# A new look at the controllability cost of linear evolution systems with a
long gaze at localized data

Roberta Bianchini  Consiglio Nazionale delle Ricerche, Istituto per le
Applicazioni del Calcolo, 00185 Rome, Italy<EMAIL_ADDRESS>,
Vincent Laheurte  IMB, Université de Bordeaux, France
<EMAIL_ADDRESS>and  Franck Sueur  IMB, Université de
Bordeaux, France<EMAIL_ADDRESS>
###### Abstract.

We revisit the classical issue of the controllability/observability cost of
linear first order evolution systems, starting with ODEs, before turning to
some linear first order evolution PDEs in several space dimensions, including
hyperbolic systems and pseudo-differential systems obtained by linearization
in fluid mechanics. In particular we investigate the cost of localized initial
data, and in the dispersive case, of initial data which are semi-classically
microlocalized.

###### Contents

  1. 1 Reminder on the cost of controllability of linear ODEs
  2. 2 Case of a transport PDE
  3. 3 A localization in space and amplitude of the cost
  4. 4 Microlocalized cost
    1. 4.1 Semiclassical Wigner matrix transform
    2. 4.2 Pseudodifferential operators.
    3. 4.3 Statement of the result
    4. 4.4 Proof of Theorem 4.14
  5. 5 Application to the incompressible Euler equations linearized at a Couette flow
  6. 6 Case of systems without crossing modes
    1. 6.1 Statement of the result
    2. 6.2 Proof of Theorem 6.4
    3. 6.3 Diagonalisation of the propagation term. Proof of Lemma 6.5
    4. 6.4 Block-diagonalisation of the amplification term. Proof of Lemma 6.6
    5. 6.5 Estimate of the off-diagonal energy terms. Proof of Lemma 6.7
  7. 7 Compressible gas-dynamics in a rapidly rotating frame

##  1\. Reminder on the cost of controllability of linear ODEs

In the setting of linear ODEs in finite dimensions, with variable
coefficients, the issue of exact controllability can be set as follows.

######  Question 1.1.

Given some time-dependent fields of matrices $A(t)$ and $B(t)$, a positive
time $T$, some initial and final states $b_{0}$ and $b_{T}$, is there a time-
dependent field $f(t)$ and a corresponding time-dependent field $b(t)$ such
that

$b^{\prime}(t)+A(t)b(t)=B(t)f(t)\text{ for }t\in[0,T]\text{ and
}(b(0),b(T))=(b_{0},b_{T})?$ (1)

Above, it is understood that the dependence on the time $t$ is smooth enough
for the system to make sense and that the matrices and vectors are compatible
in terms of their size. A positive answer to Question 1.1 means by definition
that the differential system $b^{\prime}(t)=A(t)b(t)$ is exactly controllable
in time $T$.

Let us observe that such a result is only non-trivial whenever the system is
under-actuated, that is in the case where the matrices $B(t)$ are not
invertible. Otherwise, it is sufficient to consider any smooth field $b(t)$
satisfying $(b(0),b(T))=(b_{0},b_{T})$, and then to define the control force
field $f(t)$ by

$f(t):=B(t)^{-1}\Big{(}b^{\prime}(t)+A(t)b(t)\Big{)}\text{ for }t\in[0,T],$

and then, the answer to Question 1.1 is positive.

In the case of a positive answer to Question 1.1, one may investigate the
minimal quadratic cost of such a control result.

######  Question 1.2.

What is the infimum $\mathcal{C}=\mathcal{C}(T,A,B,b_{0},b_{T})$ of

$\int_{0}^{T}|f(t)|^{2}\,dt,$

among the control force fields $f(t)$ which verify Question 1.1 ?

The Hilbert Uniqueness Method (HUM), introduced by Jacques-Louis Lions,
relates the notions of controllability and observability by means of a duality
argument, including the computation of the cost, see for example [6, Chapter
1]. In the context above, in the particular case where $b(T)=0$, the so-called
case of null-controllability, this method yields that $\mathcal{C}$ is the
minimal constant for which for any $b_{0}$, up to renaming the adjoints
$A^{*}(t)$ and $B^{*}(t)$ respectively by $A(t)$ and $B(t)$, the unique
solution to the uncontrolled equation:

$b^{\prime}(t)+A(t)b(t)=0\text{ for }t\in[0,T]\text{ and }b(0)=b_{0},$ (2)

satisfies

$|b(T)|^{2}\leqslant\mathcal{C}\int_{0}^{T}|B(t)b(t)|^{2}\,dt.$ (3)

The interpretation of the inequality above is that the energy $|b(T)|^{2}$ of
the final state reached by the uncontrolled system at time $T$ can be bounded
by the energy $\int_{0}^{T}|B(t)b(t)|^{2}\,dt$ which is observed through the
field of matrices $B(t)$ during the interval $(0,T)$, up to a constant
$\mathcal{C}$. The minimal possible constant $\mathcal{C}$ in (3) is also
called the observability cost of the differential system:
$b^{\prime}(t)+A(t)b(t)=0$ on $(0,T)$.

Before giving a characterization of $\mathcal{C}$, let us recall the following
elementary result on differential Lyapunov equations.

######  Lemma 1.3.

Let $A$ and $B$ some continuous time-dependent fields of matrices. Then there
is a unique global solution $G$ of the differential Lyapunov equation:

$Lyap(A,B):\quad G^{\prime}(t)-G(t)A(t)-A^{*}(t)G(t)=B^{*}(t)B(t)\text{ for
}t\in[0,T]\text{ and }G(0)=0.$

Moreover at any time $t$, the matrix $G(t)$ is symmetric and nonnegative.
Finally, for any time $T$,

$G(T)=\int_{0}^{T}R(t,T)^{*}B^{*}(t)B(t)R(t,T)\,\mathrm{d}t,$ (4)

where, for any time $T$, the matrix-valued function $R(\cdot,T)$ is the
solution to the linear system

$\frac{\mathrm{d}}{\mathrm{d}t}R(t,T)=-A(t)R(t,T),\quad
R(T,T)=\operatorname{Id}.$

###### Proof.

The existence and uniqueness of $G$ and of $R(\cdot,T)$ for any time $T$, as
solutions of their respective problems, follows from the linear finite-
dimensional version of the Cauchy-Lipschitz theorem. Then the symmetry follows
from the uniqueness by observing that $G$ and $G^{*}$ satisfy the same
equations. Similarly, the formula (4) follows from the uniqueness by observing
that the r.h.s. satisfies the differential Lyapunov equation. Finally, the
nonnegativity at any time $t\in[0,T]$ of the matrix $G(t)$ follows from this
representation. ∎

Let us now recall a characterization of $\mathcal{C}$ as the inverse of the
minimal eigenvalue $\lambda_{\text{min}}({G}(T))$ of the so-called
observability Gramian matrix $G(T)$.

######  Theorem 1.4.

The observability cost $\mathcal{C}$ satisfies
$\mathcal{C}=\lambda_{\text{min}}({G}(T))^{-1}$, where $G(T)$ is the value at
time $T$ of the solution $G$ to the differential Lyapunov equation
$Lyap(A,B)$.

A byproduct of Theorem 1.4 is that a necessary and sufficient condition for a
linear time-varying finite-dimensional control system to be observable is that
the observability Gramian $G(T)$ is invertible.

Let us also mention that for time-invariant finite-dimensional control
systems, this condition reduces to the famous Kalman rank condition, an
algebraic condition which does not require any integration of linear
differential equations.

