arXiv subjects

Swann Marx

Publications and source records attributed to Swann Marx.

At least 19 recordsLinked to original sources

arXiv2026

Semi-uniform stability estimates for impedance passive systems with saturated feedback *

This article investigates the long-time behavior of (possibly infinite-dimensional) impedance passive systems under saturated output feedback and external disturbances. We assume that, in the absence of saturation and disturbances, the underlying linear output feedback exponentially stabilizes the system. Our main contribution is to show that fractional Sobolev regularity of the free output is preserved by the nonlinear feedback. More precisely, if the initial condition belongs to a suitable interpolation space associated with the linear closed-loop generator, then both the output and the state of the nonlinear closed-loop system inherit the corresponding fractional regularity. This regularity is sufficiently weak to be shared by the linear and nonlinear closed-loop systems, thereby avoiding the identification of the nonlinear generator domain. Combining this regularity result with observability estimates for the linear system yields a characterization of the asymptotic behavior of the nonlinear closed-loop system in the presence of disturbances. In particular, under the impedance passivity framework and exact observability and regularity assumptions, we establish a semi-uniform input-tostate stability property. The theory is illustrated by a multidimensional wave equation with nonlinear boundary damping.

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arXiv2026

Quantitative Fredholm backstepping and rapid stabilization

In this paper, we address the existence of Fredholm backstepping transformations for self-adjoint and skew-adjoint operators $A$. Under suitable assumptions on the operator $A$ and the possibly unbounded control operator $B$, we prove the existence of a Fredholm backstepping transformation for operators of order strictly greater than $1$. This work overcomes two major limitations of the classical Fredholm backstepping framework. One of the main contributions is the explicit identification of the underlying isomorphism used in the construction of the transformation $T$, thereby bypassing the compactness arguments and Riesz basis mechanisms traditionally used in the literature. This explicit structure enables us to derive quantitative and sharp estimates for $\|T\|_{\mathcal{L}(H;H)}$ and $\|T^{-1}\|_{\mathcal{L}(H;H)}$ with respect to the decay rate $\lambda$. As a consequence, we obtain quantitative rapid stabilization and small-time null controllability results for a broad class of operators.

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arXiv2026

Control and stabilization of cascade coupled systems: application to a 1-d heat and wave coupled system

We study cascade coupled systems, for which our prototypical example is a 1-d heat equation coupled with a 1-d wave equation. The heat component is controlled through one boundary and the information is transmitted through another one to the wave component, while the wave component does not influence the heat component. Our aim is to understand the well-posedness, controllability and stabilizability properties for such a system. Establishing well-posedness is tedious using the classical energy method, which motivates us to take advantage of the cascade structure. Taking again advantage of this structure, we prove a simultaneous exact and approximate controllability result. Finally, we obtain polynomial stabilization by means of a closed-loop control defined through the solution to a Sylvester equation. These results are all discussed in an abstract LTI framework and most of our findings apply to more general situations.

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arXiv2024

Optimizing quasi-dissipative evolution equations with the moment-SOS hierarchy

We prove that there is no relaxation gap between a quasi-dissipative nonlinear evolution equation in a Hilbert space and its linear Liouville equation reformulation on probability measures. In other words, strong and generalized solutions of such equations are unique in the class of measure-valued solutions. As a major consequence, non-convex numerical optimization over these non-linear partial differential equations can be carried out with the infinite-dimensional moment-SOS hierarchy with global convergence guarantees. This covers in particular all reaction-diffusion equations with polynomial nonlinearity.

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arXiv2024

Coercive ISS-Lyapunov functionals for regular infinite-dimensional systems and applications

This paper proposes the construction of a coercive ISS-Lyapunov functional for linear regular infinite-dimensional system. Indeed, as already known, Lyapunov functionals for infinite-dimensional systems might be not coercive. Under the assumption that there exists an exactly observable output, we are able to make coercive a Lyapunov functional which is not coercive under additional regularity assumption. We discuss also about the potential applications of such a Lyapunov functional in singular perturbation theory and output regulation. The results are illustrated on a non-trivial equation, namely, the Korteweg-de Vries equation.

