PROGRAMME :
9h30-10h: Mohand Ouidir Amirat, LAGEPP
Titre : Stability analysis of 2x2 linear weakly hyperbolic systems of balance laws: application to wastewater treatment
Résumé : We investigate the exponential stability of steady states for a one-dimensional linear weakly hyperbolic system of balance laws, characterized by a non-diagonalizable coefficient matrix with equal characteristic velocities. In contrast to the well-studied strictly hyperbolic case, we shall see that the lack of diagonalizability introduces new challenges in the stability analysis, motivating a rigorous examination of suitable dissipative boundary conditions for this class of systems. Sufficient conditions ensuring exponential stability are established through analyses conducted in both the Laplace domain and the time domain, employing Lyapunov-based techniques. Numerical examples are provided to illustrate the effectiveness of the proposed approach. Finally, several perspectives are presented for extending the analysis to the nonlinear case, in particular to the stabilization of shock steady states, which are physically relevant in the context ofwastewater treatment applications.
10h-10h30 : Jérome Loheac, CRAN
Titre : Ensemble controllability of parabolic type equations
Résumé : We extend the concept of ensemble controllability introduced in 2009 by Jr-Shin Li and Navin Khaneja to a class of linear partial differential equation. More precisely, we consider some abstract parabolic equation, where the system depends on some unknown parameter which is assumed to belong to a compact interval. We investigate the possibility of approximatively reaching (in L²-norm) any target state from any initial state, with an open loop control.Here the initial and target states might depend on the unknown parameter, but the control is assumed to be parameter independent.
10h30-11h : Quentin Helman, Université de Limoges
Titre : Lipschitz parametric sensitivity of solutions to nonlinear Roesser systems
Résumé : We establish a general regularity result for nonlinear two-dimensional Roesser models with variable parameters. Uniform bounds on trajectory differences are derived for continuous settings, proving continuity with respect to boundary conditions and parameters.
11h-11h30 : Adam Braun, L2S
Titre: A Backstepping-KKL observer for a cascade of a nonlinear ODE with a heat equation
Résumé : In this presentation, we propose an observer design for a cascaded system composed of an arbitrary nonlinear ordinary differential equation (ODE) with a 1D heat equation.
The nonlinear output of the ODE imposes a boundary condition on one side of the heat equation, while the measured output is on the other side. The observer design combines an infinite-dimensional Kazantzis-Kravaris/Luenberger (KKL) observer for the ODE with a backstepping observer for the heat equation. This construction is the first extension of the KKL methodology to infinite-dimensional systems. We establish the convergence of the observer under a differential observability condition on the ODE. The effectiveness of the proposed approach is illustrated in numerical simulations.