Theorem 1.4 is well-known, see for example [6, Chapter 1]. However, we give
below a proof which sets up a method which will be repeatedly used in some
more difficult settings in the sequel.

###### Proof of Theorem 1.4.

The starting idea is that there is an extra integral over time in (3) so that
$\mathcal{C}^{-1}$ can be interpreted as a time. To restore this time in the
left hand side of (3), we introduce the following scalar field:

$E_{R}(t):=G(t)b(t)\cdot b(t),$ (5)

where the dot in (5) is the usual euclidean inner product. The quantity
$E_{R}$ will be called the recorded energy.

In particular the recorded energy starts with zero initial value, that is

$E_{R}(0)=0.$ (6)

Moreover, by Leibniz’ rule and by using the equations of $G$ and $b$, its time
derivative is given by

$\displaystyle E_{R}^{\prime}=B^{*}Bb\cdot b=|Bb|^{2},$

and then,

$\displaystyle E_{R}(T)=\int_{0}^{T}|B(t)b(t)|^{2}\,dt.$ (7)

Thus, at the final time $T$, the recorded energy is equal to the observed
energy.

On the other hand, from the definition 5, one may compare $E_{R}(T)$ with the
final energy $|b(T)|^{2}$ by

$\displaystyle\inf_{b_{0}\neq 0}\frac{G(T)b(T)\cdot
b(T)}{|b(T)|^{2}}=\inf_{b(T)\neq 0}\frac{G(T)b(T)\cdot
b(T)}{|b(T)|^{2}}=\lambda_{\text{min}}(G(T)).$

This concludes the proof of Theorem 1.4. ∎

While the proof above recalls the multiplier method, see for instance [18], it
is interesting to reinterpret in a slightly different way these computations,
more precisely the identity (7), for the sequel. In this direction, first,
from (2), we deduce that the rank-one matrix $b(t)\otimes b(t)=b(t)b(t)^{*}$,
which has to be interpreted as the energy tensor, satisfies the following ODE:

$(b\otimes b)^{\prime}+A(b\otimes b)+(b\otimes b)A^{*}=0.$ (8)

Moreover, the recorded energy reads

$E_{R}=G:(b\otimes b),$ (9)

where the notation $:$ stands for the Frobenius product, that is for two
squares matrices $A$ and $B$ of the same size,

$A:B=\operatorname{tr}(A^{*}B)=B:A.$

Recalling the following interplay between the Frobenius product and the matrix
product:

$A:(BC)=(AC^{*}):B\quad\text{ and }\quad A:(CB)=(C^{*}A):B,$ (10)

we observe that the differential Lyapunov equation $Lyap(A,B)$ is the dual
equation of (8) with source term $B^{*}B$, meaning that if one tests the
differential Lyapunov equation $Lyap(A,B)$ with a time-dependent matrix $C(t)$
by means of the Frobenius product and if one integrates by parts in time, then

$\int_{0}^{T}G:(C^{\prime}+AC+CA^{*})=(G:C)(T)-\int_{0}^{T}B^{*}B:C.$ (11)

This way, the identity (7) follows from choosing $C=b\otimes b$, recalling
(6), (8) and (9).

##  2\. Case of a transport PDE

Let us now turn to the case of a simple PDE: the following linear transport
equation:

$\partial_{t}\bm{u}+(v\cdot\nabla)\bm{u}+A\bm{u}=0,\text{ for }t\in[0,T],$
(12)

where the space variable is $x\in\mathbb{R}^{d}$ with $d\in\mathbb{N}^{*}$,
$v:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is a given regular vector field
and $A$ is a given regular field of ${N\times N}$ matrices depending on
$x\in\mathbb{R}^{d}$, with $N\in\mathbb{N}^{*}$. These matrices encode some
damping or amplification along the transportation by $v$. The final time $T>0$
is also given and the unknown is the vector field
$\bm{u}:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{N}$. In this case, the cost is
defined as follows.

######  Definition 2.1.

The cost of observability by a regular field of $n\times N$ matrices $B$
depending on $x\in\mathbb{R}^{d}$ is the minimal constant
$\mathcal{C}=\mathcal{C}(T,v,A,B)$ such that for any initial condition
$\bm{u}_{0}\in L^{2}(\mathbb{R}^{d})$, the associated solution $\bm{u}(t)$ to
(12) verifies the observability inequality:

$\|\bm{u}(T,\cdot)\|_{L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})}^{2}\leqslant\mathcal{C}\int_{0}^{T}\|B\bm{u}(t)\|_{L^{2}(\mathbb{R}^{d};\mathbb{R}^{n})}^{2}\mathrm{d}t.$
(13)

Let us highlight that, in this setting, the HUM method, which was mentioned in
the previous section, still holds and the identification of $\mathcal{C}$ with
the cost of controllability of the adjoint system as well.

The natural question that arises is whether one can give a condition for the
observability of system (12) and compute its cost, as done in Theorem 1.4 for
ODE systems.

The following result, generalizing Theorem 1.4, shows that $\mathcal{C}$ can
be computed in terms of the infimum over $x_{0}\in\mathbb{R}^{d}$ of the
minimal eigenvalues $\lambda_{\text{min}}(G_{x_{0}}(T))$ of a family of
Gramian matrices $G_{x_{0}}(T)$ associated with time-dependent Lyapunov ODEs
involving the coefficients of the PDE (12) evaluated along the characteristic
flow induced by the vector field $v$. This flow is denoted by $X_{x_{0}}(t)$
and satisfies, for any ${x_{0}}\in\mathbb{R}^{d}$,

$X^{\prime}_{x_{0}}(t)=v(X_{x_{0}}(t))\text{ for }t\in[0,T]\text{ and
}X_{x_{0}}(0)={x_{0}}.$ (14)

On the other hand, the Gramian matrices $G_{x_{0}}$ are given as solutions to
$Lyap(A_{{x_{0}}},B_{x_{0}})$ where, for any ${x_{0}}\in\mathbb{R}^{d}$ and
for any $t\in[0,T]$,

$\displaystyle
A_{{x_{0}}}(t):=A(X_{x_{0}}(t))-\frac{1}{2}(\operatorname{div}v)(X_{x_{0}}(t))\operatorname{Id}_{N}\quad\text{
and }\quad B_{x_{0}}(t):=B(X_{x_{0}}(t)).$ (15)

######  Theorem 2.2.

The observability cost $\mathcal{C}$ satisfies

$\mathcal{C}=\big{(}\inf_{{x_{0}}\in\mathbb{R}^{d}}\,\lambda_{\text{min}}(G_{x_{0}}(T)\big{)}^{-1}.$
(16)

A byproduct of Theorem 2.2 is that a necessary and sufficient condition for
(12) to be observable by the field of matrices $B$ is that the Gramian
matrices $(G_{x_{0}}(T))_{{x_{0}}\in\mathbb{R}^{d}}$ are uniformly invertible.

To prove Theorem 2.2 we will mimick the proof of Theorem 1.4 by introducing a
recorded energy to which we will apply the following lemma, where we assume
that $M$ and $F$ are regular enough.

