arXiv subjects

Frédéric Marbach

Publications and source records attributed to Frédéric Marbach.

At least 19 recordsLinked to original sources

arXiv2026

Factor-parity Hall sets and controllability

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 simultaneously and arbitrarily in small time, which yields sufficient conditions for the small-time local controllability of control-affine systems. This positive result draws on constructions of Kawski, Agrachev-Gamkrelidze and Krastanov. Conversely, we prove that each bad bracket generates an obstruction to controllability, hence a family of necessary conditions.

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arXiv2026

Some quartic control results for scalar-input systems

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 exploits simultaneously more good quartic Lie brackets than previous results. Second, we identify a family of quartic obstructions to controllability, relying on Lie brackets whose coordinates of the second kind are positive-definite functionals of the control. Third, we show that the complementarity of these results can be seen as an answer to Kawski's 1987 open problem concerning the construction of a Hall basis which somehow separates good and bad quartic brackets. We give examples and remarks illustrating some of the intricacies of these notions.

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arXiv2026

A family of interpolation inequalities involving products of low-order derivatives

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 interpolation factor of the right-hand side is replaced by a Lebesgue norm of a pointwise product of derivatives of the function. We provide a short, independent proof, recovering the usual one-dimensional inequality as a particular case. We give examples of applications to control theory and PDE analysis, and discuss related open problems.

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arXiv2026

Convergent realizations of Lie subalgebras

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 subalgebras of formal vector fields over formal power series. In this note, we characterize the Lie subalgebras which admit a convergent realization in the sense of locally analytic vector fields. We give generalizations of these properties for the problem of output realization. We give reformulations and applications of these algebraic results in the context of control theory. In particular, we recover and clarify previous results on the realization of Chen--Fliess series for control-affine systems, the equivalence of control systems, and the existence of embedded or canonical systems.

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arXiv2026

Local convergence of Sussmann's infinite product for analytic vector fields

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 induction. He proved that this product converges locally in time for bilinear control systems associated with matrices or bounded operators. In this note, we prove that Sussmann's product also converges locally in time for control-affine systems driven by analytic vector fields. The difficulty is that one must keep track of the structure of the involved Lie brackets, as the natural decay of the explicit coefficients is not enough to counterbalance the natural growth of the iterated Lie brackets.

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arXiv2026

Small-time local control of a Schrödinger equation: a negative and a positive quadratic result

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 where the linearized system is not controllable, necessitating a second-order analysis. We prove two complementary results. The negative result provides a new PDE instance of Sussmann's classical quadratic obstruction, corresponding to a non-vanishing Lie bracket. The positive result appears to be the first to establish STLC at the quadratic order for a physical PDE with a single scalar control. Both proofs rely on a Fourier-based approach, which is crucial because the integral kernel of the second-order term lacks the regularity required by standard integration-by-parts arguments. Along the way, we develop tools valid in a more general setting to analyze such quadratic forms. In particular, we prove results that allow for the multiplication of a kernel by a modulation function.

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arXiv2026

Control theory and splitting methods

Our goal is to highlight some deep connections between numerical splitting methods and control theory. We consider evolution equations of the form $\dot{x} = f_0(x) + f_1(x)$, where $f_0$ encodes non-reversible dynamics, motivating schemes that involve only forward flows of $f_0$. In this context, a splitting method can be interpreted as a trajectory of the control-affine system $\dot{x}(t)=f_0(x(t))+u(t)f_1(x(t))$, associated with a control $u$ that is a finite sum of Dirac masses. The goal is then to find a control such that the flow generated by $f_0 + u(t)f_1$ is as close as possible to the flow of $f_0+f_1$. Using this interpretation and classical tools from control theory, we revisit well-known results on numerical splitting methods and prove several new ones. First, we show that there exist numerical schemes of arbitrary order involving only forward flows of $f_0$, provided one allows complex coefficients for $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to small-time local controllability. Second, for real-valued coefficients, we show that the well-known order restrictions are linked to so-called "bad" Lie brackets from control theory, which are known to obstruct small-time local controllability. We investigate the conditions under which high-order methods exist, thanks to a basis of the free Lie algebra that we recently constructed.

