Spectral inequalities for pseudo-differential operators and control theory on compact manifolds

Spain 21.11.2022

Spectral inequalities for pseudo-differential operators and control theory on compact manifolds

Author: Duvan Cardona

In this post, we explain some results related to the null-controllability of diffusion models on closed manifolds, which is a problem of wide interest in control theory, and its relation with the validity of spectral inequalities for differential and pseudo-differential operators. We summarise our results as follows.

• We extend some spectral inequalities for elliptic positive pseudo-differential operators on compact manifolds (criteria are based on the symbols defined by local coordinate systems). Hörmander classes $\Psi^m_{\rho,\delta}(M),$ $0\leq \delta \lt \rho \leq 1,$ $\rho \geq 1-\delta.$ • Spectral inequalities for elliptic positive pseudo-differential operators on compact Lie groups (criteria are based on the matrix-valued symbols of operators). Hörmander classes $\Psi^m_{\rho,\delta}(G),$ $0 \leq \delta \lt \rho \leq 1.$ • Applications to control theory: null-controllability for the fractional heat equation for an elliptic operator.

Below, we will explain our setting and we will give some preliminaries about the theory of pseudo-differential operators. In the end, we present our results about spectral inequalities and our applications to control theory.

 

1. Our setting: compact Lie groups and general manifolds without boundary

• Lie groups = manifolds with symmetries.
• compact Lie groups = are diffeomorphic to closed subgroups of $\textnormal{U}(N)=\{M\in \mathbb{C}^{N\times N}:M^*=M^{-1}\}$ for $N$ large enough.

• Examples: the torus $\mathbb{T}^n\cong (\R/\mathbb{Z})^n,$ linear Lie groups (groups of matrices), $\textnormal{SU}(n),$ $\textnormal{SO}(n),$ etc. In particular, $\textnormal{SU}(2)\cong \mathbb{S}^3;$ $0\leq \delta \lt \rho\leq 1.$ • If $M$ is a closed, connected and simply connected, then $M\cong \mathbb{S}^3.$ (The Poincaré conjecture proved by Perelman). Our approach induces global spectral inequalities on $M$ for any $0\leq \delta \lt \rho\leq 1.$ • General compact manifolds $0\leq \delta \lt \rho\leq 1,$ $\rho\geq 1-\delta.$

 

2. Some preliminary information

• In the late 1980 { H. Donnelly and C. Fefferman} in their celebrated Inventiones’ paper proved the doubling property

$\sup_{B(2R)}|\phi|\leq e^{C_1\lambda +C_2} \sup_{B(R)}|\phi|$

(1)

for any eigenfunction of the Laplacian $\Delta_g$ on $M,$ that is, $-\Delta_g \phi=\lambda^2 \phi,$ where $B(2R)$ and $B(R)$ represent concentric balls (associated to the geodesic distance) where the constants $C_1$ and $C_2$ are independent of $R>0,$ and depending only on $M.$ • Charles Fefferman, Fields medal, 1978.
• The doubling property

$\sup_{B(2R)}|\kappa|\leq e^{C_1\lambda +C_2} \sup_{B(R)}|\kappa|$

(2)

remains valid for sums of eigenfunctions

$\kappa=\sum_{\lambda_k\leq \lambda}a_k\phi_k\in \textnormal{span}\{\phi_k:\lambda_k^2\leq \lambda^2\}$

of the positive Laplacian $\Delta_g.$

 

3. Another spectral inequality (that implies the Donnelly-Fefferman doubling property

• Let $M$ be a compact Riemmanian manifold with (or without) smooth boundary $\partial M.$ Let $(\rho_j,\lambda_j^2)$ be the corresponding spectral data of the Laplacian $-\Delta_g:$ $-\Delta_g\rho_j=\lambda_j^2\rho_j.$ Then, for any non-empty open subset $\omega\subset M,$ we have the loss of orthogonality estimate

$\Vert \varkappa\Vert_{L^2(M)}\leq C_1e^{C_2 {\lambda}}\Vert \varkappa\Vert_{L^2(\omega)},\,\,\,\varkappa\in \textnormal{span}\{\rho_j:\lambda_j\leq \lambda\}.$

(3)

Moreover, the growth constant $C_1e^{C_2 {\lambda}}$ is sharp. This inequality was proved by Jerison-Lebeau/Lebeau-Robbiano/Lebeau-Zuazua.