######  Lemma 2.3.

Let $\bm{u}(t)$ be a solution of (12) associated with the initial condition
$\bm{u}_{0}\in L^{2}(\mathbb{R}^{d})$. Let $M$ satisfying the following PDE

$\partial_{t}M+(v\cdot\nabla)M+(\operatorname{div}v)M-A^{*}M-MA=F,$ (17)

where $F(t,x)$ is a time-dependent family of matrices. Let

$\displaystyle E[M,u]:=\int_{\mathbb{R}^{d}}(\bm{u}\otimes\bm{u}):M.$ (18)

Then its time derivative satisfies for all $t\in[0,T]$,

$\displaystyle E[M,u]^{\prime}=\int_{\mathbb{R}^{d}}(\bm{u}\otimes\bm{u}):F.$

###### Proof.

By Leibniz’ rule, (12) twice, integration by parts and a transposition, we
obtain that the time derivative $E[M,u]^{\prime}$ of $E[M,u]$ is given by

$\displaystyle E[M,u]^{\prime}$
$\displaystyle=\int_{\mathbb{R}^{d}}\Big{(}\partial_{t}M+(v\cdot\nabla)M+(\operatorname{div}v)M-A^{*}M-MA\Big{)}\bm{u}\cdot\bm{u}\,dx.$
(19)

Then it is sufficient to use (17) to conclude. ∎

Another elementary ingredient in the proof of Theorem 2.2 is the following
result.

######  Lemma 2.4.

Let $m:\mathbb{R}^{d}\to\mathbb{R}^{N\times N}$ be a continuous field of
symmetric non negative matrices. Then

$\inf_{\|u\|_{L^{2}}=1}\int_{\mathbb{R}^{d}}m(x)u(x)\cdot
u(x)\,\mathrm{d}x=\inf_{x\in\mathbb{R}^{d}}\lambda_{\rm{min}}(m(x)),$

where the infimum on the left hand side is taken over all the vector fields
$u$ in $L^{2}(\mathbb{R}^{d})$ with $\|u\|_{L^{2}}=1$.

###### Proof.

Let

$C:=\inf_{x\in\mathbb{R}^{d}}\lambda_{\rm{min}}(m(x)).$

On the one hand, for any $u\in L^{2}(\mathbb{R}^{d})$, for any
$x\in\mathbb{R}^{d}$,

$m(x)u(x)\cdot u(x)\geq C|u(x)|^{2},$

so that, by integration,

$\int_{\mathbb{R}^{d}}m(x)u(x)\cdot u(x)\,\mathrm{d}x\geq C\|u\|_{L^{2}}.$
(20)

On the other hand, for any $\varepsilon>0$, we choose
$x_{\varepsilon}\in\mathbb{R}^{d}$ such that
$\lambda_{\rm{min}}(m(x_{\varepsilon}))\leq C+\varepsilon$, as well as
$b_{\varepsilon}\in\mathcal{S}^{N-1}$ an eigenvector of $m(x_{\varepsilon})$
associated with $\lambda_{\rm{min}}(m(x_{\varepsilon}))$, so that

$m(x_{\varepsilon})b_{\varepsilon}=\lambda_{\rm{min}}(m(x_{\varepsilon}))b_{\varepsilon}.$

Finally, we set, for any $x\in\mathbb{R}^{d}$,

$u_{\varepsilon}(x):=(\pi\varepsilon)^{-\frac{d}{4}}b_{\varepsilon}e^{-\frac{|x-x_{\varepsilon}|^{2}}{2\varepsilon}}.$

Then, for any $\varepsilon>0$, $\|u_{\varepsilon}\|_{L^{2}}=1$ and, since $m$
is continuous,

$\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{d}}m(x)u_{\varepsilon}(x)\cdot
u_{\varepsilon}(x)\,\mathrm{d}x=C.$

Combining this equality with (20), we conclude the proof of the lemma. ∎

We are now equipped to start the proof of Theorem 2.2.

###### Proof of Theorem 2.2.

Let $M$ be given by setting for any $t$ and $x$,

$G_{x}(t)=M(t,X_{x}(t)).$ (21)

We highlight that, for any $t$, the mapping $x\mapsto X_{x}(t)$ is invertible,
since its Jacobian satisfies a linear first order ODE with $1$ as initial
condition. Then we observe that it follows from the fact that the Gramian
matrices $G_{x}$ are solutions to $Lyap(A_{{x}},B_{x})$ and from the chain
rule that $M(t,x)$ is a nonnegative symmetric matrix for any $t$ and $x$, that
$M$ satisfies an equation of the type (17) with $F=B^{*}B$ and that
$M(0,\cdot)=0$. Therefore, by the lemma 2.3 and an integration on $[0,T]$, the
recorded energy defined by $E_{R}:=E[M,u]$ satisfies

$\displaystyle E_{R}(T)$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{R}^{d}}|B\bm{u}(t,x)|^{2}\,\mathrm{d}x\mathrm{d}t.$
(22)

On the other hand, since the map $\bm{u}_{0}\mapsto\bm{u}(T,\cdot)$ is a
bijection from $L^{2}(\mathbb{R}^{d})$ to itself, by Lemma 2.4, we obtain

$\inf_{\bm{u}_{0}\neq
0}\frac{E_{R}(T)}{\|\bm{u}(T,\cdot)\|_{L^{2}}^{2}}=\inf_{x\in\mathbb{R}^{d}}\lambda_{\rm{min}}(M(T,x))=\inf_{x\in\mathbb{R}^{d}}\lambda_{\rm{min}}(G_{x}(T)),$
(23)

since $x\mapsto X_{x}(T)$ is a bijection from $\mathbb{R}^{d}$ to itself. We
conclude the proof of Theorem 2.2 by gathering (22) and (23). ∎

##  3\. A localization in space and amplitude of the cost

While the definition of observability cost is rather _global_ in space, we
consider below the observability cost

$\mathcal{C}[\bm{u}_{0}]:=\frac{\|\bm{u}(T,\cdot)\|_{L^{2}}^{2}}{\int_{0}^{T}\|B\bm{u}(t)\|_{L^{2}}^{2}\,\mathrm{d}t},$
(24)

of the initial data $\bm{u}_{0}$ (at time $T$) whose local energy tensor
$\bm{u}_{0}\otimes\bm{u}_{0}$ are _localized_ , for the weak-* topology of the
space $\mathcal{M}(\mathbb{R}^{d};\mathbb{R}^{N\times N})$ of the matrix-
valued measures, at $(b_{0}\otimes b_{0})\delta_{x_{0}}$, for some
$x_{0}\in\mathbb{R}^{d}$ and $b_{0}\in\mathbb{R}^{N}$. Above $\bm{u}$ is the
solution of the transport PDE (12) associated with $\bm{u}_{0}$. Since it is a
linear equation, the cost $\mathcal{C}[\bm{u}_{0}]$ is homogeneous of degree
$0$ with respect to the initial data $\bm{u}_{0}$.

######  Theorem 3.1.

Let $x_{0}\in\mathbb{R}^{d}$ and set $G_{T}:=G_{x_{0}}(T)$, where $G_{x_{0}}$
is the solution to the differential Lyapounov equation
$Lyap(A_{x_{0}},B_{x_{0}})$, where we recall the notation (15). Let
$b_{0}\in\mathbb{R}^{N}\setminus\\{0\\}$ and denote by

$b_{T}^{\mathcal{N}}:=\frac{b_{x_{0},b_{0}}(T)}{|b_{x_{0},b_{0}}(T)|},$

where $b_{x_{0},b_{0}}\in\mathbb{R}^{N}\setminus\\{0\\}$ is the solution of
the linear ordinary differential equation:

$b_{x_{0},b_{0}}^{\prime}=-A_{x_{0}}b_{x_{0},b_{0}},\quad\text{ with }\quad
b_{x_{0},b_{0}}(0)=b_{0}.$ (25)

Let $(\bm{u}_{0}^{n})_{n}$ be a sequence in $C^{\infty}(\mathbb{R}^{d})\cap
L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})$ such that