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arXiv2026

Time-iteration methods for controllability

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 course presents three important strategies in control theory, formulated as time-iteration methods, where each time step brings the state of the system closer to the desired target. For linear PDEs, we survey the classical Lebeau-Robbiano method and its more recent developments. This method combines spectral inequalities and dissipation estimates to prove null controllability of a dissipative linear system. For nonlinear PDEs, we reinterpret the Liu-Takahashi-Tucsnak method, which establishes local controllability of a nonlinear system by analyzing the control cost of its linearization. We provide an easily applicable black-box formulation of their method. Finally, for nonlinear ODEs, we present the tangent vectors method, which establishes local exact controllability starting from approximately reachable directions.

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arXiv2025

Nonlinear Forward-Backward Problems

We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we consider have opposite signs in the lower and upper half of the domain, this is a quasilinear forward-backward parabolic problem, which changes type across a critical curved line within the domain. In particular, lateral boundary conditions can be imposed only where the characteristics are inwards. There are several difficulties associated with this problem. First, the forward-backward geometry depends on the solution itself. This requires to be quite careful with the approximation procedure used to construct solutions. Second, and more importantly, the linearized equations solved at each step of the iterative scheme admit a finite number of singular solutions, of which we provide an explicit construction. This is similar to well-known phenomena in elliptic problems in nonsmooth domains. Hence, the solutions to the equation are regular if and only if the source terms satisfy a finite number of orthogonality conditions. A key difficulty of this work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. To tackle this deceivingly simple problem, we develop a methodology which we believe to be both quite natural and adaptable to other situations in which one wishes to prove the existence of regular solutions to a nonlinear problem for suitable data despite the existence of singular solutions at the linear level. This paper is a shorter version of [3].

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arXiv2025

Linear and nonlinear parabolic forward-backward problems

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 similarities and differences between them. The epitomal linear example will be the stationary Kolmogorov equation $y\partial_x u -\partial_{yy} u=f$ in a rectangle. We first prove that this equation admits a finite number of singular solutions, of which we provide an explicit construction. Hence, the solutions to the Kolmogorov equation associated with a smooth source term are regular if and only if $f$ satisfies a finite number of orthogonality conditions. This is similar to well-known phenomena in elliptic problems in polygonal domains. We then extend this theory to a Vlasov--Poisson--Fokker--Planck system, and to two quasilinear equations: the Burgers type equation $u \partial_x u - \partial_{yy} u = f$ in the vicinity of the linear shear flow, and the Prandtl system in the vicinity of a recirculating solution, close to the line where the horizontal velocity changes sign. We therefore revisit part of a recent work by Iyer and Masmoudi. For the two latter quasilinear equations, we introduce a geometric change of variables which simplifies the analysis. In these new variables, the linear differential operator is very close to the Kolmogorov operator $y\partial_x -\partial_{yy}$. Stepping on the linear theory, we prove existence and uniqueness of regular solutions for data within a manifold of finite codimension, corresponding to some nonlinear orthogonality conditions.

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arXiv2024

A unified approach of obstructions to small-time local controllability for scalar-input systems

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 controllability as resulting from interpolation inequalities between the functionals associated with the formal Lie brackets of the system. Using this approach, we give compact unified proofs of all known necessary conditions, we prove a conjecture of 1986 due to Kawski, and we derive entirely new obstructions. Our work doubles the number of previously-known necessary conditions, all established in the 1980s. In particular, for the third quadratic bracket, we derive a new necessary condition which is complementary to the Agrachev-Gamkrelidze sufficient ones. We rely on a recent Magnus-type representation formula for the state, a new Hall basis of the free Lie algebra over two generators, an appropriate use of Sussmann's infinite product to compute the Magnus expansion, and Gagliardo-Nirenberg interpolation inequalities.

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arXiv2023

On expansions for nonlinear systems, error estimates and convergence issues

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 splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied. First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann's infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation. Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input. Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann's infinite product expansion. Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.

