arXiv subjects

Armand Koenig

Publications and source records attributed to Armand Koenig.

10 recordsLinked to original sources

arXiv2025

Control of the half-heat equation

In this paper we investigate null-controllable initial states of the half heat equation controlled from a sub-arc $\omega$ of the unit circle. We also study the projection on positive frequencies of the half-heat equation. For this projected half-heat equation, we obtain necessary as well as sufficient conditions for an initial condition to be null-controllable. These conditions, which are almost sharp, are expressed in term of projections on positive frequencies of functions supported on $\omega$. From these results, and with the help of classical results on sum of holomorphic and anti-holomorphic functions, we also treat the (unprojected) half-heat equation. Surprisingly, without using our conditions on null-controllable states, we are able to show that the space of null-controllable functions does not depend on time by using a result of separation of singularities for holomorphic functions.

math.AP↗

arXiv2023

Null-controllability for weakly dissipative heat-like equations

We study the null-controllability properties of heat-like equations posed on the whole Euclidean space $\mathbb R^n$. These evolution equations are associated with Fourier multipliers of the form $\rho(\vert D_x\vert)$, where $\rho\colon[0,+\infty)\rightarrow\mathbb C$ is a measurable function such that $\Re\rho$ is bounded from below. We consider the ``weakly dissipative'' case, a typical example of which is given by the fractional heat equations associated with the multipliers $\rho(\xi) = \xi^s$ in the regime $s\in(0,1)$, for which very few results exist. We identify sufficient conditions and necessary conditions on the control supports for the null-controllability to hold. More precisely, we prove that these equations are null-controllable in any positive time from control supports which are sufficiently thick at all scales. Under assumptions on the multiplier $\rho$, in particular assuming that $\rho(\xi) = o(\xi)$, we also prove that the null-controllability implies that the control support is thick at all scales, with an explicit lower bound of the thickness ratio in terms of the multiplier $\rho$.Finally, using Smith-Volterra-Cantor sets, we provide examples of non-trivial control supports that satisfy these necessary or sufficient conditions.

math.AP↗

arXiv2023

Null-controllability of underactuated linear parabolic-transport systems with constant coefficients

The goal of the present article is to study controllability properties of mixed systems of linear parabolic-transport equations, with possibly non-diagonalizable diffusion matrix, on the one-dimensional torus. The equations are coupled by zero or first order coupling terms, with constant coupling matrices, without any structure assumptions on them. The distributed control acts through a constant matrix operator on the system, so that there might be notably less controls than equations, encompassing the case of indirect and simultaneous controllability. More precisely, we prove that in small time, such kind of systems are never controllable in appropriate Sobolev spaces, whereas in large time, null-controllability holds, for sufficiently regular initial data, if and and only if a spectral Kalman rank condition is verified. We also prove that initial data that are not regular enough are not controllable. Positive results are obtained by using the so-called fictitious control method together with an algebraic solvability argument, whereas the negative results are obtained by using an appropriate WKB construction of approximate solutions for the adjoint system associated to the control problem. As an application to our general results, we also investigate into details the case of $2\times2$ systems (i.e., one pure transport equation and one parabolic equation).

math.OC↗

arXiv2022

Null-controllability for weakly dissipative heat-like equations

We study the null-controllability properties of heat-like equations posed on the whole Euclidean space $\mathbb R^n$. These evolution equations are associated with Fourier multipliers of the form $\rho(\vert D_x\vert)$, where $\rho\colon[0,+\infty)\rightarrow\mathbb C$ is a measurable function such that $\Re\rho$ is bounded from below. We consider the ``weakly dissipative'' case, a typical example of which is given by the fractional heat equations associated with the multipliers $\rho(\xi) = \xi^s$ in the regime $s\in(0,1)$, for which very few results exist. We identify sufficient conditions and necessary conditions on the control supports for the null-controllability to hold. More precisely, we prove that these equations are null-controllable in any positive time from control supports which are sufficiently thick at all scales. Under assumptions on the multiplier $\rho$, in particular assuming that $\rho(\xi) = o(\xi)$, we also prove that the null-controllability implies that the control support is thick at all scales, with an explicit lower bound of the thickness ratio in terms of the multiplier $\rho$. Finally, using Smith-Volterra-Cantor sets, we provide examples of non-trivial control supports that satisfy these necessary or sufficient conditions.