$\bm{u}_{0}^{n}\otimes\bm{u}_{0}^{n}\rightharpoonup(b_{x_{0},b_{0}}\otimes
b_{x_{0},b_{0}})\delta_{x_{0}}\quad\text{ weak-* in
}\,\mathcal{M}(\mathbb{R}^{d};\mathbb{R}^{N\times N}),\quad\text{ as
}\,n\rightarrow+\infty.$ (26)

Then,

$\mathcal{C}[\bm{u}_{0}^{n}]^{-1}\rightarrow(b_{T}^{\mathcal{N}})^{*}\,G_{T}\,b_{T}^{\mathcal{N}},\quad\text{
as }\,n\rightarrow+\infty.$

To prove Theorem 3.1, it is interesting to follow the view-point introduced at
the end of Section 1 and to consider the local energy tensor $u\otimes u$. By
elementary operations, we deduce from (12) that $u\otimes u$ satisfies the
transport PDE:

$(\partial_{t}+v\cdot\nabla)(u\otimes u)+A(u\otimes u)+(u\otimes u)A^{*}=0.$
(27)

While we consider some vector fields $u$ with
$L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})$ regularity in space, the equation (27)
makes sense and the Cauchy problem is well-posed, even if $u\otimes u$ is a
matrix-valued measure $\mu$ (or even more generally for a distribution) rather
than a $L^{1}(\mathbb{R}^{d};\mathbb{R}^{N\times N})$ function. As we are
interested in the case where the local energy tensor $u\otimes u$ concentrates
in a Dirac mass, it is useful to recall the following elementary result on the
Cauchy problem for transport equations with measure solutions.

######  Lemma 3.2.

Let $n,N\in\mathbb{N}^{*}$ and $T>0$. Let $V$ be a regular field of vector
field from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ and $A,\tilde{A}$ some smooth
vector fields of matrices in $\mathbb{R}^{N\times N}$. Let
$\mathcal{M}(\mathbb{R}^{n};\mathbb{R}^{N\times N})$ be the space of matrix-
valued measures and

$\mathfrak{C}:=C([0,T];\mathcal{M}(\mathbb{R}^{n};\mathbb{R}^{N\times
N})-w*),$

the space of the measures depending on time in a continuous way when
$\mathcal{M}(\mathbb{R}^{n};\mathbb{R}^{N\times N})$ is endowed by its weak-*
topology. Then

  1. (1)

For any $\mu_{0}$ in $\mathcal{M}(\mathbb{R}^{n};\mathbb{R}^{N\times N})$,
there exists a unique solution $\mu\in\mathfrak{C},$ to the PDE:

$\partial_{t}\mu+\operatorname{div}(\mu V)+A\mu+\mu\tilde{A}=0,$ (28)

with initial value $\mu_{0}$.

  2. (2)

If the initial datum $\mu_{0}$ is of the form
$\mu_{0}^{y_{0},P_{0}}=P_{0}\delta_{y_{0}}$ with $y_{0}\in\mathbb{R}^{n}$ and
$P_{0}\in\mathbb{R}^{N\times N}$ then the unique corresponding solution is
given by

$\mu^{y_{0},P_{0}}(t,\cdot)=P(t)\delta_{y(t)},$ (29)

where $y$ solves the nonlinear ODE:

$y^{\prime}=V(y)\quad\text{ and }\quad y(0)=y_{0},$ (30)

and $P\in C^{1}([0,T];\mathbb{R}^{N\times N})$ solves the linear ODE:

$P^{\prime}+(A\circ y)P+P(\tilde{A}\circ y)=0\quad\text{ and }\quad
P(0)=P_{0}.$ (31)

  3. (3)

Finally, if $(\mu_{0}^{n})_{n}$ a sequence in
$\mathcal{M}(\mathbb{R}^{n};\mathbb{R}^{N\times N})$ which converges weakly-*
to $\mu_{0}^{y_{0},P_{0}}$, then the corresponding solution $\mu^{n}$
converges to $\mu^{y_{0},P_{0}}$ in $\mathfrak{C}$ as $n\rightarrow+\infty$.

###### Proof.

The proof is very standard and we only give here a few words of comments. The
existence follows from the characteristics method. Indeed solutions can be
represented as superpositions of the atomic case considered in the second
item. The uniqueness and stability parts of the statement can be proved by a
duality argument, in the spirit of [9], or by a quantitative approach, thanks
to matricial Wasserstein distances, see [4, 20]. ∎

Equipped with this result, we are now ready to prove Theorem 3.1.

###### Proof of Theorem 3.1.

Let $X_{x_{0}}$ be given by (14) and $b_{x_{0},b_{0}}$ by (25). Then, by
Leibniz’ rule,

$(b_{x_{0},b_{0}}\otimes
b_{x_{0},b_{0}})^{\prime}=-A_{x_{0}}(b_{x_{0},b_{0}}\otimes
b_{x_{0},b_{0}})-(b_{x_{0},b_{0}}\otimes b_{x_{0},b_{0}})A_{x_{0}}^{*}.$

where we recall that $A_{x_{0}}$ is given by (15). Let

$\mu(t,\cdot):=(b_{x_{0},b_{0}}(t)\otimes
b_{x_{0},b_{0}}(t))\delta_{X_{x_{0}}(t)}.$

Therefore, by the second item of Lemma 3.2, $\mu$ is the unique solution to

$\partial_{t}\mu+\operatorname{div}(\mu
V)+\Big{(}A-\frac{1}{2}(\operatorname{div}v)\operatorname{Id}\Big{)}\mu+\mu\Big{(}A^{*}-\frac{1}{2}(\operatorname{div}v)\operatorname{Id}\Big{)}=0,$
(32)

with the initial data $\mu_{0}=(b_{0}\otimes b_{0})\delta_{x_{0}}$.

Let $\bm{u}^{n}$ be the solution of the transport PDE (12) associated with the
initial data $\bm{u}_{0}^{n}$, and set $\mu^{n}:=\bm{u}^{n}\otimes\bm{u}^{n}$
the corresponding local energy tensor. Then by (27),

$\partial_{t}\mu^{n}+\operatorname{div}(\mu^{n}V)+\Big{(}A-\frac{1}{2}(\operatorname{div}v)\operatorname{Id}\Big{)}\mu^{n}+\mu^{n}\Big{(}A^{*}-\frac{1}{2}(\operatorname{div}v)\operatorname{Id}\Big{)}=0.$
(33)

Then, by (26) and the third item of Lemma 3.2, we obtain that $\mu^{n}$
converges to $\mu$ in $\mathfrak{C}$ as $n\rightarrow+\infty$.

Now, considering again $M$ given by (21) one may re-use (22), which can
actually be re-obtained as a consequence of the fact that (17) is the dual
equation of (27) with $B^{*}B$ as a source term, that is:

$\displaystyle\int_{0}^{T}\|B\bm{u}^{n}(t)\|_{L^{2}}^{2}\,\mathrm{d}t=\int_{\mathbb{R}^{d}}M(T,x):\mu^{n}(T,x)\,\mathrm{d}x,$
(34)

while

$\displaystyle\|\bm{u}^{n}(T,\cdot)\|_{L^{2}}^{2}=\int_{\mathbb{R}^{d}}\operatorname{Id}:\mu^{n}(T,x)\,\mathrm{d}x.$
(35)

Thus the proof of Theorem 3.1 follows by passing to the limit in these two
expressions, recalling (21) for the first one. ∎

##  4\. Microlocalized cost

As we have seen before, the concept of energy is at the center of the
observability issue. In the previous sections, in the case of some transport
equations, we considered the energy tensor $\bm{u}\otimes\bm{u}$, which
encodes the localization of the energy in space, with respect to the
components of $\bm{u}$. However, for more general evolution linear first order
PDEs, in several space dimensions, dispersive effects occur and the
propagation of energy depends not only on the space variable $x$, but also on
the frequency variable $\xi$. Hence we need a tool to encode the distribution
of energy in the phase space, generalizing the energy tensor
$\bm{u}\otimes\bm{u}$. To this end, the natural object to consider is the
matrix Wigner transform, which we consider in its semiclassical version,
focusing on high frequencies.

