14:00h – 14:40h

Observability and control of parabolic equations on networks

During the last decades, the use of networks has been very helpful and effective in the
study of pipes, neural systems, the flow of traffic on roads, the global economy or the
human circulatory systems.
In this talk I would like to show you a contribution to this area from the fields of control
theory and inverse problems. We will consider the propagation of diffusion on a network
with loops. Our objective is to control these networks by acting on the system that
models the process of the heat diffusion in them, both by open-loop and closed-loop
controls, extending in this way the results of [2] and [3] to networks with loops.
The observability of the entire network will be achieved under certain hypotheses about
the position of the observation domain. This will be done using a Carleman inequality.
Then, we will use that observability to prove the null controllability of the network and
to obtain the Lipschitz stability for an inverse problem consisting of retrieving a
stationary potential in the heat equation from measurements on the observation
domain.
This work has been done in collaboration with Jon Asier Bárcena-Petisco, from the
University of the Basque Country, and is based on the article [1].

References
[1] J. Apraiz and J. A. Bárcena-Petisco, Observability and control of parabolic equations
on networks with loops, Journal of Evolution Equations 23, 37, 2023,
https://doi.org/10.1007/s00028-023-00882-2 .
[2] J. A. Bárcena-Petisco, M. Cavalcante, G. M. Coclite, N. Nitti and E. Zuazua, Control of
hyperbolic and parabolic equations on networks and singular limits, hal-03233211
(2021).
[3] L. Ignat, A. F. Pazoto and L. Rosier, Inverse problem for the heat equation and the
Schrödinger equation on a tree, Inverse Prob. 28, 1, (2011), 015011.

14:50h – 15:30h

Homographic minimal motions in the N body problem: Scattering and Instabilty

14:00h – 14:40h

Existence of Travelling Waves for a class of Higher order Boussinesq system

14:50h – 15:30h

Forcing rapid dissipation of Navier-Stokes flows

14:00h – 14:40h

Nash strategies for boundary data completion coupled to shape and distribuited parameter identification problems

 Inverse problems such as data recovery, obstacle detection, or the identification of coefficients in distributed systems are known to be severely ill-posed in the sense of Hadamard, and a major effort of current research in this field focuses on the development and study of efficient and stable algorithms. In this talk, not very far from the well-trodden paths of regularization and decomposition, we investigate the capabilities of game theory in addressing such problems. We show how this formalism is used to design original and effective algorithms for the numerical solution of these problems, including those that were previously inaccessible due to the lack of an ad hoc formulation which accounts for the coupling of ill-posed inverse problems. This work is based on approaches developed in [1, 2, 3, 4].

[1] Ouni, Marwa, Abderrahmane Habbal, and Moez Kallel.
A three-player Nash game for point-wise source identification in Cauchy–Stokes problems.
  Journal of Computational and Applied Mathematics 417 (2023): 114613.

[2] Chamekh, R., Habbal, A., Kallel, M. and Zemzemi, N.
A Nash game algorithm for the solution of coupled conductivity identification and data completion in cardiac electrophysiology,
  Mathematical Modelling of Natural Phenomena 14, 2, February 2019, 15 pages.

[3] A. Habbal, M. Kallel,  and M. Ouni,
Nash strategies for the inverse inclusion Cauchy-Stokes problem,
  Inverse Problems and Imaging (IPI) 13 , 4, 2019, pp. 827-862.

[4] A. Habbal and M. Kallel.
Neumann-Dirichlet Nash strategies for the solution of elliptic Cauchy problems,
  SIAM Journal on Control and Optimization, 2013, vol. 51, no 5,  pp. 4066-4083.

14:50h – 15:30h

Natural System Management: A Mathematician’s Perspective

14:00h – 14:40h

Understanding and measuring evolutionary dynamics in cancer through the lens of mathematics: Examples.

14:50h – 15:30h

Decay of solutions for some dissipative systems in critical spaces