4. The Lebeau-Robbiano result of the null-controllability of the heat equation
• Consequences: let { $\omega$} be a non-empty open subset of $M.$ Then, the heat equation for the positive Laplacian $\Delta_g$

$\begin{cases}u_t(x,t)+ \Delta_g u(x,t)=g(x,t)\cdot 1_\omega (x) ,& (x,t)\in M\times (0,T), \\u(0,x)=u_0,\end{cases}$

is null-controllable at any time $T>0,$ that is, for any initial condition $u_0,$ there is an input function $g\in L^2(M\times (0,T))$} such that the solution to (4) vanishes in time $T,$ that is $u(x,T)=0,$ $x\in M.$

 

5. Some remarks

• In general pseudo-differential operators are non-local and the use of Carleman estimates, which is the analytical tool by excellence in the proof of the Lebeau-Robbiano spectral inequality and their subsequent generalisations, are not valid.
• It is natural to ask if the doubling property

$\sup_{B(2R)}|\phi|\leq e^{C_1\lambda +C_2} \sup_{B(R)}|\phi|$

(5)

remains valid for sums of eigenfunctions

$\phi=\sum_{\lambda_k\leq \lambda}a_k\phi_k\in \textnormal{span}\{\phi_k:\lambda_k^2\leq \lambda^2\}$

of the positive elliptic pseudo-differential operators on compact manifolds (with or without boundary).

 

6. A motivating problem

• The extension of the Lebeau-Robbiano/Jerison-Lebeau/Lebeau-Zuazua spectral inequality

$\Vert \varkappa\Vert_{L^2(M)}\leq C_1e^{C_2 {\lambda}}\Vert \varkappa\Vert_{L^2(\omega)},\,\,\,\varkappa\in \textnormal{span}\{\rho_j:\lambda_j^\nu\leq \lambda^\nu\}.$

(6)

to positive elliptic pseudo-differential operator $E(x,D)$ of order $\nu>0,$ implies the null-controllability for its corresponding diffusion model

$\begin{cases}u_t(x,t)+ E(x,D) u(x,t)=g(x,t)\cdot 1_\omega (x) ,& (x,t)\in M\times (0,T), \\u(0,x)=u_0,\end{cases}$

(7)

at any time $T>0.$ Here $\omega \subset M,$ $\omega\neq \emptyset,$ is the controllability sensor. A natural question/motivating problem is to verify if this inequality remains valid for pseudo-differential operators.

 

7. Short overwiew about pseudo-differential operators

Pseudo-differential operators on $\Bbb R^n$ [Kohn+Nirenberg 1965, Hörmander 1967]:

$\widehat{f}(\xi) = \int_{\Bbb R^n} f(x)\ {\sf e}^{-2\pi \sf i} x\cdot\xi\ {\rm d}x, \quad Af(x) = \int_{\R^n} {\sf e}^{2\pi \sf i x\cdot\xi} \sigma_A(x, \xi) \widehat{f}(\xi) {\rm d}\xi,$

$\left| \partial_\xi^\alpha \partial_x^\beta \sigma_A(x,\xi) \right| \leq C_{\alpha\beta}\ \langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}, \langle{\xi}\rangle=(1+|\xi|^2)^{1/2}, \xi\in\R^n.$

$\Psi$DOs on the torus $\Bbb T^n=\Bbb R^n/\Bbb Z^n:$ Fourier coefficients with $\xi\in\mathbb{Z}^n$,

