Ambroise Vest’s research while affiliated with University of Waterloo and other places

This works contains two independent parts, each one dealing with the control of partial differential equations. In the first part, we study an explicit and already known feedback law that applies to linear, time-reversible systems, with a possibly unbounded control operator. We prove the well-posedness of the closed-loop problem in the semi-group framework and we study the decay rate of the solutions. In the second part, we give conditions on the choice of some time instants, such that the positions of a vibrating string (or beam) at these times enable to recover the initial data. The method relies on Diophantine approximation results. Using a duality method, we give a related exact controllability result.

In this paper, we obtain new observability inequalities for the vibrating string. This work was motivated by a recent paper by A. Szij\'art\'o and J. Heged\H{u}s in which the authors ask the question of determining the initial data by only knowing the position of the string at two distinct time instants. The choice of the observation instants is crucial and the estimations rely on the Fourier series expansion of the solutions and results of Diophantine approximation.

In this paper, we obtain new observability inequalities for the vibrating string. This work was motivated by a recent paper by A. Szij\'art\'o and J. Heged\H{u}s in which the authors ask the question of determining the initial data by only knowing the position of the string at two distinct time instants. The choice of the observation instants is crucial and the estimations rely on the Fourier series expansion of the solutions and results of Diophantine approximation.

We investigate some structural properties of an efficient feedback law that stabilize linear time-reversible systems with an arbitrarily large decay rate. After giving a short proof of the generation of a group by the closed-loop operator, we focus on the domain of the infinitesimal generator in order to illustrate the difference bewteen a distributed control and a boundary control, the latter being technically more complex. We also give a new proof of the exponential decay of the solutions and we provide an explanation of the higher decay rate observed in some experiments.

We prove the well-posedness of a linear closed-loop system with an explicit (already known) feedback leading to arbitrarily large decay rates. We define a mild solution of the closed-loop problem using a dual equation and we prove that the original operator perturbed by the feedback is (up to the use of an extension) the infinitesimal generator of a strongly continuous group. We also give a justification to the exponential decay of the solutions. Our method is direct and avoids the use of optimal control theory.