Mathias Dus

• On \( L^\infty \) stabilization of diagonal semilinear hyperbolic systems by saturated boundary control, (ESAIM: Control, Optimisation and Calculus of Variations, 2020) (with F. Ferrante et C. Prieur). Available on Hal.
• BV exponential stability for systems of scalar conservation laws using saturated controls, (SIAM SICON, 2021). Available on Hal.
• Exponential stability of a general slope limiter scheme for scalar conservation laws subject to a dissipative boundary condition, (MCSS, 2021). Available on Hal
• Spectral stabilization of linear transport equations with boundary and in-domain couplings, (to appear, CRM, 2021). Available on Hal.
• The discretized backstepping method: an application to a general system of 2×2 linear balance laws (to appear, MCRF, 2022). Available on Hal.

In this thesis, we study the problem of boundary stabilization of general hyperbolic systems of partial differential equations. More precisely, the analysis focuses on systems where the transport term is scalar and for which the information propagates in a fixed direction. In addition, the chosen control is most of the time a state feedback law for which a saturation is possibly applied. The work is divided into two distinct parts, one focusing on Lyapunov techniques while the other one uses the linearity of the problem.

In the first part of the thesis, two main works are presented. In the first one, only linear transport equations with positive velocities are considered. The main goal is to design a saturated linear feedback in order to stabilize exponentially the open-loop system in \(L^\infty\). The method consists of using classical Lyapunov techniques to exhibit a basin of attraction for which a fine estimate is given. We also extend this work to nonlinear scalar conservation laws in a BV framework. In the other work, thanks to a slope limiter scheme, a system of scalar conservation laws is discretized. Inspired by "continuous" Lyapunov methods, a discrete Lyapunov functional is studied to prove the exponential $BV$ stabilization of the discrete solution using a linear feedback.

When the system under consideration is linear, two works are exposed as well. On the one hand, we study systems of linear transport equations of arbitrary dimension, coupled on the domain and at the boundary. Designing a controller from a pole placement algorithm, the exponential stabilization is proved in \(L^2\). On the other hand, we develop a numerical Backstepping theory in order to stabilize in finite time a numerical scheme modeling a \(2 \times 2\) linear system with in domain and boundary couplings.