\widehat{f}(\xi)= \int_{\Bbb T^n} f(x) {\sf e}^{-{\sf i}2\pi x\cdot\xi} \rm dx, \quad Af(x) = \sum_{\xi\in\Bbb Z^n} {\sf e}^\sf i 2\pi x\cdot\xi \sigma_A(x,{ \xi}) \widehat{f}(\xi),

$\left| \triangle_\xi^\alpha \partial_x^\beta \sigma_A(x,\xi) \right| \leq C_{\alpha\beta}\ \langle\xi\rangle^{ m-\rho|\alpha|+\delta|\beta| }, \xi\in\Bbb Z^n.$

 
[Agranovich 1990], [McLean 1991], [Turunen 2000],[Ruzhansky+Turunen, JFAA, 2010].

$\Psi$DOs on a compact Lie group $G$:
[Ruzhansky+Turunen, Birkhaüser book, 2010]

$\widehat{f}(\xi) = \int_G f(x)\ \xi(x)^\ast\ {\rm d}x, \; Af(x) = \sum_{[\xi]\in\widehat G} d_{\xi} \ {\sf Tr}\left( \xi(x)\ { \sigma_A(x,\xi)}\ \widehat{f}(\xi) \right),$

${ \|} { \triangle_\xi^\alpha} X^\beta \sigma_A(x,\xi) { \|_{op}} \leq C_{\alpha\beta}\ { \langle\xi\rangle}^{m-\rho|\alpha|+\delta|\beta|},\; {\xi\in \widehat G,} \ \langle{\xi}\rangle =e.v.,\ \Delta_\xi=\textnormal{diff.op.},\ \cdots.$

 

8. Pseudo-differential operators on compact manifolds without boundary Kohn-Nirenberg+Hörmander

1. There is a well-known formulation of pseudo-differential operators on compact manifolds, (and so on compact Lie groups) by using symbols defined by charts.
2. If $U\subset \R^n$ is open, the symbol $a:U\times \R^n\rightarrow \mathbb{C},$ belongs to the Hörmander class $S^{ m}_{\rho},\delta({U\times \R^n}),$ $0\leqslant \rho,\delta\leqslant 1,$ if for every compact subset $K\subset U,$ the symbol inequalities,

$|\partial_{x}^\beta\partial_{\xi}^\alpha a(x,\xi)|\leqslant C_{\alpha,\beta,K}(1+|\xi|)^{ m} -\rho|\alpha|+\delta|\beta|,$

hold true uniformly in $x\in K$ and $\xi\in \R^n.$ 3. Then, a continuous linear operator $A:C^\infty_0(U) \rightarrow C^\infty(U)$ is a pseudo-differential operator of order $m,$ of $(\rho,\delta)$-type, if there exists
a function $a\in S^m_{\rho,\delta}(U\times \R^n),$ satisfying

$Af(x)=\int\limits_{\R^n}e^{2\pi i x\cdot \xi}a(x,\xi)(\mathscr{F}_{\R^n}{f})(\xi)d\xi,$

for all $f\in C^\infty_0(U),$ where

$(\mathscr{F}_{\R^n}{f})(\xi):=\int\limits_Ue^{-i2\pi x\cdot \xi}f(x)dx,$

is the Euclidean Fourier transform of $f$ at $\xi\in \R^n.$ 4. The class $S^m_{\rho,\delta}(U\times \R^n)$ on the phase space $U\times \R^n,$ is invariant under coordinate changes only if $\rho\geqslant 1-\delta,$ while a symbolic calculus (closed for products, adjoints, parametrices, etc.) is only possible for $\delta \lt \rho$ and $\rho\geqslant 1-\delta.$ •
5. $A:C^\infty_0(M)\rightarrow C^\infty(M)$ is a pseudo-differential operator of order $m,$ of $(\rho,\delta)$-type, $\rho\geqslant 1-\delta,$ if for every local coordinate patch $\omega: M_{\omega}\subset M\rightarrow U\subset \R^n,$ and for every $\phi,\psi\in C^\infty_0(U),$ the operator

