the limited nature of the measurements available for the information reconstruction and the difficulty to characterize several of the electrochemical parameters without extensive (and specialized) experiments.

Besides the main goal of creating/consolidating scientific connections between Chile and France, we have the following scientific goals:

• Study of the control properties of some nonlinear PDEs, and use of nonlinear control laws (as saturating inputs) for systems describing the propagation of surface and internal waves (Korteweg-de Vries and Schrodinger equations).
• Use of backstepping techniques for the boundary control of hyperbolic systems in presence of destabilizing boundary conditions and/or distributed damping along the media. This subject is clearly connected with the drilling application that has been presented in Section 5.1.
• We will consider also control systems where communication takes place over non-transparent communication links. To do that, we will model networks by systems of conservation laws in presence of quantization (since the transmitted information are often quantized) and samplings in both outputs and inputs (since it is often not possible to control at any time).
• Study of observability properties and observer design to extend existing results on state estimation for coupled systems of parabolic PDEs with few available measurements, with a particular focus on electrochemical battery models.