math.AP↗

arXiv2022

Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets

We consider the null-controllability problem for the generalized Baouendi-Grushin equation $(\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = 1_\omega u$ on a rectangular domain. Sharp controllability results already exist when the control domain $\omega$ is a vertical strip, or when $q(x) = x$. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $\omega$. In some geometries for $\omega$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability. Our proof relies on several tools: known results when $\omega$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schr\"odinger operator $-\partial_x^2 + \nu^2 q(x)^2$ when $\Re(\nu)>0$, pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound.

math.OC↗

arXiv2020

On the small-time local controllability of a KdV system for critical lengths

This paper is devoted to the local null-controllability of the nonlinear KdV equation equipped the Dirichlet boundary conditions using the Neumann boundary control on the right. Rosier proved that this KdV system is small-time locally controllable for all non-critical lengths and that the uncontrollable space of the linearized system is of finite dimension when the length is critical. Concerning critical lengths, Coron and Cr\'{e}peau showed that the same result holds when the uncontrollable space of the linearized system is of dimension 1, and later Cerpa, and then Cerpa and Cr\'epeau established that the local controllability holds at a finite time for all other critical lengths. In this paper, we prove that, for a class of critical lengths, the nonlinear KdV system is {\it not} small-time locally controllable.

math.AP↗

arXiv2019

Null-controllability of linear parabolic-transport systems

Over the past two decades, the controllability of several examples of parabolic-hyperbolic systems has been investigated. The present article is the beginning of an attempt to find a unified framework that encompasses and generalizes the previous results. We consider constant coefficients heat-transport systems with coupling of order zero and one, with a locally distributed control in the source term, posed on the one dimensional torus. We prove the null-controllability, in optimal time (the one expected because of the transport component) when there is as much controls as equations. When the control acts only on the transport (resp. parabolic) component, we prove an algebraic necessary and sufficient condition, on the coupling term, for the null controllability. The whole study relies on a careful spectral analysis, based on perturbation theory. The negative controllability result in small time is proved on solutions localized on high hyperbolic frequencies, that solve a pure transport equation up to a compact term. The proof of the positive result in large time relies on a spectral decomposition into low, and asymptotically parabolic or hyperbolic frequencies.

math.AP↗

arXiv2018

Control of the Grushin equation: non-rectangular control region and minimal time

This paper is devoted to the study of the internal null-controllability of the Grushin equation. We determine the minimal time of controllability for a large class of non-rectangular control region. We establish the positive result thanks to the fictitious control method and the negative one by interpreting the associated observability inequality as an $L^2$ estimate on complex polynomials.

math.AP↗

arXiv2018

Lack of null-controllability for the fractional heat equation and related equations

We consider the equation $(\partial_t + \rho(\sqrt{-\Delta}))f(t,x) = \mathbf 1_\omega u(t,x)$, $x\in \mathbb R$ or $\mathbb T$. We prove it is not null-controllable if $\rho$ is analytic on a conic neighborhood of $\mathbb R_+$ and $\rho(\xi) = o(|\xi|)$. The proof relies essentially on geometric optics, i.e.\ estimates for the evolution of semiclassical coherent states. The method also applies to other equations. The most interesting example might be the Kolmogorov-type equation $(\partial_t -\partial_v^2 + v^2\partial_x)f(t,x,v) = \mathbf 1_\omega u(t,x,v)$ for $(x,v)\in \Omega_x\times \Omega_v$ with $\Omega_x = \mathbb R$ or $\mathbb T$ and $\Omega_v = \mathbb R$ or $(-1,1)$. We prove it is not null-controllable in any time if $\omega$ is a vertical band $\omega_x\times \Omega_v$. The idea is to remark that, for some families of solutions, the Kolmogorov equation behaves like the rotated fractional heat equation $(\partial_t + \sqrt i(-\Delta)^{1/4})g(t,x) = \mathbf 1_\omega u(t,x)$, $x\in \mathbb T$.

math.AP↗

arXiv2017

Non null controllability of the Grushin equation in 2D

We are interested in the exact null controllability of the equation $\partial_t f - \partial_x^2 f - x^2 \partial_y^2f = \mathbf 1_\omega u$, with control $u$ supported on $\omega$. We show that, when $\omega$ does not intersect a horizontal band, the considered equation is never null-controllable. The main idea is to interpret the associated observability inequality as an $L^2$ estimate on entire functions, which Runge's theorem disproves. To that end, we study in particular the first eigenvalue of the operator $-\partial_x^2 + (nx)^2$ with Dirichlet conditions on $(-1,1)$ and we show a quite precise estimation it satisfies, even when $n$ is in some complex domain.

math.AP↗