###  4.1. Semiclassical Wigner matrix transform

Below $\varepsilon\in(0,1]$. However the regime of interest later on
corresponds to this parameter $\varepsilon$ converging to $0$, its inverse
standing for the typical size of the frequencies under consideration.

######  Definition 4.1.

For $u$ and $v$ in $L^{2}(\mathbb{R}^{d};\mathbb{C}^{N})$, we define the
Wigner matrix $W^{\varepsilon}[u,v](x,y)$, for any
$x,\xi\in\mathbb{R}^{d}\times\mathbb{R}^{d}$, by

$W^{\varepsilon}[u,v](x,\xi):=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}u\left(x-\varepsilon\frac{y}{2}\right)\otimes\overline{v}\left(x+\varepsilon\frac{y}{2}\right)e^{iy\cdot\xi}\,\mathrm{d}y.$

The mapping $(u,v)\mapsto W^{\varepsilon}[u,v]$ is bilinear and verifies.

$W^{\varepsilon}[u,v]=\left(W^{\varepsilon}[v,u]\right)^{*}.$ (36)

In the case where $u=v$ we simply denote by $W^{\varepsilon}[u]$ the Wigner
transform $W^{\varepsilon}[u,u]$, which is a field of symmetric matrices. The
Wigner matrix encodes the energy in the phase space in the following sense:
the marginals of $W^{\varepsilon}[u,v]$ respectively satisfy, formally, for
any $x\in\mathbb{R}^{d}$, for any $\xi\in\mathbb{R}^{d}$,

$\displaystyle\int_{\mathbb{R}^{d}}W^{\varepsilon}[u,v]\,\mathrm{d}\xi$
$\displaystyle=u(x)\otimes\overline{v}(x),$
$\displaystyle\int_{\mathbb{R}^{d}}W^{\varepsilon}[u,v]\,\mathrm{d}x$
$\displaystyle=\frac{1}{(2\pi\varepsilon)^{d}}\widehat{u}\left(\frac{\xi}{\varepsilon}\right)\otimes\left(\widehat{v}\right)\left(\frac{\xi}{\varepsilon}\right),$

where $\hat{u},\hat{v}$ denote the Fourier transforms of $u$ and $v$. In
particular,

$\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Big{(}W^{\varepsilon}[u]:\operatorname{Id}\Big{)}\,\mathrm{d}x\mathrm{d}\xi=\|u\|_{L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})}^{2},$
(37)

where we recall that the notation $:$ stands for the Frobenius product.
We also have, for any $1\leq j,k\leq N$,

$\left|W_{j,k}^{\varepsilon}[u,v](x,\xi)\right|\leq\frac{1}{(\pi\varepsilon)^{d}}\|u_{j}\|_{L^{2}(\mathbb{R}^{d};\mathbb{C})}\|v_{k}\|_{L^{2}(\mathbb{R}^{d};\mathbb{C})},$

where $W_{j,k}^{\varepsilon}[u,v]$ denotes the coefficient $(j,k)$ of the
Wigner matrix, and $u_{j}$ and $v_{k}$ respectively denote the $j$-th
component of $u$ and the $k$-th component of $v$. This implies that the Wigner
matrix $W[u,v]$ is in $L^{\infty}$, from $\mathbb{R}^{d}\times\mathbb{R}^{d}$
to $\mathbb{R}^{N\times N}$ equipped with the Frobenius norm, and that

$\|W^{\varepsilon}[u,v]\|_{L^{\infty}}\leq\frac{1}{(\pi\varepsilon)^{d}}\|u\|_{L^{2}}\|v\|_{L^{2}}.$

The Wigner matrix $W^{\varepsilon}[u,v]$ is also in
$L^{2}(\mathbb{R}^{d}\times\mathbb{R}^{d};\mathbb{R}^{N\times N})$, with

$\|W^{\varepsilon}[u,v]\|_{L^{2}}=\|u\|_{L^{2}}\|v\|_{L^{2}}.$

For more details about those properties, we refer, for instance, to [19].

######  Remark 4.2.

The Wigner matrix $W^{\varepsilon}[u]$ carries information about the energy
tensor $u\otimes u$, but is not necessarily nonnegative. However, for
positivity, one may consider the Husimi transform instead, or Wigner measures,
which are weak limits of Wigner matrices as $\varepsilon$ goes to 0, in some
sense. For our purpose, a sign is not required so we will not use those tools.

Below we recall some classical examples.

######  Example 4.3.

An important example is given by the coherent states defined, for any
$x_{0},\xi_{0}\in\mathbb{R}^{d}$, for any $\mathbf{b}_{0}\in\mathbb{R}^{N}$,
as follows:

$\bm{u}^{\varepsilon}_{x_{0},\xi_{0},\mathbf{b}_{0}}(x):=(\pi\varepsilon)^{-\frac{d}{4}}\mathbf{b}_{0}e^{i\frac{(x-x_{0})\cdot\xi_{0}}{\varepsilon}}e^{-\frac{|x-x_{0}|^{2}}{2\varepsilon}}.$

Then one may compute

$W^{\varepsilon}[\bm{u}^{\varepsilon}_{x_{0},\xi_{0},\mathbf{b}_{0}}](x,\xi)=(\pi\varepsilon)^{-d}(\mathbf{b}_{0}\otimes\mathbf{b}_{0})e^{-\frac{|x-x_{0}|^{2}+|\xi-\xi_{0}|^{2}}{\varepsilon}}.$

In particular, the matrix Wigner transform
$\mathcal{W}^{\varepsilon}[\bm{u}^{\varepsilon}_{x_{0},\xi_{0},\mathbf{b}_{0}}]$
weakly converges, as $\varepsilon\rightarrow 0$, in the sense of distributions
to $(\mathbf{b}_{0}\otimes\mathbf{b}_{0})\delta_{x_{0},\xi_{0}}$.