$Tu:=\psi(\omega^{-1})^*A\omega^{*}(\phi u),\,\,u\in C^\infty(U), ^1$

is a pseudo-differential operator with symbol in $S^m_{\rho,\delta}(U\times \R^n).$ $^1$ As usually, $\omega^{*}$ and $(\omega^{-1})^*$ are the pullbacks induced by the maps $\omega$ and $\omega^{-1},$ respectively.}

 

9. Pseudo-differential operators on compact Lie groups (Ruzhansky-Turunen classes)

9.1. Basics on Representation theory
• Unitary representation $\xi$ of a group $G$ is $\xi:G\to \mathcal{L}(\mathcal{H}_{\xi})$, where $\mathcal{H}_{\xi}$ is a Hilbert representation space, such that $\xi(x)^*=\xi(x)^{-1}$ (unitary) and {$\xi(xy)=\xi(x)\ \xi(y)$} (preserves group structure).
• It is irreducible if $\xi\not=\xi_{1}\oplus\xi_{2}$ for some unitary representations $\xi_{1},\xi_{2}$.
• If $G$ is compact, it is enough to consider finite-dimensional $\mathcal{H}_{\xi}$, i.e. $\xi:G\to {\mathbb C}^{d_{\xi}\times d_{\xi}}$ for $d_{\xi}=\dim \mathcal{H}_{\xi}$ the dimension of $\xi$.

• Example: For $\mathbb{T}^n$, $\xi_k(x)=e^{2\pi i x\cdot k}$, $k\in\mathbb{Z}^n$. Then $\xi_k:\mathbb{T}^n\to\mathbb{C}^{1\times 1}$, $d_{\xi_k}=1$.
• The unitary dual $\widehat{G}$ is defined (omitting equivalent classes) as

$\widehat{G}$ = continuous irreducible unitary representations of G

• Fourier coefficient $\widehat{f}(\xi)$ of $f \in L^{1}(G)$ at $\xi \in \widehat{G}$ is

$\widehat{f}(\xi) = \int_G f(x)\ \xi(x)^*\ {\rm d}x.$

Note that $\widehat{f}(\xi)\in\mathcal{L}(\mathcal{H}_{\xi})$ is now an operator; a matrix if $G$ is compact.

• We define the weight $\langle{\xi}\rangle$ to measure the growth of the Fourier coefficients. This is essential if talking about function spaces and about the symbol classes associated to the unique bi-invariant Riemannian structure on $G$ (Laplacian).

For each $\xi\in\widehat{G}$, we have for the Laplacian:

$-{\mathcal{L}}_{G} \xi=\lambda_\xi^2 \xi$

In other words,

$-{\mathcal{L}}_G \xi_{ij}(x)=\lambda_{\xi}^{2}\xi_{ij}(x),\quad 1\leq i,j\leq d_\xi.$

We define

$\langle {\xi}\rangle :=(1+\lambda_{\xi}^2)^{1/2}$

These are the eigenvalues of the first order elliptic operator $(1-\mathcal{L}_{G})^{1/2}$, and we note that $\langle {\xi}\rangle$ has here multiplicity $d_{\xi}^{2}.$

9.2. Operators on compact Lie groups

• If $A:C^\infty(G)\to C^\infty(G)$ cont. \& linear, then we can define its symbol as

$(x,\xi)\mapsto \sigma_A(x,\xi),\quad \sigma_A(x,\xi) := \xi(x)^* (A\xi)(x),$

where $(A\xi)_{ij}=A(\xi_{ij})$ acts on components. This (full) symbol of $A$ is matrix-valued: $\sigma_A(x,\xi)\in\mathbb{C}^{d_\xi\times d_\xi}$. Then we can show that

$Af(x) = \sum_{\xi\in\widehat{G}} d_\xi {\rm Tr}\left( \xi(x) \sigma_A(x,\xi) \widehat{f}(\xi) \right).$

This symbol is well-defined
on $G\times\widehat{G}$ (non-commutative phase space).

