# Registered Data

## [00854] Control and stabilization of PDEs: recent advances and applications

**Session Date & Time**:- 00854 (1/3) : 2C (Aug.22, 13:20-15:00)
- 00854 (2/3) : 2D (Aug.22, 15:30-17:10)
- 00854 (3/3) : 2E (Aug.22, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: As control problems arise naturally from engineering and physics, control theory has attracted a lot of attention in the last century. In recent decades, the study of control problems from PDEs' point of view has developed quickly, finding natural connections with fluid mechanics, microlocal analysis, stochastic analysis, and many applications such as traffic flow regulations, lane manufacturing, crowd motion, and biology, etc. We focus on these new developments, which cover a wide range of problems from observability to quantitative stabilization and finite-time stabilization across many different types of systems from subelliptic equations to conservation laws to hybrid systems.**Organizer(s)**: Shengquan Xiang, Maria Teresa Chiri, Amaury Hayat, Qi Lü**Classification**:__35L04__,__76B15__,__49J20__,__93D15__,__93C95__**Speakers Info**:**Shengquan Xiang**(Peking University)- Maria Teresa Chiri (Queen’s University)
- Amaury Hayat (Ecole des Ponts Paristech)
- Ludovick Gagnon (Inria and Université de Lorraine)
- Nathalie T. Khalil (University of Porto (FEUP))
- Bahman Gharesifard (UCLA)
- Zhiqiang Wang (Fudan University)
- Pierre Lissy (Université Paris-Dauphine)
- Yue Wang (Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU))
- Chenmin Sun (CNRS and Universtité Paris-Est Crétei)
- Gengsheng Wang (Tianjin University)
- Fabio Ancona (University of Padova)

**Talks in Minisymposium**:**[02522] Geometry of observable sets****Author(s)**:**gengsheng wang**(Tianjin University)

**Abstract**: We introduce several observability inequalities in some abstract setting. Then for some concrete evolution equations, such as the heat equation and the Schrodinger equations on the whole space, we give the characterizations of the observable sets such that the aforementioned inequalities hold. We further give some comments.

**[02524] Quantitative rapid stabilization of some PDE models****Author(s)**:**Shengquan Xiang**(Peking University)

**Abstract**: Quantitative stabilization is an active research topic in PDEs’ control theory, namely to construct explicit feedback laws as a control to make the closed-loop system stable together with quantitative estimates. In this presentation, we will talk about some recent progress in this topic including the Frequency Lyapunov method for the stabilization of Navier-Stokes equations and the Fredholm backstepping method for the stabilization of water waves.

**[02863] Generalized Fredholm-backstepping transformation****Author(s)**:**Ludovick Gagnon**(Inria Nancy)- Christophe Zhang (Inria Nancy)
- Amaury Hayat (Ecole des Ponts Paristech)
- Shengquan Xiang (Peking University)
- Swann Marx (CNRS)

**Abstract**: The backstepping method for PDEs, introduced over 20 years ago, is a powerful method to prove the rapid stabilisation of linear PDEs. The Fredholm alternative, introduced by Coron and Lü, quickly proved to provide a systematic approach to the backstepping method based on the spectral behaviour of the PDE as well as controllability assumptions. We present in this talk recent advances on sufficient conditions for the Fredholm-backstepping method in an abstract setting.

**[02943] Feedback stabilization and inverse problem for a nonlocal transport equation****Author(s)**:**Zhiqiang Wang**(Fudan University)

**Abstract**: In this talk, we will show some results on feedback stabilization and inverse problems for a transport equation with nonlocal velocity. This model arises in the control of semiconductor manufacturing systems which have a highly re-entrant character. Firstly we obtain a semi-global stabilization result by using a time-varying feedback control. Secondly with the help of certain feedback control, we recover the velocity function from the measurements.

**[02944] Indirect controllability of linear constant coefficients parabolic-transport systems****Author(s)**:**Pierre Lissy**(Université Paris-Dauphine)

**Abstract**: I will present controllability properties of mixed systems of linear parabolic-transport equations, with possibly nondiagonalizable diffusion matrix, on the 1D torus, coupled by constant coupling terms. The distributed control acts through a constant matrix, with possibly less controls than equations. In small time or for not regular enough initial data, these systems are never controllable, whereas in large time, null-controllability holds, for regular initial data, iff a spectral Kalman rank condition is verified.