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arXiv2022

Growth of structure constants of free Lie algebras relative to Hall bases

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 growth, especially as a function of the length of the involved Lie brackets. First, using the classical recursive decomposition algorithm, we obtain a rough upper bound valid for all Hall bases. We then introduce new notions (which we call alphabetic subsets and relative foldings) related to structural properties of the Lie brackets created by the algorithm, which allow us to prove a sharp upper bound for the general case. We also prove that the length of the relative folding provides a strictly decreasing indexation of the recursive rewriting algorithm. Moreover, we derive lower bounds on the structure constants proving that they grow at least geometrically in all Hall bases. Second, for the celebrated historical length-compatible Hall bases and the Lyndon basis, we prove tighter sharp upper bounds, which turn out to be geometric in the length of the brackets. Third, we construct two new Hall bases, illustrating two opposite behaviors in the two-indeterminates case. One is designed so that its structure constants have the minimal growth possible, matching exactly the general lower bound, linked with the Fibonacci sequence. The other one is designed so that its structure constants grow super-geometrically. Eventually, we investigate asymmetric growth bounds which isolate the role of one particular indeterminate. Despite the existence of super-geometric Hall bases, we prove that the asymmetric growth with respect to each fixed indeterminate is uniformly at most geometric in all Hall bases.

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arXiv2019

Controllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force

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 any positive time, for any finite energy initial data, there exist controls on the left and right boundaries and a distributed force, which can be chosen arbitrarily small in any Sobolev norm in space, such that the corresponding solution is at rest at the given final time. Our work improves earlier results where the distributed force is small only in a negative Sobolev space. It is a further step towards an answer to Jacques-Louis Lions' question about the small-time global exact boundary controllability of the Navier-Stokes equation with the no-slip boundary condition, for which no distributed force is allowed. Our analysis relies on the well-prepared dissipation method already used for Burgers and for Navier-Stokes in the case of the Navier slip-with-friction boundary condition. In order to handle the larger boundary layers associated with the no-slip boundary condition, we perform a preliminary regularization into analytic functions with arbitrarily large analytic radius and prove a long-time nonlinear Cauchy-Kovalevskaya estimate relying only on horizontal analyticity.

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arXiv2019

Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations

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 authors recently classified the possible quadratic behaviors. Quadratic terms introduce coercive drifts in the dynamics, quantified by integer negative Sobolev norms, which are linked to Lie brackets and which prevent smooth small-time local controllability for the full nonlinear system. In the context of nonlinear parabolic equations, we prove that the same obstructions persist. More importantly, we prove that two new behaviors occur, which are impossible in finite dimension. First, there exists a continuous family of quadratic obstructions quantified by fractional negative Sobolev norms or by weighted variations of them. Second, and more strikingly, small-time local null controllability can sometimes be recovered from the quadratic expansion. We also construct a system for which an infinite number of directions are recovered using a quadratic expansion. As in the finite dimensional case, the relation between the regularity of the controls and the strength of the possible quadratic obstructions plays a key role in our analysis.

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arXiv2019

On the controllability of the Navier-Stokes equation in a rectangle, with a little help of a distributed phantom force

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 authors, concerning the controllability of the Navier-Stokes equation. We refer the interested readers to the original paper for the full technical details of the proof, which will be omitted here, to focus on the main underlying ideas.

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arXiv2018

High frequency analysis of the unsteady Interactive Boundary Layer model

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 steady boundary layer flows, and compares favorably to the Prandtl one, especially past separation. We consider here the unsteady version of the IBL, and study its linear well-posedness, namely the linear stability of shear flow solutions to high frequencyperturbations. We show that the IBL model exhibits strong unrealistic instabilities, that are in particular distinct from the Tollmien-Schlichting waves. We also exhibit similar instabilities for a Prescribed Displacement Thickness model (PDT), which is one of the building blocks of numerical implementations of the IBL model.

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arXiv2017

Quadratic obstructions to controllability: from ODEs to PDEs

We investigate the small-time local controllability of systems in the vicinity of an equilibrium. Given a small time, an initial data and a final data close from the equilibrium, is it possible to find a control (a source term) that guides the solution from the initial state to the wished final state at the given time? The natural method is to start by studying the controllability of the linearized system near the equilibrium. When this system is not controllable, it is necessary to continue the expansion with the quadratic order.In this note, we highlight the links between different recent results around this topic, in the particular case where the control is a single scalar input. These results tend to prove that, in this particular case, the quadratic order can only yield obstructions to controllability. We especially comment an exhaustive result obtained in collaboration with Karine Beauchard in finite dimension and a result contained in the thesis of the author, which involves a new kind of phenomenon in infinite dimension.

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