27 citations · 60 across the 21 of their papers we have counts for
22 papers
Factor-parity Hall sets and controllability
Karine Beauchard, Frédéric Marbach
We introduce a class of Hall sets, which we call factor-parity Hall sets, whose elements split into good and bad brackets. We prove that the good brackets can be steered simultaneo…
Some quartic control results for scalar-input systems
Karine Beauchard, Frédéric Marbach
We investigate the role of quartic terms in the small-time local controllability of scalar-input systems. First, we prove a new sufficient condition for controllability, which expl…
Convergent realizations of Lie subalgebras
Karine Beauchard, Jérémy Le Borgne, Frédéric Marbach
It has been known since the seminal work of Guillemin and Sternberg that Lie subalgebras of finite codimension of an arbitrary real or complex Lie algebra can be realized as subalg…
Local convergence of Sussmann's infinite product for analytic vector fields
Jérémy Le Borgne, Frédéric Marbach
In 1986, Sussmann proposed an expansion of the Chen series as an infinite product of exponentials of Hall basis elements multiplied by coefficients defined through a simple inducti…
Time-iteration methods for controllability
Frédéric Marbach
These notes are based on a short course delivered at the Summer School EUR MINT 2025 "Control, Inverse Problems and Spectral Theory", held in June 2025 in Toulouse, France. The cou…
Nonlinear Forward-Backward Problems
Anne-Laure Dalibard, Frédéric Marbach, Jean Rax
We prove the existence and uniqueness of strong solutions to the equation in the vicinity of the linear shear flow, subject to perturbations of the source term…
Small-time local control of a Schrödinger equation: a negative and a positive quadratic result
Karine Beauchard, Frédéric Marbach, Thomas Perrin
We study the small-time local controllability (STLC) of a bilinear Schrödinger equation with Neumann boundary conditions near its ground state. We focus on the degenerate case wher…
Control theory and splitting methods
Karine Beauchard, Adrien Busnot Laurent, Frédéric Marbach
Our goal is to highlight some deep connections between numerical splitting methods and control theory. We consider evolution equations of the form , wher…
A family of interpolation inequalities involving products of low-order derivatives
Frédéric Marbach
Gagliardo-Nirenberg interpolation inequalities relate Lebesgue norms of iterated derivatives of a function. We present a generalization of these inequalities in which the low-order…
A unified approach of obstructions to small-time local controllability for scalar-input systems
Karine Beauchard, Frédéric Marbach
We present a unified approach for determining and proving obstructions to small-time local controllability of scalar-input control systems. Our approach views obstructions to contr…
Linear and nonlinear parabolic forward-backward problems
Anne-Laure Dalibard, Frédéric Marbach, Jean Rax
The purpose of this paper is to investigate the well-posedness of several linear and nonlinear equations with a parabolic forward-backward structure, and to highlight the similarit…
Growth of structure constants of free Lie algebras relative to Hall bases
Karine Beauchard, Jérémy Le Borgne, Frédéric Marbach
We derive a priori bounds on the size of the structure constants of the free Lie algebra over a set of indeterminates, relative to its Hall bases. We investigate their asymptotic g…
On expansions for nonlinear systems, error estimates and convergence issues
Karine Beauchard, Jérémy Le Borgne, Frédéric Marbach
Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator…
On the controllability of the Navier-Stokes equation in a rectangle, with a little help of a distributed phantom force
Jean-Michel Coron, Frédéric Marbach, Franck Sueur +1
This note echoes the talk given by the second author during the Journées EDP 2018 in Obernai. Its aim is to provide an overview and a sketch of proof of the result obtained by the…
Controllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force
Jean-Michel Coron, Frédéric Marbach, Franck Sueur +1
We consider the 2D incompressible Navier-Stokes equation in a rectangle with the usual no-slip boundary condition prescribed on the upper and lower boundaries. We prove that for an…
Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations
Karine Beauchard, Frédéric Marbach
We consider scalar-input control systems in the vicinity of an equilibrium, at which the linearized systems are not controllable. For finite dimensional control systems, the author…
High frequency analysis of the unsteady Interactive Boundary Layer model
Anne-Laure Dalibard, Helge Dietert, David Gérard-Varet +1
The present paper is about a famous extension of the Prandtl equation, the so-called Interactive Boundary Layer model (IBL). This model has been used intensively in the numerics of…
Quadratic obstructions to small-time local controllability for scalar-input differential systems
Karine Beauchard, Frédéric Marbach
We consider nonlinear scalar-input differential control systems in the vicinity of an equilibrium. When the linearized system at the equilibrium is controllable, the nonlinear syst…
On the controllability of the Navier-Stokes equation in spite of boundary layers
Jean-Michel Coron, Frédéric Marbach, Franck Sueur
In this proceeding we expose a particular case of a recent result obtained by the authors regarding the incompressible Navier-Stokes equations in a smooth bounded and simply connec…
Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions
Jean-Michel Coron, Frédéric Marbach, Franck Sueur
In this work, we investigate the small-time global exact controllability of the Navier-Stokes equation, both towards the null equilibrium state and towards weak trajectories. We co…
An obstruction to small time local null controllability for a viscous Burgers' equation
Frédéric Marbach
In this work, we are interested in the small time local null controllability for the viscous Burgers' equation on the line segment , with null…
Fast global null controllability for a viscous Burgers' equation despite the presence of a boundary layer
Frédéric Marbach
In this work, we are interested in the small time global null controllability for the viscous Burgers' equation y_t - y_xx + y y_x = u(t) on the line segment [0,1]. The second-hand…