MFG with large discount: uniqueness of solutions and convergence to agent based models
Elisa Continelli
We focus on a class of Mean Field Games with discount. When the discount factor is sufficiently large, these systems behave similarly to agent based models. Indeed, in [Bardi, Cardaliaguet (2021)] it is shown that solutions to Mean Field Games with discount converge to solutions to certain agent based models, as the discount factor goes to infinity. Motivated by this, we establish quantitative versions of convergence results in [Bardi, Cardaliaguet (2021)]. Moreover, we show that, for large values of the discount factor, solutions to the considered class of MFG systems are unique, identifying a uniqueness regime that falls outside the usual ones involving monotonicity.
Joint work with Marco Cirant.
Geometric Inverse Problems for PDEs
Anna Doubova
This talk delves into geometric inverse problems for Partial Differential Equations (PDEs), aiming to identify subdomains within multidimensional sets. We explore two crucial aspects: uniqueness and numerical reconstruction. Several geometric inverse problems will be considered, including cases involving unknown initial data. We establish uniqueness results based on observations from the boundary or an interior domain, thereby deriving information about the unknown geometry. The primary analytical tools for the proofs include unique continuation, time analyticity of solutions, and semigroup theory. Additionally, we present numerical techniques for the reconstruction of the unknown domain.
This work is joint work with J. Apraiz, E. Fernández-Cara, and M. Yamamoto.
The New SCOLE: Controlling Hybrid Systems of Flexible Beams with Mixed-Order Strategies
Sarah Ismail
In this talk, we introduce a SCOLE-type hybrid PDE-ODE system consisting of two EulerBernoulli beams serially-connected at an interior joint, where the joint carries both a point mass and a rotational inertia, and the coupling is imposed through dynamic transmission conditions, while each sub-beam preserves classical Euler-Bernoulli dynamics.
For a class of boundaryinterface velocity feedback laws combining lower-order and higher-order traces at the joint, we establish a complete classification of the system’s asymptotic behavior.
First, we consider the case where boundary dissipation is active. When at least one higher-order interface feedback term is present, the associated C_0-semigroup is exponentially stable, without arithmetic restrictions on material parameters, feedback gains, or the interface location. In contrast, if higher-order interface feedback is replaced by lower-order damping alone, exponential stability fails and the total energy decays at the sharp polynomial rate t^(−1).
Then, we consider the case where boundary dissipation is violated. In this scenario, exponential decay is achieved only when both controllers are higher-order. Any other admissible pairing (that is, any combination that does not consist exclusively of higher-order feedback at both the boundary and the interface) yields only polynomial decay of order t^(−1) or t^(−2).
The proofs rely on sharp frequency-domain resolvent estimates within the Borichev–Tomilov framework, which is shown to be essential due to the failure of classical spectral, multiplier, and Riesz basis methods for this class of interface-inertial generators. These findings refine the SCOLE paradigm and provide new insights into the distributed control of large-scale networks of flexible structures.
This talk is based on joint work with M. Akil (Université Polythechniques Hauts-de-France, France), A. Ö. Özer, K. Ashburn , Z. Brown (Western Kentucky University, USA), and G. Fragnelli (Università degli Studi di Siena, Italy).
Stability for mixed operators in peridynamics
Dimitri Mugnai
We consider new problems modelling some situations in civil engineering. Stability issues will be considered both in linear and nonlinear cases.
A two-layer Energy Balance Climate model
Cristina Urbani
Energy Balance Models (EBMs) provide a simple yet effective framework for describing the Earth's climate. Introduced by Budyko and Sellers in the late 1960s, classical EBMs describe the evolution of the zonally averaged surface temperature through a one-dimensional diffusive equation.
A natural extension of these models is to include vertical resolution by accounting for the energy exchange between the Earth's surface and the atmosphere. In this talk, I will present the mathematical analysis of a two-layer diffusive EBM. I will first briefly review the main results obtained for the spatially homogeneous model, including well-posedness, positivity, and long-time behaviour [1].
I will then focus on the spatially dependent system [2]. The main ingredient is the establishment of new maximum and comparison principles for coupled degenerate cooperative parabolic systems. These results allow us to prove global well-posedness, preservation of positivity, and the existence of distinguished equilibria. Finally, I will discuss the asymptotic dynamics of the model by proving the existence of a global attractor.
References:
[1] P. Cannarsa, V. Lucarini, P. Martinez, C. Urbani, J. Vancostenoble, Analysis of a Two-Layer Energy Balance Model: Long-Time Behaviour and Greenhouse Effect, Chaos, 33 (2023), 113111
[2] P. Cannarsa, V. Lucarini, P. Martinez, C. Urbani, J. Vancostenoble, Comparison Principles and Long-Time Behaviour for a Diffusive Energy Balance Model with Vertical Resolution, (2026) submitted
Lifting controllability from characteristics to continuity equation
Arianna Vicari
We address the problem of steering a probability measure to another through the flow of a controlled continuity equation, with dynamics linear in the control. The answer to controllability questions strongly depends on the setting: the regularity of the initial and target measures, the structure of the admissible controls and geometric properties of the vector fields generating the dynamics (e.g. Hörmander-type conditions) may lead to different results.
The aim of this talk is to show that, despite this variety of outcomes, a common strategy emerges. It consists in lifting controllability from the characteristic ODE, where the control acts on trajectories, to the continuity equation at the level of probability measures. This viewpoint provides a unified framework for approaching controllability questions for the continuity equation under different assumptions. Finally, we briefly discuss ongoing work aimed at extending this lifting procedure to nonlocal dynamics.