arXiv subjects

Lucas Davron

Publications and source records attributed to Lucas Davron.

4 recordsLinked to original sources

arXiv2026

Output tracking: an application for irrational SISO transfer functions and a survey for MIMO systems

In this paper we consider (MIMO) finite dimensional linear and time-invariant systems and irrational single-input single-output (SISO) transfer functions. In both cases our aim is to describe as best as possible the range of the input-output map $u(\cdot) \mapsto y(\cdot)$. The theory in finite dimension is quite satisfying but rather scarce, the first aim of this paper is to collect the main results in this direction in a comprehensive way. Our second aim is to improve a recent result on the tracking problem for irrational SISO transfer functions, allowing one to completely describe the outputs of such system for inputs $u \in L^2(0,\infty)$. An application is given for a heat equation with polynomial coefficients.

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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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arXiv2025

Exact output tracking for the one-dimensional heat equation and applications to the interpolation problem in Gevrey classes of order 2

This paper provides a complete characterization of the Dirichlet boundary outputs that can be exactly tracked in the one-dimensional heat equation with Neumann boundary control. The problem consists in describing the set of boundary traces generated by square-integrable controls over a finite or infinite time horizon. We show that these outputs form a precise functional space related to Gevrey regularity of order 2. In the infinite-time case, the trackable outputs are precisely those functions whose successive derivatives satisfy a weighted summability condition, which corresponds to specific Gevrey classes. For finite-time horizons, an additional compatibility condition involving the reachable space of the system provides a full characterization. The analysis relies on Fourier-Laplace transform, properties of Hardy spaces, the flatness method, and a new Plancherel-type theorem for Hilbert spaces of Gevrey functions. Beyond control theory, our results yield an optimal solution to the classical interpolation problem in Gevrey-$2$ classes, which improves results of Mitjagin on the optimal loss factor. The techniques developed here also extend to variants of the heat system with different boundary conditions or observation points.

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arXiv2024

On the control of LTI systems with rough control laws

The theory of linear time invariant systems is well established and allows, among other things, to formulate and solve control problems in finite time. In this context the control laws are typically taken in a space of the form L^p(0,T;U). In this paper we consider the possibility of taking control laws in (H^1(0,T;U))* , which induces non-trivial issues. We overcome these difficulties by adapting the functional setting, notably by considering a generalized final state for the systems under consideration. In addition we collect time regularity properties and we pretend that in general it is not possible to consider control laws in H^{-1}(0,T;U). Then, we apply our results to propose an interpretation of the inifinite order of defect for an observability inequality, in terms of controllability properties.

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