######  Example 4.4.

Another classical example is given by the WKB states defined, for any
$\alpha\in(0,1]$, any $\xi_{0}\in\mathbb{R}^{d}$, and any amplitude function
$a(x)\in L^{2}(\mathbb{R}^{d})$, by

$u^{\varepsilon}_{a,\alpha}(x):=a(x)e^{i\frac{x\cdot\xi_{0}}{\varepsilon^{\alpha}}}.$

Then, as $\varepsilon\to 0$, for any $\alpha\in(0,1)$,

$W^{\varepsilon}[u^{\varepsilon}_{a,\alpha}]\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
w*}}}{\to}}a(x)\otimes a(x)\delta_{0}(\xi)\text{ in
}\mathcal{S}^{\prime}(\mathbb{R}^{d}\times\mathbb{R}^{d}),$

while in the case where $\alpha=1$,

$W^{\varepsilon}[u^{\varepsilon}_{a,\alpha}]\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
w*}}}{\to}}a(x)\otimes a(x)\delta_{\xi_{0}}(\xi)\text{ in
}\mathcal{S}^{\prime}(\mathbb{R}^{d}\times\mathbb{R}^{d}).$

###  4.2. Pseudodifferential operators.

Our analysis allows to consider not only PDEs but also pseudo-differential
equations, what is of interest to cover equations from incompressible fluid
mechanics. In microlocal analysis, different kinds of quantizations are
available, see for instance to [22, Chapter 4]. Here we consider the Weyl
quantization, which associates with a symbol $a=a(x,\xi)$ in a suitable class
the semiclassical pseudo-differential operator
$\operatorname{op^{w}_{\varepsilon}}[a]$ defined through its action on
functions $u\in L^{2}(\mathbb{R}^{d})$, by

$\displaystyle(\operatorname{op^{w}_{\varepsilon}}[a]u)(x):=(2\pi\varepsilon)^{-d}\int
e^{\frac{i(x-y)\cdot\xi}{\varepsilon}}a\left(\frac{x+y}{2},\xi\right)u(y)\;\mathrm{d}\xi\,\mathrm{d}y.$

Throughout this paper, we will simply assume every symbol to be in the class
$S(1)$ of functions bounded as well as all their derivatives, defined by:

$S(1):=\\{a\in
C^{\infty}(\mathbb{R}^{2d}):\forall\alpha\in\mathbb{N}^{2d},\exists
C_{\alpha}>0,\forall(x,\xi)\in\mathbb{R}^{2d},\|\partial^{\alpha}a\|_{\infty}\leq
C_{\alpha}\\}.$ (38)

Wider classes of symbols may be studied, but they are not required for the
analysis which we perform in this paper. With that choice of class, the
pseudo-differential operator given above defines a linear continuous operator
from $L^{2}(\mathbb{R}^{d};\mathbb{C}^{N})$ to itself, and there exists a
constant $C>0$, uniform in $\varepsilon$, as well as a universal constant
$\gamma$, such that, for all $a\in S(1),$ it holds

$\|\operatorname{op^{w}_{\varepsilon}}[a]\|_{\mathcal{L}(L^{2};L^{2})}\leq
C\sup_{|\alpha|+|\beta|\leq\gamma
d}\|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a\|_{\infty},$ (39)

where $\mathcal{L}(L^{2};L^{2})$ denotes the space of linear bounded operators
from $L^{2}$ to itself, endowed with the usual operator norm. From now on, we
denote for $k$ in $\mathbb{N}$ by $\mathcal{N}_{k}$ the norm on $S(1)$:

$\mathcal{N}_{k}(\cdot):=\sup_{|\alpha|+|\beta|\leq
k}\|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\cdot\|_{\infty}.$ (40)

We list below some useful properties of calculus of the Weyl quantization.
More details, other choices of quantizations and other classes of symbols can
be found in [19], [22].

######  Lemma 4.5 (Adjoint operator.).

Let $a\in S(1)$ be a symbol. Then the adjoint operator of
$\operatorname{op^{w}_{\varepsilon}}[a]$ is given by

$(\operatorname{op^{w}_{\varepsilon}}[a])^{*}=\operatorname{op^{w}_{\varepsilon}}[a^{*}].$
(41)

######  Lemma 4.6 (Composition of operators.).

Let $a,b\in S(1)$ be two symbols. Then the following composition law holds:

$\operatorname{op^{w}_{\varepsilon}}[a]\operatorname{op^{w}_{\varepsilon}}[b]=\operatorname{op^{w}_{\varepsilon}}[ab]+\frac{\varepsilon}{2i}\operatorname{op^{w}_{\varepsilon}}[\\{a,b\\}]+O(\varepsilon^{2})_{\mathcal{L}(L^{2};L^{2})},$
(42)

where $\\{\cdot,\cdot\\}$ denotes the Poisson bracket

$\\{a,b\\}=(\partial_{\xi}a)(\partial_{x}b)-(\partial_{x}a)(\partial_{\xi}b).$

The Wigner transform is associated with the semiclassical Weyl quantization by
the relation:

$\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Big{(}W^{\varepsilon}[u,v]:a(x,\xi)\Big{)}\,\mathrm{d}x\mathrm{d}\xi=\langle\operatorname{op^{w}_{\varepsilon}}[a^{*}]u,v\rangle_{L^{2}},$
(43)

for any $a(x,\xi)$ in $S(1)$, and any $u,v\in
L^{2}(\mathbb{R}^{d};\mathbb{C}^{N})$. This property allows us to view Wigner
matrices as tempered distributions in $S(1)^{\prime}$, the dual space of
$S(1)$. Indeed, for any $u\in L^{2}(\mathbb{R}^{d};\mathbb{C}^{N})$, combining
the property (43) with the estimate (39) yields the existence of a constant
$C=C(u)>0$, uniform in $\varepsilon$, such that for any $a\in S(1)$ it holds:

$\left|\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}W^{\varepsilon}[u](x,\xi):a(x,\xi)\,\mathrm{d}x\,\mathrm{d}\xi\right|\leq
C\mathcal{N}_{\gamma d}(a),$

which allows us to consider $W^{\varepsilon}[u]$ as a distribution of order at
most $\gamma d$ (where $W^{\varepsilon}[u]=W^{\varepsilon}[u,u]$ as in (37)).
More precisely, for any $u,v\in L^{2}(\mathbb{R}^{d};\mathbb{C}^{N})$, we have
$W^{\varepsilon}[u,v]\in S(1)^{\prime}_{\gamma d}(\mathbb{R}^{2d})$, where

$S(1)^{\prime}_{k}(\mathbb{R}^{2d})=\left\\{\psi(x,\xi)\in
S(1)^{\prime}:\exists C>0,\forall\varphi\in
S(1),\left|\langle\psi,\varphi\rangle_{\mathcal{S}^{\prime},\mathcal{S}}\right|\leq
C\mathcal{N}_{k}(\varphi)\right\\}.$

We endow $S(1)^{\prime}_{k}(\mathbb{R}^{2d})$ with the norm

$\|\psi\|_{\mathcal{S}^{\prime}_{k}}:=\sup_{\phi\in
S(1),\mathcal{N}_{k}(\phi)\leq
1}\langle\psi,\phi\rangle_{S(1)^{\prime},S(1)}.$

For any $u,v\in L^{2}(\mathbb{R}^{d};\mathbb{C}^{N})$, and $a\in S(1),$

$W^{\varepsilon}[\operatorname{op^{w}_{\varepsilon}}[a]u,v]=aW^{\varepsilon}[u,v]+\frac{\varepsilon}{2i}\\{a,W^{\varepsilon}[u,v]\\}+\varepsilon^{2}\mathcal{R}_{\varepsilon},$
(44)

where the remainder term $\mathcal{R}_{\varepsilon}\in S(1)^{\prime}_{\gamma
d+2}(\mathbb{R}^{2d})$ verifies

$\|\mathcal{R}_{\varepsilon}\|_{S(1)^{\prime}_{\gamma d+2}}=O(1).$

###  4.3. Statement of the result

With the previous material, it is natural to extend our setting to hyperbolic
pseudo-differential equations of the form

$\displaystyle\varepsilon\partial_{t}\bm{u}+\operatorname{op^{w}_{\varepsilon}}[iH(x,\xi)\operatorname{Id}+\varepsilon
A(x,\xi)]\bm{u}=0,\quad\text{on }[0,T],$ (45)

where the principal part $H$ is a real scalar symbol in $S(1)$ and the sub-
principal part $A$ is a matrix-valued symbol in $S(1)$.

