# 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.

\PsiDOs 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].

\PsiDOs 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,

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.

References
 D. Cardona, Spectral inequalities for elliptic pseudo-differential operators on closed manifolds, arXiv:2209.10690.
 D. Cardona, J. Delgado, M. Ruzhansky. Estimates for sums of eigenfunctions of elliptic pseudo-differential operators on compact Lie groups, arXiv:2209.12092.}
 H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93, 161–183, (1988).
 V. Fischer, M. Ruzhansky. Quantization on nilpotent Lie groups, Progress in Mathematics, Vol. 314, Birkhauser, 2016. xiii+557pp.
 D. Jerison, G. Lebeau. Nodal sets of sums of eigenfunctions. Harmonic analysis and partial differential equations (Chicago, IL, 1996), Chicago Lectures in Math, 223–239, (1999).
 L, Hörmander. The Analysis of the linear partial differential operators} Vol. III. Springer-Verlag, (1985).
 G. Lebeau, L. Robbiano, Controle exact de l’equation de la chaleur, Comm. Partial Diff. Equations., 20, 335–356, (1995).
 G. Lebeau, E. Zuazua. Null-Controllability of a System of Linear Thermoelasticity. Arch. Rational Mech. Anal. 141(4), 297–329, (1998).
 L. Miller. On the controllability of anomalous diffusions generated by the fractional Laplacian. Math. Control Signals Systems 18(3), 260–271, (2006).
 L. Miller. On the cost of fast controls for thermoelastic plates, Asymptot. Anal. 51, 93–100, (2007).
 M. Ruzhansky, V. Turunen. Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics, Birkhauser, Basel, 2010. 724pp.