There are many familiar features, e.g. if

$Af(x)=\int_G K(x,y) f(y) dy=\int_G f(y) R_A(x,y^{-1}x)\ dy,$

then $\sigma_A(x,\xi)=\int_G R_A(x,y)\ \xi(y)^* dy$ (i.e. symbol is F.T. of the kernel).

We have (full) symbolic calculus for this quantization, and (!) with formulae resembling the familiar formulae on $\R^n$.

• As usual, $\Psi^m_{\rho,\delta}(G)$ is the Hörmander class of pseudo-differential operators on $G$, i.e. such that their localisations have symbols in $S^m_{\rho,\delta}(\R^n)$.

Let $\langle{\xi}\rangle$ be the eigenvalue of $(I-\mathcal{L}_G)^{1/2}$, $\mathcal{L}_G$ Laplacian, corresponding to $\xi$.

Now, we define the class of symbols $\sigma_A\in S^m_{\rho,\delta}(G\times \widehat{G})$ by

$\| \triangle_\xi^\alpha X^{\beta}_{x} \sigma_A(x,\xi)\|_{op} \le C_{\alpha\beta} \langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|}$

Independent of the choice of a strongly admissible collection for $1\leq \delta \lt \rho\leq 1.$

• Ruzhansky-Turunen-Wirth Theorem: Let $0\leq \delta \lt \rho\leq 1,$ and $\rho\geq 1-\delta.$ Then, $A\in\Psi^m_{\rho,\delta}(G)$ if and only if $\sigma_A\in S^m_{\rho,\delta}(G\times \widehat{G}).$

 

10. Elliptic pseudo-differential operators on compact Lie groups and on general compact manifolds

• Let $M$ be a closed manifold.
We say that an operator $A\in \Psi^m_{\rho,\delta}(M)$ is elliptic of order $m\in \R,$ if in any local coordinate system $U,$ and for any compact subset $K\subset{U},$ its symbol satisfies the inequality

$C_1 (1+|\theta|)^m\leq |a(x,\theta)|\leq C_2 (1+|\theta|)^m, |\theta|\geq R,$

(8)

for some $R>0,$ uniformly in $x\in K,$ and $\theta\in \R^n.$ • We say that a matrix-valued symbol $a:G\times \widehat{G}\rightarrow \cup_{\ell}\mathbb{C}^{\ell\times \ell},$ is elliptic of order $m\in \R$ if
– $\exists R>0,\, a(x,[\theta])\in \textnormal{GL}(d_\theta,\mathbb{C})$ is an invertible matrix for all $\langle \theta\rangle\geq R$.
– The following symbol inequality holds uniformly in $x\in G,$ when $\langle \theta\rangle\geq R:$

$\|( a(x,[\theta]))^{-1}\|_{\textnormal{op}}\leqslant C\langle\theta\rangle^{-m}.$

 

11. Main results: Spectral inequalities for pseudo-differential operators and applications to control theory

• D. Cardona, 2022. Let $\nu>0,$ and let $0\leq \delta \lt \rho\leq 1$ be such that $\rho\geq 1-\delta.$ Let $E(x,D)\in \Psi^\nu_{\rho,\delta}(M)$ be an elliptic positive pseudo-differential operator of order $\nu>0.$ Let $(x,\xi)\in T^*M,$ and assume that for any $\xi\neq 0,$ $E(x,\xi)>0$ is strictly positive. Then, for any non-empty open subset $\omega\subset M,$ we have

$\Vert \varkappa\Vert_{L^2(M)}\leq C_1e^{C_2 {\lambda}}\Vert \varkappa\Vert_{L^2(\omega)},\,\,\,\varkappa\in \textnormal{span}\{\rho_j:\lambda_j\leq \lambda\}.$

(9)

• D. Cardona, 2022. For any $R>0$ let $B(x,R)$ be a ball defined by the geodesic distance, of radius $R>0$ and centred at $x.$ Then,