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arXiv2024

Intrinsic Derivation of the Equations of a Snake Robot based on a Cosserat Beam Model

In this paper, we present an intrinsic derivation of the equations ruling the dynamics motion of a snake robot dynamics. Based on a Cosserat beam model, we first show that the extended configuration space is a Lie group. Endowing it with an appropriate left invariant metric, the corresponding Euler-Poincar\'e equations can be reduced to a system of hyperbolic PDEs in the Lie algebra $\mathfrak{se}(3)$. We also provide the constitutive law describing the actuation in this system of PDEs.

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arXiv2023

A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws

We propose a numerical method to solve parameter-dependent hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre's hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel-Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.

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arXiv2023

Stability analysis of a linear system coupling wave and heat equations with different time scales

In this paper we consider a system coupling a wave equation with a heat equation through its boundary conditions. The existence of a small parameter in the heat equation, as a factor multiplying the time derivative, implies the existence of different time scales between the constituents of the system. This suggests the idea of applying a singular perturbation method to study stability properties. In fact, we prove that this method works for the system under study. Using this strategy, we get the stability of the system and a Tikhonov theorem, which allows us to approximate the solution of the coupled system using some appropriate uncoupled subsystems.

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arXiv2023

Frequency domain approach for the stability analysis of a fast hyperbolic PDE coupled with a slow ODE

This paper deals with the exponential stability of systems made of a hyperbolic PDE coupled with an ODE with different time scales, the dynamics of the PDE being much faster than that of the ODE. Such a difference of time scales is modeled though a small parameter $\varepsilon$ multiplying the time derivative in the PDE, and our stability analysis relies on the singular perturbation method. More precisely, we define two subsystems: a reduced order system, representing the dynamics of the full system in the limit $\varepsilon = 0$, and a boundary-layer system, which represents the dynamics of the PDE in the fast time scale. Our main result shows that, if both the reduced order and the boundary-layer systems are exponentially stable, then the full system is also exponentially stable for $\varepsilon$ small enough, and our strategy is based on a spectral analysis of the systems under consideration. Our main result improves a previous result in the literature, which was proved using a Lyapunov approach and required a stronger assumption on the boundary-layer system to obtain the same conclusion.

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arXiv2023

Event-triggered boundary damping of a linear wave equation

This article presents an analysis of the stabilization of a multidimensional partial differential wave equation under a well designed event-triggering mechanism that samples the boundary control input. The wave equation is set in a bounded domain and the control is performed through a boundary classical damping term, where the Neumann boundary condition is made proportional to the velocity. First of all, existence and regularity of the solution to the closed-loop system under the event-triggering mechanism of the control are proven. Then, sufficient conditions based on the use of a specific Lyapunov functional are proposed in order to ensure that the solutions converge into a compact set containing the origin, that can be tuned by the designer. Furthermore, as expected, any Zeno behavior of the closed-loop system is avoided.

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arXiv2022

Singular perturbation analysis for a coupled KdV-ODE system

Asymptotic stability is with no doubts an essential property to be studied for any system. This analysis often becomes very difficult for coupled systems and even harder when different timescales appear. The singular perturbation method allows to decouple a full system into what are called the reduced order system and the boundary layer system, to get simpler stability conditions for the original system. In the infinite-dimensional setting, we do not have a general result making sure this strategy works. This papers is devoted to this analysis for some systems coupling the Korteweg-the Vries equation and an ordinary differential equation with different timescales. More precisely, We obtain stability results and Tikhonov-type theorems.

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arXiv2022

Sliding mode control for a class of linear infinite-dimensional systems

This paper deals with the stabilization of a class of linear infinite-dimensional systems with unbounded control operators and subject to a boundary disturbance. We assume that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable in the absence of disturbance. To achieve our objective, we follow a sliding mode strategy and we add another term to this controller in order to reject the disturbance. We prove the existence of solutions to the closed-loop system and its global asymptotic stability, while making sure the disturbance is rejected.

