研究会 (2025 年 07 月 12 日)

概要
1. This talk focuses on the stability analysis of ODE systems coupled with PDEs,
   like the heat equation, and the role of its boundary conditions. The
   complexity lies in addressing the PDE's infinite-dimensional nature and
   precisely accounting for the dynamics at its boundaries. By combining
   Integral Quadratic Constraints with projection methods, we derive Linear
   Matrix Inequality conditions that make the stability analysis more tractable.

   Bio: Sara Callegari is a second-year PhD student at LAAS-CNRS in Toulouse
   under the supervision of Dimitri Peaucelle and Frédéric Gouaisbaut. She
   holds a Bachelor's degree in Computer Engineering and a Master's degree in
   Control Systems Engineering from the University of Padova, Italy. Her
   research focuses on coupled ODE–PDE systems and robust control.

2. Mathematical models of epidemics are crucial for understanding disease
   dynamics and informing public health strategies. In this talk, I will
   present a network-based extension of the classical SIR epidemic model,
   capturing heterogeneous interactions among multiple subpopulations. I will
   show how such models can exhibit complex behaviors, including multimodal
   infection curves. In particular, I will discuss the role of rank-1
   interaction matrices and derive explicit conditions ensuring the occurrence
   of multiple infection peaks. I will then introduce behavioral feedback
   mechanisms, where the transmission rate evolves in response to the epidemic
   state, highlighting their impact on both transient and long-term dynamics.
   This framework allows for a more realistic representation of adaptive social
   behavior, and I will describe results obtained both in single-population and
   network settings. I will also discuss related work in optimal control, and
   conclude with ongoing research on adoption-opinion models for sustainable
   innovations.

   Bio: Martina Alutto received her B.Sc. and M.Sc. (cum laude) in Mathematical
   Engineering from Politecnico di Torino, Italy, in 2018 and 2021. She is
   currently completing a PhD in Pure and Applied Mathematics at the Department
   of Mathematical Sciences, Politecnico di Torino, and is a research fellow at
   the CNR - Istituto di Elettronica e di Ingegneria dell’Informazione e delle
   Telecomunicazioni (CNR-IEIIT), Turin. Her research focuses on the analysis
   and control of networked systems, with applications to epidemic modeling and
   social dynamics. Starting in September 2025, she will join the Division of
   Decision and Control Systems at the School of Electrical Engineering and
   Computer Science (EECS), KTH Royal Institute of Technology, as a postdoctoral
   researcher. 

3. In this lecture, we talk about the absolute stability analysis problem of
   both continuous-time and discrete-time feedback systems comprising LTI
   systems and nonlinearities that are slope-restricted and repeated. By using
   O'Shea-Zames-Falb multipliers in the framework of Integral Quadratic
   Constraints (IQCs), we can obtain linear matrix inequality (LMI) conditions
   to ensure the absolute stability. However since these conditions are
   generally sufficient, if LMIs turn out to be numerically infeasible, then we
   can conclude nothing about the absolute stability. To address such problems,
   we consider the dual of IQC-based LMIs that are feasible if and only if the
   (primal) LMIs are infeasible. As a result, we show that if the dual solution
   satisfies a certain rank condition, then  we can determine both a
   destabilizing slope-restricted nonlinearity and a non-zero equilibrium point,
   thereby proving that the target system is not absolutely stable.

   Bio: Hibiki Gyotoku was born in Tokyo, Japan, in 2000. He received his
   Bachelor's degree from Kyushu University in March 2024, and entered the
   Graduate School of Information Science and Electrical Engineering at Kyushu
   University  in April of the same year. He is currently a second-year Master's
   student, conducting research on the absolute stability analysis of nonlinear
   feedback systems using dual linear matrix inequalities (LMIs).