# Amaury Hayat - Academic webpage

Welcome on my academic webpage ! Here you will find:

Some info and a CV

I am currently a Postdoctoral Associate at Rutgers University, working with Benedetto Piccoli. Prior to this, I was a PhD Candidate at Sorbonne Université (UPMC) in the Laboratory Jacques-Louis Lions, under the supervision of Jean-Michel Coron, and joint supervision of Sébastien Boyaval (École Nationale des Ponts et Chaussées, Laboratory of Hydraulic Saint-Venant).

Here is a CV .

You can contact me at: amaury.hayat « at »rutgers.edu

### Current research subject:

I study the stabilization of partial differential equations (PDE) and systems of PDEs. My aim is to make system stable by acting on the boundary conditions. The results are mostly applied to fluid mechanics, but not only.

## Research subjects

I currently study the stabilization of hyperbolic partial differential equations (PDE) and systems of PDE. In particular, I work on:

### The stabilization of inhomogeneous systems.

Most physical systems are inhomogeneous, however this inhomogeneity leads to some difficulties when it couples equations with different propagation speeds…

### The stabilization of shock steady-states in fluid mechanics.

For instance for the Burgers' or the Saint-Venant equations (approximation of Navier-Stokes in shallow waters). An example.

### The stabilization of PDEs using a Propotionnal integral (PI) control.

PI controls are among the most useful controllers but they are hard to handle mathematically when applied to nonlinear PDEs, and thus seldom studied. A method for a scalar equation. An example of system: Saint-Venant equations .

### Traffic model

Interestingly, the usual entropic solutions are generally not the right physical solutions of these systems. Nonclassical shocks and new behaviours can appear, especially when looking at the influence of an autonomous vehicle...

## Articles and preprints

### In mathematics (at Sorbonne University, Laboratory Jacques-Louis Lions, ENPC and Rutgers University):

1. Global exponential stability and Input-to-State Stability of semilinear hyperbolic systems for the $$L^{2}$$ norm, 2020. Preprint.
2. Feedforward boundary control of $$2 \times 2$$ nonlinear hyperbolic systems with application to Saint-Venant equations, (with Georges Bastin and Jean-michel Coron), 2020. Preprint, abstract.
3. Input-to-State Stability in sup norms for hyperbolic systems with boundary disturbances, (with Georges Bastin and Jean-Michel Coron), 2020. Preprint, abstract.
4. Exponential stability of density-velocity systems with boundary conditions and source term for the $$H^{2}$$ norm, (with Peipei Shang), 2019. Preprint, abstract.
5. PI controller for the general Saint-Venant equations, 2018. Preprint, abstract.
6. Boundary feedback stabilization of hydraulic jumps, (with Georges Bastin, Jean-Michel Coron and Peipei Shang), accepted in IFAC Journal of systems and Control, 2018. Preprint.
7. PI controllers for 1D nonlinear transport equation, (with Jean-Michel Coron), accepted in IEEE: TAC, 2018. Preprint, abstract.
8. On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the $$C^1$$ norm, accepted in ESAIM: COCV, 2018. Preprint.
9. Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation, (with Georges Bastin, Jean-Michel Coron and Peipei Shang), accepted in M3AS, 2017. Preprint, abstract.
10. Exponential stability of general 1-D quasilinear systems with source terms for the $$C^{1}$$ norm under boundary conditions, 2019, accepted in SIAM J. Control. Optim. Preprint (2017), abstract.
11. A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope (avec Peipei Shang), accepted in Automatica, 2017. Preprint.

### In machine learning

1. Deep Differential System Stability -- Learning advanced computations from examples, (with François Charton and Guillaume Lample), 2020. Preprint, abstract.

### In plant science (at Cambridge University, DPMMS / Friend Group)

1. Modeling Tree Growth Taking into Account Carbon Source and Sink Limitations, (with Andrew Friend, Andrew J. Hacket-Pain, Hans Pretzsch, Tim T. Rademacher) Front. Plant Sci. 2017, 8:182.doi: 10.3389/fpls.2017.00182. Article.

### In optics (at Harvard University, Capasso Group)

1. Lateral chirality-sorting optical forces, (with J.P. Balthasar Mueller and Federico Capasso) Proceedings of the National Academy of Sciences Oct 2015, 112 (43) 13190-13194; DOI: 10.1073/pnas.1516704112. Article.