**[02947] Advances on structural controllability of ensembles****Author(s)**:**Bahman Gharesifard**(UCLA)

**Abstract**: Ensemble control studies the problem of steering the state of a large population of systems, or a continuum, using a finite number of controllers. The problem has historic ties to control of partial differential equations, with many applications including quantum ensembles in spectroscopy, control of large limits of complex networks with few inputs, and recently in the study of universal approximation of neural networks. The purpose of this talk is to study a suit of structural controllability results for linear ensemble systems, where the objective is to identify structural properties that generically render the state of the ensemble — or some statistical properties of the profile over the parametrization space, for instance the average or higher moments — controllable. The latter provides a necessary and critical complement to full structural ensemble controllability which is hard to achieve in multiparameter settings. For the case where the statistical property of interest is the average, we provide a graph-theoretic characterization of structural controllability. Along the way of establishing this result, we hint at a conjecture on minimal ``complexity’’ controllers and relate it a conjecture on invertibility of a sparse version of Hilbert matrices, which is open for most parts.

**[03018] Stabilization of 1D evolution systems: new approaches****Author(s)**:**Amaury Hayat**(Ecole des Ponts Paristech)

**Abstract**: We discuss recent advances in the F-equivalence (or Fredholm backstepping) method. This consists in reformulating the stabilization problem and to find a control operator such that the PDE system can be inversely mapped to a simpler PDE system. Surprisingly powerful, this approach offers the possibility to treat very general classes of systems. We will also examine traffic flow stabilization. Finally, we will briefly discuss some results on AI for mathematics.

**[03282] Second microlocalization and optimal decay for the Bouendi-Grushin damped wave equation****Author(s)**:**Chenmin Sun**(CNRS(Université Paris Est Créteil))- Victor Arnaiz (Nantes Université)

**Abstract**: The Bouendi-Grushin damped-wave operator is a hypoelliptic operator with a distributional damping. This presentation introduces the second microlocalization method for obtaining the optimal resolvent estimate for the operator under varying damping conditions.

**[03571] Optimal Control of Moving Sets****Author(s)**:**Maria Teresa Chiri**(Queen's University)

**Abstract**: We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. We first analyze the optimal control of 1-dimensional traveling wave profiles. Using Stokes’ formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. Then we introduce a family of optimization problems for a moving set and show how these can be derived from the original parabolic problems, by taking a sharp interface limit. Assuming that the initial contaminated set is convex, we prove that an eradication strategy is optimal if an only if at each given time the control is active along the portion of the boundary where the curvature is maximal. We then consider the eradication problem with geographical constraints, and derive necessary and sufficient conditions for the existence of a solution.

**[04252] The controllability of a special class of coupled wave systems****Author(s)**:**Jingrui Niu**(Sorbonne Université)- Pierre Lissy (Université Paris-Dauphine)

**Abstract**: I will present an exact controllability result for coupled wave systems with two distinct speeds. A distributed scalar control function is effective in a subdomain satisfying the geometric control conditions and acts on only one speed. We establish compatibility conditions, which are associated with the particular coupling structure. Furthermore, the exact controllability holds in these compatible spaces if and only if the coupling structure satisfies an operator Kalman rank condition.

**[04439] On the boundary controllability of conservation laws with boundary and source controls****Author(s)**:**Fabio Ancona**(University of Padova)- Khai Tien Nguyen (North Carolina State University)

**Abstract**: We will discuss local and global controllability results for hyperbolic conservation laws on a bounded domain, where the control acts through a time dependent source term in combination with the boundary controls. We shall investigate first this problem for scalar conservation laws with a not necessarily convex flux. Next, we shall address the problem of extending these results to the case of rich systems of conservation laws.

**[04573] Controls on Networks: Modeling, Learning and Applications****Author(s)**:**Yue Wang**(Friedrich-Alexander-Universität Erlangen-Nürnberg)

**Abstract**: This talk is an introduction of controllability properties and methods for networked 1D hyperbolic systems based on results obtained by the speaker and her collaborators in recent years. Modelling, analysis of the underlying dynamics and exact controllability of several physical models will be presented at first. Some recent numerical experiments with Physics-Informed Neural Networks (PINNs) show interesting possibilities for future research on the interface between control and machine learning.