Our setting also allows to consider a wider class of observability operators
given as well by pseudodifferential operators associated with a symbol matrix
$B$ in $S(1)$. Thus we are led to the following extension of the notion of
observability cost $\mathcal{C}[\bm{u}_{0}]$ of an initial data $\bm{u}_{0}$
at time $T$ for the equation (45) by the pseudodifferential observability
operator of symbol $B$ in $S(1)$.

######  Definition 4.7.

Let $T>0$. Let
$(x_{0},\xi_{0},b_{0})\in\mathbb{R}^{d}\times(\mathbb{R}^{d}\setminus\\{0\\})\times\mathbb{R}^{N}$
and $\bm{u}_{0}$ in $L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})$. The observability
cost $\mathcal{C}[\bm{u}_{0}]$ of the initial data $\bm{u}_{0}$ at time $T$
for the equation (45) by the observability operator of symbol $B$ in $S(1)$ is
defined by

$\mathcal{C}[\bm{u}_{0}]:=\frac{\|\bm{u}(T,\cdot)\|_{L^{2}}^{2}}{\int_{0}^{T}\|\operatorname{op^{w}_{\varepsilon}}[B]\bm{u}(t)\|_{L^{2}}^{2}\,\mathrm{d}t},$
(46)

where $\bm{u}$ is the solution of (45) associated with $\bm{u}_{0}$.

The purpose of this section is to study the observability cost of initial data
which are localized at $(x_{0},\xi_{0},\mathbf{b})$ in the following sense.

######  Definition 4.8.

A family $(\bm{u}_{0}^{\varepsilon})_{\varepsilon\in(0,1]}$ in
$L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})$ is said to be localized at
$(x_{0},\xi_{0},b_{0})\in\mathbb{R}^{d}\times(\mathbb{R}^{d}\setminus\\{0\\})\times\mathcal{S}^{N-1}$
if its Wigner matrix satisfies

$\mathcal{W}^{\varepsilon}[\bm{u}_{0}^{\varepsilon}]\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
w*}}}{\to}}(b_{0}\otimes b_{0})\delta_{x_{0},\xi_{0}}\quad\text{ in
}\,\mathcal{(}S(1))^{\prime},\quad\text{ as }\,\varepsilon\rightarrow 0.$

The question at stake is then the following.

######  Question 4.9.

Let $T>0$. Let
$(x_{0},\xi_{0},b_{0})\in\mathbb{R}^{d}\times(\mathbb{R}^{d}\setminus\\{0\\})\times\mathbb{R}^{N}$
and a family $(\bm{u}_{0}^{\varepsilon})_{\varepsilon\in(0,1]}$ in
$L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})$ localized at $(x_{0},\xi_{0},b_{0})$.
Can we determine the limit of the observability cost
$\mathcal{C}[\bm{u}_{0}^{\varepsilon}]$ as $\varepsilon\rightarrow 0$ ?

The result below provides an answer in terms of the Hamiltonian flow induced
by the principal symbol $H$, the microlocalized observability Gramian
matrices, the bicharacteristic equations and the bicharacteristic amplitude
equation whose definitions are now given.

######  Definition 4.10.

Let $H$ be a real scalar symbol in $S(1)$ and $v_{H}$ the divergence free
vector field from $\mathbb{R}^{d}\times\mathbb{R}^{d}$ to
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ defined by

$v_{H}:=(\nabla_{\xi}H,-\nabla_{x}H).$ (47)

For any $x_{0},\xi_{0}\in\mathbb{R}^{d}$, we define the Hamiltonian flow
$(X_{x_{0},\xi_{0}}(t),\Xi_{x_{0},\xi_{0}}(t))$ induced by $H$ as the unique
solution to the nonlinear ODE:

$\big{(}X_{x_{0},\xi_{0}},\Xi_{x_{0},\xi_{0}}\big{)}^{\prime}=v_{H}\big{(}(X_{x,\xi},\Xi_{x_{0},\xi_{0}})\big{)},\quad\left(X_{x_{0},\xi_{0}},\Xi_{x_{0},\xi_{0}}\right)(0)=(x_{0},\xi_{0}).$
(48)

######  Remark 4.11.

It is worth noticing that the determinant
$J(t):=\det(D(X_{x_{0},\xi_{0}}(t),\Xi_{x_{0},\xi_{0}}(t)))$ of the Jacobian
matrix associated with the Hamiltonian flow satisfies $J(t)=1$ for all
$t\in[0,T]$ since
$J^{\prime}(t)=\operatorname{\bf{div}}\left(\nabla_{\xi}H,-\nabla_{x}H\right)J(t)=0$
and $J(0)=0$. Here and below we use the notation $\operatorname{\bf{div}}$ to
denote the divergence operator in $\mathbb{R}^{d}\times\mathbb{R}^{d}$.

We now define the microlocalized observability Gramian matrices, as solutions
of some differential Lyapunov equations involving the coefficients of (45)
along the Hamiltonian flow.

######  Definition 4.12.

For any $x_{0},\xi_{0}\in\mathbb{R}^{d}$, we define the Gramian matrix
microlocalized in $(x_{0},\xi_{0})$ as the unique solution $G_{x_{0},\xi_{0}}$
to the differential Lyapunov equation
$Lyap(A_{x_{0},\xi_{0}},B_{x_{0},\xi_{0}})$ where the time-dependent matrices
$A_{x_{0},\xi_{0}}$ and $B_{x_{0},\xi_{0}}$ are given by

$A_{x_{0},\xi_{0}}(t):=A(X_{x_{0},\xi_{0}}(t),\Xi_{x_{0},\xi_{0}}(t))\quad\text{
and }\quad
B_{x_{0},\xi_{0}}(t):=B(X_{x_{0},\xi_{0}}(t),\Xi_{x_{0},\xi_{0}}(t)).$

Finally we define the bicharacteristic amplitude.

######  Definition 4.13.

For any $x_{0},\xi_{0}\in\mathbb{R}^{d}$, for any $\mathbf{b}_{0}$ in
$\mathbb{R}^{N}$, we define the bicharacteristic amplitude
$\mathbf{b}_{x_{0},\xi_{0},\mathbf{b}_{0}}(t)$ associated with $H$ and $A$ as
the unique solution to the linear ODE:

$\mathbf{b}_{x_{0},\xi_{0},\mathbf{b}_{0}}^{\prime}=-A_{x_{0},\xi_{0}}\,\mathbf{b}_{x_{0},\xi_{0},\mathbf{b}_{0}},\quad\mathbf{b}_{x_{0},\xi_{0},\mathbf{b}_{0}}(0)=\mathbf{b}_{0}.$
(49)

The result below answers to Question 4.9 and identifies the limit of the
observability cost $\mathcal{C}[\bm{u}_{0}^{\varepsilon}]$ of a family
$(\bm{u}_{0}^{\varepsilon})_{\varepsilon\in(0,1]}$ in
$L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})$ localized at $(x_{0},\xi_{0},b_{0})$ as
$\varepsilon\rightarrow 0$ in terms of the Rayleigh quotient of the
microlocalized observability Gramian matrix along the bicharacteristic flow
with the bicharacteristic amplitude, at the final time.

