[Paper Review] Control of the Grushin equation: non-rectangular control region and minimal time

Michel Duprez, Armand Koenig|arXiv (Cornell University)|Jul 3, 2018

Stability and Controllability of Differential EquationsEngineering24 references19 citations

TL;DR

This paper establishes the minimal time for null-controllability of the Grushin equation with non-rectangular control regions using the fictitious control method and complex polynomial $L^2$ estimates. It proves that if the control region contains an $varepsilon$-neighborhood of a path from bottom to top boundary, null-controllability holds for all $T > a^2/2$, where $a$ is the maximum horizontal deviation of the path, and shows this time is sharp via observability inequalities.

ABSTRACT

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.

Motivation & Objective

• To determine the minimal time for null-controllability of the Grushin equation when the control region is non-rectangular and does not touch the degeneracy line $x=0$.
• To extend known controllability results beyond vertical strips to more general control domains containing paths from bottom to top boundary of the domain.
• To establish sharpness of the minimal time $T > a^2/2$ via observability inequalities interpreted as $L^2$ estimates on complex polynomials.
• To generalize the minimal time result to non-rectangular regions by constructing a cutoff function and using energy estimates.

Proposed method

• The fictitious control method is employed to reduce the null-controllability problem to an observability inequality.
• A cutoff function $\theta$ is constructed using mollification of the indicator function of the right component of $\mathbb{R}^2 \setminus \tilde{\gamma}(\mathbb{R})$, ensuring $\operatorname{supp}(\nabla\theta) \subset \omega_0 \subset \omega$.
• The solution is decomposed into free and internal parts using a partition of unity, and the control is derived from energy estimates in the weighted space $V(\Omega)$.
• The proof relies on the spectral structure of the Grushin operator and the use of complex analysis techniques, particularly Runge's theorem, to construct counterexamples to observability.
• The minimal time $T > a^2/2$ is shown to be sharp by interpreting the observability inequality as an $L^2$ estimate on complex polynomials and constructing a counterexample in the $L^\infty$ norm.

Experimental results

Research questions

• RQ1 What is the minimal time required to achieve null-controllability of the Grushin equation when the control region is not a vertical strip but contains a path from bottom to top boundary of the domain?
• RQ2 Can the minimal time $T = a^2/2$ be achieved for non-rectangular control regions, and is this time sharp?
• RQ3 How does the geometry of the control region, particularly its distance from the degeneracy line $x=0$, affect the minimal controllability time?
• RQ4 Can the observability inequality for the Grushin equation be characterized as an $L^2$ estimate on complex polynomials, and what does this imply for controllability?
• RQ5 Is the minimal time $T > a^2/2$ sharp for control regions that are not vertical strips, and how is this proven?

Key findings

• If the control region $\omega$ contains an $\varepsilon$-neighborhood of a continuous path $\gamma$ from $(-1,1)\times\{0\}$ to $(-1,1)\times\{\pi\}$, then the Grushin equation is null-controllable for all $T > a^2/2$, where $a = \max_s |\operatorname{abscissa}(\gamma(s))|$.
• The minimal time $T > a^2/2$ is sharp: for any $T \leq a^2/2$, the observability inequality fails, implying null-controllability is impossible.
• The proof of the negative result relies on interpreting the observability inequality as an $L^2$ estimate on complex polynomials, and constructing a counterexample using Runge's theorem.
• The control is localized in $\omega$ via a carefully constructed cutoff function $\theta$ with $\operatorname{supp}(\nabla\theta) \subset \omega_0 \subset \omega$, ensuring the control acts only on the desired region.
• The result generalizes previous findings on vertical strips and symmetric strips, showing that the minimal time depends only on the maximal horizontal extent of the path, not on its vertical variation.
• The sharpness of the minimal time is established by showing that the $L^2$ norm of a polynomial on a disk cannot be bounded by its $L^2$ norm on a set $U$ that does not approach the boundary, violating the required observability estimate.