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arXiv2022

Path differentiability of ODE flows

We consider flows of ordinary differential equations (ODEs) driven by path differentiable vector fields. Path differentiable functions constitute a proper subclass of Lipschitz functions which admit conservative gradients, a notion of generalized derivative compatible with basic calculus rules. Our main result states that such flows inherit the path differentiability property of the driving vector field. We show indeed that forward propagation of derivatives given by the sensitivity differential inclusions provide a conservative Jacobian for the flow. This allows to propose a nonsmooth version of the adjoint method, which can be applied to integral costs under an ODE constraint. This result constitutes a theoretical ground to the application of small step first order methods to solve a broad class of nonsmooth optimization problems with parametrized ODE constraints. This is illustrated with the convergence of small step first order methods based on the proposed nonsmooth adjoint.

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arXiv2021

Forwarding-Lyapunov design for the stabilization of coupled ODEs and exponentially stable PDEs

This paper is about the stabilization of a cascade system composed by an infinite-dimensional system, that we suppose to be exponentially stable, and an ordinary differential equation (ODE), that we suppose to be marginally stable. The system is controlled through the infinite-dimensional system. Such a structure is particularly useful when applying the internal model approach on infinite-dimensional systems. Our strategy relies on the forwarding method, which uses a Lyapunov functional and a Sylvester equation to build a feedback-law. Under some classical assumptions in the output regulation theory, we prove that the closed-loop system is globally exponentially stable.

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arXiv2021

ISS-Lyapunov strictification via observer design and integral action control for a Korteweg-de-Vries equation

The article deals with the output regulation of a Korteweg-de-Vries (KdV) system subject to a distributed disturbance. The control input and the output are located at the boundary. To achieve this objective, we follow a Lyapunov approach. For that, inspired by the strictication methodology proposed in [42] in the finite-dimensional context, we construct an ISS-Lyapunov functional for the KdV equation thanks to the use of an observer designed via the backstepping approach. Then, thanks to this Lyapunov functional, we apply the forwarding method in order to solve the desired output regulation problem.

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arXiv2020

Minimizing rational functions: a hierarchy of approximations via pushforward measures

This paper is concerned with minimizing a sum of rational functions over a compact set of high-dimension. Our approach relies on the second Lasserre's hierarchy (also known as the upper bounds hierarchy) formulated on the pushforward measure in order to work in a space of smaller dimension. We show that in the general case the minimum can be approximated as closely as desired from above with a hierarchy of semidefinite programs problems or, in the particular case of a single fraction, with a hierarchy of generalized eigenvalue problems. We numerically illustrate the potential of using the pushforward measure rather than the standard upper bounds hierarchy. In our opinion, this potential should be a strong incentive to investigate a related challenging problem interesting in its own; namely integrating an arbitrary power of a given polynomial on a simple set (e.g., unit box or unit sphere) with respect to Lebesgue or Haar measure.

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arXiv2020

Forwarding design for stabilization of a coupled transport equation-ODE with a cone-bounded input nonlinearity

We propose a new design technique for the stabilization of coupled ODE-PDE systems in feedforward form. In particular, we address the stabilization problem of a one-dimensional transport equation driven by a scalar ODE which is controlled via a cone-bounded nonlinearity. The unforced transport equation is conservative but not asymptotically stable. The proposed technique is inspired by the forwarding approach early introduced in the 90's. Well-posedness and asymptotic stability of the closed-loop system are discussed.

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arXiv2020

A Fredholm transformation for the rapid stabilization of a degenerate parabolic equation

This paper deals with the rapid stabilization of a degenerate parabolic equation with a right Dirich-let control. Our strategy consists in applying a backstepping strategy, which seeks to find an invertible transformation mapping the degenerate parabolic equation to stabilize into an exponentially stable system whose decay rate is known and as large as we desire. The transformation under consideration in this paper is Fredholm. It involves a kernel solving itself another PDE, at least formally. The main goal of the paper is to prove that the Fredholm transformation is well-defined, continuous and invertible in the natural energy space. It allows us to deduce the rapid stabilization.

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