[Paper Review] Null-controllability of evolution equations associated with fractional Shubin operators through quantitative Agmon estimates

Paul Alphonse|arXiv (Cornell University)|Dec 8, 2020

Stability and Controllability of Differential Equations30 references4 citations

TL;DR

This paper establishes null-controllability for evolution equations driven by fractional anisotropic Shubin operators on $\mathbb{R}^n$ using quantitative Agmon estimates and Gelfand-Shilov regularity. It proves that such equations are null-controllable from thick control supports in any positive time under the condition $2sm > 1$, with sharp smoothing estimates in Gelfand-Shilov spaces for the associated semigroups, generalizing results for fractional harmonic oscillators.

ABSTRACT

We consider the anisotropic Shubin operators $(-Δ)^m + \vert x\vert^{2k}$ acting on the space $L^2(\mathbb R^n)$, with $k, m \geq1$ some positive integers. We provide sharp quantitative estimates in Gelfand-Shilov spaces for the eigenfunctions of these selfadjoint differential operators, that is, exponential decay estimates both for these functions and their Fourier transforms in $L^2(\mathbb R^n)$. The strategy implemented is based on the classical approach to obtain Agmon estimates in spectral theory. By using a Weyl law for the eigenvalues of the anisotropic Shubin operators, we also describe the smoothing properties of the semigroups generated by the fractional powers of these operators, with precise estimates in short times. This description allows us to prove positive null-controllability results for the associated evolution equations posed on the whole space $\mathbb R^n$, from control supports which are thick with respect to densities and in any positive time. We generalize in particular results known for the evolution equations associated with fractional harmonic oscillators.

Motivation & Objective

• To establish null-controllability of evolution equations associated with fractional anisotropic Shubin operators on the whole space $\mathbb{R}^n$.
• To analyze the smoothing properties of the semigroups generated by fractional powers of these operators using Gelfand-Shilov regularity.
• To extend known null-controllability results for fractional harmonic oscillators to the more general class of anisotropic Shubin operators.
• To characterize the minimal geometric conditions on control supports ensuring null-controllability, particularly thickness and density-based thickness.
• To derive precise quantitative estimates for the regularity of solutions in short time via spectral theory and Weyl laws.

Proposed method

• Derives sharp quantitative Agmon-type estimates for eigenfunctions of anisotropic Shubin operators using spectral theory and functional calculus.
• Applies a Weyl law for eigenvalues of $H_{k,m} = (-\Delta)^m + |x|^{2k}$ to describe the short-time smoothing behavior of the semigroup $e^{-tH^s_{k,m}}$.
• Establishes that $e^{-tH^s_{k,m}}g$ belongs to Gelfand-Shilov spaces $S^{ u_{s,k,m}}_{ u_{s,k,m}}(\mathbb{R}^n)$ with explicit exponents $\nu_{s,k,m} = \max\left(\frac{1}{2sk}, \frac{m}{k+m}\right)$ and $\mu_{s,k,m} = \max\left(\frac{1}{2sm}, \frac{k}{k+m}\right)$.
• Uses a cutoff function approach with Weyl quantization to prove convergence of operators $\chi_j^w$ to identity in $L^2$, and controls commutator norms via symbolic calculus.
• Applies a quantitative version of the Calderón-Vaillancourt theorem to bound operator norms uniformly in the cutoff parameter $j$, ensuring stability in convergence.
• Combines these estimates with duality arguments and Carleman inequalities to prove null-controllability from thick control supports under the condition $2sm > 1$.

Experimental results

Research questions

• RQ1 Under what geometric and analytic conditions is the evolution equation $\partial_t f + H^s_{k,m}f = h\mathbbm{1}_\omega$ null-controllable on $\mathbb{R}^n$?
• RQ2 What are the precise Gelfand-Shilov regularity properties of the semigroup $e^{-tH^s_{k,m}}$ for fractional anisotropic Shubin operators?
• RQ3 How do the smoothing effects of $H^s_{k,m}$ in short time depend on the parameters $s$, $k$, and $m$?
• RQ4 Can null-controllability be established from control supports that are thick with respect to densities, particularly in the isotropic case $k=m=l$?
• RQ5 What is the sharp threshold for null-controllability in terms of the parameter $s$ when $k=m=1$?

Key findings

• The semigroup $e^{-tH^s_{k,m}}$ maps $L^2(\mathbb{R}^n)$ into the Gelfand-Shilov space $S^{ u_{s,k,m}}_{ u_{s,k,m}}(\mathbb{R}^n)$ for all $t > 0$, with $\nu_{s,k,m} = \max\left(\frac{1}{2sk}, \frac{m}{k+m}\right)$ and $\mu_{s,k,m} = \max\left(\frac{1}{2sm}, \frac{k}{k+m}\right)$.
• For short times $0 < t \ll 1$, the semigroup exhibits quantitative smoothing in Gelfand-Shilov norms, with explicit decay estimates for $\|x^\alpha \partial^\beta_x e^{-tH^s_{k,m}}g\|_{L^2}$.
• Null-controllability holds from any control support $\omega \subset \mathbb{R}^n$ that is thick (in the sense of [18, 41]) whenever $2sm > 1$, in any positive time $T > 0$.
• In the isotropic case $k = m = l$, null-controllability is achieved from control supports that are thick with respect to densities, extending results from [32] to fractional Shubin operators.
• For the special case $k = m = 1$ and $s > 1$, null-controllability holds from any measurable control support $\omega$ with positive Lebesgue measure, regardless of geometric thickness.
• The paper generalizes known null-controllability results for fractional harmonic oscillators ($H^s_{1,1}$) to the broader class of anisotropic Shubin operators, providing a unified framework based on Agmon estimates and spectral theory.