$\sup_{B(x,2R)}|\varkappa|\leq e^{C_1' {\lambda}+C_2'} \sup_{B(x,R)}|\varkappa|,\,\,\,\varkappa\in \textnormal{span}\{\rho_j:\lambda_j\leq \lambda\},$

(10)

with $C_1'=C_{1}'(R)$ and $C_2'=C_2'(R)$ are dependent only on the radius $R>0$ but not on $\varkappa.$ • D. Cardona, 2022. Let $\nu>0,$ and let $0\leq \delta \lt \rho\leq 1$ be such that $\rho\geq 1-\delta.$ Let $E(x,D)\in \Psi^\nu_{\rho,\delta}(M)$ be a positive elliptic pseudo-differential operator of order $\nu>0.$ Let $(x,\xi)\in T^*M,$ and assume that for any $\xi\neq 0,$ $E(x,\xi)>0$ is strictly positive. Then,
for any $\alpha>1/\nu,$ the fractional diffusion problem

$\begin{cases}u_t(x,t)+ E(x,D)^\alpha u(x,t)=g(x,t)\cdot 1_\omega (x) ,& (x,t)\in M\times (0,T), \\u(0,x)=u_0,\end{cases}$

(11)

is null-controllable at any time $T>0,$ for any non-empty open subset $\omega\subset M.$ • D. Cardona, J. Delgado, M. Ruzhansky, 2022. Let $0\leq \delta \lt \rho\leq 1.$ Let $A\in \Psi^m_{\rho,\delta}(G\times \widehat{G})$ be a positive elliptic pseudo-differential operator of order $m>0.$ Assume that $\sigma_A(x,\xi)\geq 0$ for all $(x,[\xi])\in G\times \widehat{G}.$ Let $(e_j,\lambda_j^m), \lambda_j\geq 0,$ be the corresponding spectral data of $A,$ determined by the eigenvalue problem $Ae_j=\lambda_j^me_j$ with the eigenfunctions $e_j$ being $L^2$-normalised. Then the following spectral estimates are valid:
– For any non-empty open subset $\omega\subset G,$ we have

$\Vert \varkappa\Vert_{L^2(G)}\leq C_1e^{C_2 {\lambda}}\Vert \varkappa\Vert_{L^2(\omega)},\,\,\,\varkappa\in \textnormal{span}\{e_j:\lambda_j\leq \lambda\},$

(12)

with $C_1=C_1(\omega)$ and $C_2=C_2(\omega)$ depending on $\omega,$ but not on $\varkappa.$ – For any $R>0$ let $B(x,R)$ be a ball defined by the geodesic distance, of radius $R>0$ and centred at $x.$ Then,

$\sup_{B(x,2R)}|\varkappa|\leq e^{C_1' {\lambda}+C_2'} \sup_{B(x,R)}|\varkappa|,\,\,\,\varkappa\in \textnormal{span}\{e_j:\lambda_j\leq \lambda\},$

(13)

with $C_1'=C_{1}'(R)$ and $C_2'=C_2'(R)$ depending only on the radius $R>0$ but not on $\varkappa.$ \end{itemize}
• D. Cardona, J. Delgado, M. Ruzhansky, 2022. Let $A$ be a positive and elliptic pseudo-differential operator of order $m > 0$ in the Hörmander class $\Psi^ m _{\rho,\delta}(G\times \widehat{G})$ and let $u_0\in L^2(G)$ be an initial datum. Asuma that $\sigma_A(x,[\xi])\geq 0$ for all $(x,[\xi])\in G\times \widehat{G}.$ Then, for any $\alpha>1/ m ,$ the fractional diffusion model

$\begin{cases}u_t(x,t)+ A^\alpha u(x,t)=g(x,t)\cdot 1_\omega (x) ,& (x,t)\in G\times (0,T), \\u(0,x)=u_0,\end{cases}$

(14)

is null-controllable at any time $T>0,$ that is, there exists an input function $g=g(x,t)\in L^2(G)$ such that for any $x\in G, u(x,T)=0.$

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