arXiv subjects

Thomas Perrin

Publications and source records attributed to Thomas Perrin.

9 recordsLinked to original sources

arXiv2026

Integer quadratic obstructions to the local controllability of the Burgers equation

We consider the Burgers equation with a scalar control acting through a fixed spatial profile, in a regime where the linearized system is not controllable. We identify natural conditions on the control profile leading to quadratic obstructions to small-time local null-controllability, quantified by negative Sobolev norms of the control of arbitrary integer order. We show that these conditions are related to iterated Lie brackets. The proof relies on repeated integrations by parts in the quadratic expansion combined with nonlinear remainder estimates. Finally, we construct smooth spatial profiles that produce obstructions at every negative integer order.

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arXiv2026

Fractional quadratic obstructions to the local controllability of the Burgers equation

We study the local controllability near zero of the Burgers equation with a scalar control and a fixed space-dependent source profile, in the case where the linearized system fails to be controllable and a second-order analysis is therefore required. We prove that quadratic obstructions to finite-time controllability can be quantified by Sobolev norms of the control with fractional negative exponents ranging over a full interval. To our knowledge, this is the first example, for a natural physical PDE, of a continuous scale of fractional quadratic obstructions, and the first such continuous scale for finite-time local controllability. Our explicit constructions shed light on the origin of fractional obstructions for partial differential equations, by relating the obstruction exponent to the regularity, and in some cases to the physical-space singularity, of the source profile. We identify the natural structural conditions on the source profile leading to obstructions quantified by the $H^{-1}$ and $H^{-5/4}$ norms of the control, thereby providing a general framework in which the previously studied case of a constant source profile fits naturally. In this constant-profile case, we improve existing results by identifying the arithmetic condition on the Fourier mode which ensures that a small-time obstruction actually persists in finite time. Finally, we derive sharp nonlinear remainder estimates adapted to the precise regularity of the source profile. These estimates make most of our obstruction results optimal with respect to the smallness assumption imposed on the control.

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arXiv2026

Approximate controllability of a bilinear wave equation and minimum time

We study the global approximate controllability (GAC) of a Klein-Gordon wave equation, posed on the torus $\mathbb{T}^d$ of arbitrary dimension $d\in \mathbb{N}^*$, with bilinear control potentials supported on the first $(2d+1)$-Fourier modes. Let $Z(W_0)\subset \mathbb{T}^d$ be the set of essential zeroes of the initial state $W_0\in H^1\times L^2(\mathbb{T}^d)$, and $r(W_0)\geq 0$ be the maximum radius of a ball of $\mathbb{T}^d$ contained in $Z(W_0)$. Due to finite speed of propagation, the minimum control time starting from $W_0$ is necessarily larger than or equal to $r(W_0)$. We prove the following three facts. In low dimensions $d \in \{1,2\}$: the minimum time for GAC from $W_0 \neq 0$ is equal to $r(W_0)$. In any dimensions $d\geq 3$: the minimum time for GAC from $W_0$ is zero if $Z(W_0)$ has zero Lebesgue measure; and the GAC in sufficiently large time from all $W_0\neq 0$. The proof strategy consists in combining Lie bracket techniques \emph{\`a la Agrachev-Sarychev} with the propagation of well-prepared positive states.

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arXiv2025

Blow-up, decay, and convergence to equilibrium for focusing damped cubic Klein-Gordon and Duffing equations

We study long-time dynamics of the damped focusing cubic Klein-Gordon equation on a compact three-dimensional Riemannian manifold, together with its space-independent reduction, the damped focusing Duffing equation. Under the geometric control condition on the damping and an assumption on the set of stationary solutions, we establish a sharp trichotomy for initial data with energy slightly above that of the ground state: every solution either blows up in finite time, decays exponentially to zero, or converges to a ground state. We provide a complete classification of the Duffing dynamics above the energy of the constant solution, use it to construct Klein-Gordon solutions realising each of the three behaviours in the case of a domain without boundary, and derive a simple spectral criterion ensuring that the ground states are nonconstant - and hence that different types of behaviour can indeed occur for solutions with initial energy above that of the ground state.

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arXiv2025

Small-time local control of a Schr\"odinger equation: a negative and a positive quadratic result

We study the small-time local controllability (STLC) of a bilinear Schr\"odinger 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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arXiv2024

Uniform estimates for solutions of nonlinear focusing damped wave equations

For a damped wave (or Klein-Gordon) equation on a bounded domain, with a focusing power-like nonlinearity satisfying some growth conditions, we prove that a global solution is bounded in the energy space, uniformly in time. Our result applies in particular to the case of a cubic equation on a bounded domain of dimension 3.

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arXiv2023

Change of regularity in controllability and observability of systems of wave equations

Solutions of a system of wave equations are constructed for both homogeneous and inhomogeneous Dirichlet boundary conditions at every regularity level. We prove that boundary observability, and thus boundary exact controllability, at some regularity level is equivalent to boundary observability at all levels. The main ingredient is the ellipticity of a time-derivative on the Neumann trace of the solution, which is proved by microlocal techniques.

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arXiv2023

Local controllability around a regular solution and null-controllability of scattering solutions for semilinear wave equations

On a Riemannian manifold with or without boundary, and whether bounded or unbounded, we consider a semilinear wave (or Klein-Gordon) equation with a subcritical nonlinearity (either defocusing or focusing). We establish local controllability around a partially analytic solution, under the Geometric Control Condition. Specifically, some blow-up solutions can be controlled. In the case of a Klein-Gordon equation on a non-trapping exterior domain of small dimension, we prove the null-controllability of scattering solutions. The proof is based on local energy decay and global-in-time Strichartz estimates. Several consequences are presented, including the null-controllability of a solution initiated near the ground state in some focusing cases, and exact controllability in some defocusing cases.

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arXiv2023

The damped focusing cubic wave equation on a bounded domain

For the focusing cubic wave equation on a compact Riemannian manifold of dimension $3$, the dichotomy between global existence and blow-up for solutions starting below the energy of the ground state is known since the work of Payne and Sattinger. In the case of a damped equation, we prove that the dichotomy between global existence and blow-up still holds. In particular, the damping does not prevent blow-up. Assuming that the damping satisfies the geometric control condition, we then prove that any global solution converges to a stationary solution along a time sequence, and that global solutions below the energy of the ground state can be stabilised, adapting the proof of a similar result in the defocusing case.

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