Christophe Zhang

University of Lorraine | UdL · INRIA SPHINX team

We consider the internal control of linear parabolic equations through on-off shape controls, i.e., controls of the form M(t) \chi_{\omega(t)} with M(t) \geq 0 and \omega(t) with a prescribed maximal measure. We establish small-time approximate controllability towards all possible final states allowed by the comparison principle with nonnegative co...

We consider the internal control of linear parabolic equations through on-off shape controls, i.e., controls of the form M (t)χ ω(t) with M (t) ≥ 0 and ω(t) with a prescribed maximal measure. We establish small-time approximate controllability towards all possible final states allowed by the comparison principle with nonnegative controls. We manage...

We give convergence and cost estimates for a data-driven system identification method: given an unknown dynamical system, the aim is to recover its vector field and its flow from trajectory data. It is based on the so-called Koopman operator, which uses the well-known link between differential equations and linear transport equations. Data-driven m...

We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. This classical framework allows us to present the backstepping method with Fredholm transformations for the Laplace operator in a sharp functional setting, which is the main objective of this work. We first...

In this article we study the so-called water tank system. In this system, the behavior of water contained in a one dimensional tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform steady-states are not controllable, the uncontrollable part being of infinite dimens...

Fredholm-type backstepping transformation, introduced by Coron and L\"u, has become a powerful tool for rapid stabilization with fast development over the last decade. Its strength lies in its systematic approach, allowing to deduce rapid stabilization from approximate controllability. But limitations with the current approach exist for operators o...

We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present th...

In this article we study the so-called water tank system. In this system, the behavior of water contained in a 1-D tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform steady-states are not controllable, the uncontrollable part being of infinite dimension. Here we...

We use a variant the backstepping method to study the stabilization of a 1-D linear transport equation on the interval \begin{document}$ (0,L) $\end{document}, by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order...

We consider a 1-D linear transport equation on the interval (0,L), with an internal scalar control. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized in finite time, and we give an explicit feedback law.

In this thesis we study controllability and stabilization questions for some hyperbolic systems in one space dimension, with an internal control. The first question we study is the indirect internal controllability of a system of two coupled semilinear wave equations, the control being a function of time and space. Using the so-called fictitious co...

We use the backstepping method to study the stabilization of a 1-D linear transport equation on the interval (0, L), by controlling the scalar amplitude of a piecewise regular function of the space variable in the source term. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be sta...

We study systems of two coupled wave equations in one space dimension, with one control, spatially supported on an arbitrarily small interval. We obtain the controllability of such systems under certain conditions on the coupling. To do this we apply the “fictitious control method” in two cases: general systems with a controllable linearized system...

We prove the internal controllability of some systems of two coupled wave equations in one space dimension, with one control, under certain conditions on the coupling. To do this we apply the "fictitious control method" in two cases: general systems with a "non-degenerate" coupling, and a particular case where the coupling is "degenerate", namely a...