######  Theorem 4.14.

Let $T>0$. Let
$x_{0}\in\mathbb{R}^{d},\xi_{0}\in\mathbb{R}^{d}\setminus\\{0\\}$ and set
$G_{T}:=G_{x_{0},\xi_{0}}(T),$ where $G_{x_{0},\xi_{0}}$ is the Gramian matrix
microlocalized in $(x_{0},\xi_{0})$ as given by Definition 4.12. Let
$\mathbf{b}_{0}\in\mathcal{S}^{N-1}$ and set

$\mathbf{b}_{T}^{\mathcal{N}}:=\frac{\mathbf{b}_{x_{0},\xi_{0},\mathbf{b}_{0}}(T)}{|\mathbf{b}_{x_{0},\xi_{0},\mathbf{b}_{0}}(T)|},$

where $\mathbf{b}_{x_{0},\xi_{0},\mathbf{b}_{0}}$ is the transported amplitude
associated with $H$ and $A$ given by Definition 4.13. Let
$(\bm{u}_{0}^{\varepsilon})_{\varepsilon\in(0,1]}$ in
$C^{\infty}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})$ be
localized at $(x_{0},\xi_{0},b_{0})$ in the sense of Definition 4.8. Then the
observability cost $\mathcal{C}[\bm{u}_{0}^{\varepsilon}]$ satisfies

$\mathcal{C}[\bm{u}_{0}^{\varepsilon}]^{-1}\rightarrow(\mathbf{b}_{T}^{\mathcal{N}})^{*}\,G_{T}\,\mathbf{b}_{T}^{\mathcal{N}},\quad\text{
as }\,\varepsilon\rightarrow 0.$

Let us highlight that the interest of the result above is that the cost of
observability of initial data which are localized at $(x_{0},\xi_{0},b_{0})$
converges to a limit which can determined by ODEs. It can be considered as a
broad extension of the result obtained in the case of transport equations in
Theorem 3.1, to general hyperbolic pseudo-differential equations of the form
(45), in the high frequency limit, despite the extra dispersive effects. As a
matter of fact, the equation (12) can be recast as an equation of the form
(45) with

$H(x,\xi)=v(x)\cdot\xi\quad\text{and}\quad
A(x,\xi):=A(x)-\frac{1}{2}(\operatorname{div}v(x))\operatorname{Id}.$ (50)

Then the the bicharacteristic equations (48) reduce to

$\displaystyle X^{\prime}_{x_{0},\xi_{0}}(t)=v(X_{x_{0},\xi_{0}}(t)),\,\text{
and
}\,\Xi^{\prime}_{x_{0},\xi_{0}}(t)=-(\nabla_{x}v(X_{x_{0},\xi_{0}}(t)))^{*}\Xi_{x_{0},\xi_{0}}(t)\quad\text{for}\;t\in(0,T).$
(51)

In particular, the first equation does not depend on $\xi$, translating the
lack of dispersion in the case of transport equations. Moreover the matrix
$A(x,\xi)$ defined in (50) is the one which is involved in the Lyapounov
equation satisfied by the Gramian matrix $G_{x_{0}}$, see (15).

###  4.4. Proof of Theorem 4.14

For any $(x,\xi)$, let $G_{x,\xi}$ is the Gramian matrix microlocalized in
$(x,\xi)$ as given by Definition 4.12. We define the symbolic multiplier
$M(t,x,\xi)$ by setting, for any $(t,x,\xi)$,

$G_{x,\xi}(t)=:M(t,(X_{x,\xi}(t),\Xi_{x,\xi}(t))).$ (52)

As the Hamiltonian flow is invertible, see Remark 4.11, the above $M$ is
uniquely defined. Similarly to the reformulation in (34) and (34) performed in
the non-dispersive case in Section 3, we are going to use a reformulation of
the observability cost in terms of a recorded energy by the use of a
multiplier. The latter is precisely chosen as the symbolic multiplier
$M(t,x,\xi)$, and the corresponding recorded energy is then set as

$\displaystyle\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathcal{W}^{\varepsilon}(t,x,\xi):M(t,x,\xi)\,\mathrm{d}x\mathrm{d}\xi=\langle\operatorname{op^{w}_{\varepsilon}}[M(t,x,\xi)]\bm{u},\bm{u}\rangle_{L^{2}(\mathbb{R}^{d})},$
(53)

where we use the shorthand notation $\mathcal{W}^{\varepsilon}$ for the Wigner
matrix transform $\mathcal{W}^{\varepsilon}[\bm{u}^{\varepsilon}]$, recalling
the property (43).

Let $T>0$ and $(\bm{u}_{0}^{\varepsilon})_{\varepsilon\in(0,1]}$ be bounded in
$L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})$. A key point of the proof of Theorem
4.14 is to establish that as $\varepsilon\rightarrow 0$, the observability
cost $\mathcal{C}[\bm{u}_{0}^{\varepsilon}]$ satisfies

$\mathcal{C}[\bm{u}_{0}^{\varepsilon}]=\frac{\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathcal{W}^{\varepsilon}(T,\cdot):\operatorname{Id}\,\mathrm{d}x\,\mathrm{d}\xi}{\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathcal{W}^{\varepsilon}(T,\cdot):M(T,\cdot)\,\mathrm{d}x\,\mathrm{d}\xi}+o(1).$
(54)

Recalling (37), we already have:

$\displaystyle\|\bm{u}(T,\cdot)\|_{L^{2}}^{2}$
$\displaystyle=\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\mathcal{W}^{\varepsilon}(T,\cdot):\operatorname{Id}\,\mathrm{d}x\mathrm{d}\xi.$
(55)

The reformulation of the denominator is more subtle. First, by (43), the
properties of the Frobenius product and (44), we observe that, at any time,

$\displaystyle\|\operatorname{op^{w}_{\varepsilon}}[B]\bm{u}\|_{L^{2}}^{2}=\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Big{(}\mathcal{W}^{\varepsilon}:B^{*}B\Big{)}\,\mathrm{d}x\mathrm{d}\xi+O(\varepsilon).$
(56)

Then, using the chain rule and (48), we observe that

$\displaystyle
G^{\prime}_{x,\xi}(t)=\Big{(}\partial_{t}M+\operatorname{\bf{div}}(Mv_{H})\Big{)}(t,X_{x,\xi}(t),\Xi_{x,\xi}(t)),$

so that, since $G_{x,\xi}$ verifies the differential Lyapunov equation
$Lyap(A_{x,\xi},B_{x,\xi})$, we infer that the symbolic multiplier $M$
satisfies the following PDE:

$\left\\{\begin{aligned}
\partial_{t}M+\operatorname{\bf{div}}(Mv_{H})-MA-A^{*}M&=B^{*}B,\\\
M(0)&=0,\end{aligned}\right.$ (57)

where $B^{*}B$ appears as a source term. On the other hand, the dynamics of
the Wigner matrix $\mathcal{W}^{\varepsilon}$ is given by the following
result.

######  Lemma 4.15.

There exists a remainder term $\mathcal{R}_{\varepsilon}$ in $(S(1))_{\gamma
d+2}^{\prime}$ with
$\|\mathcal{R}_{\varepsilon}\|_{\mathcal{S}^{\prime}_{\gamma
d+2}}=O(\varepsilon)$ such that

$\partial_{t}\mathcal{W}^{\varepsilon}+\operatorname{\bf{div}}(\mathcal{W}^{\varepsilon}v_{H})+A\mathcal{W}^{\varepsilon}+\mathcal{W}^{\varepsilon}A^{*}=\mathcal{R}_{\varepsilon}.$
(58)

###### Proof.

By bilinearity and the symmetry property (36), the Wigner matrix transform
$\mathcal{W}^{\varepsilon}$ satisfies

$\displaystyle\partial_{t}\mathcal{W}^{\varepsilon}$
$\displaystyle=W^{\varepsilon}[\partial_{t}\bm{u},\bm{u}]+W^{\varepsilon}[\partial_{t}\bm{u},\bm{u}]^{*}.$
(59)

Moreover, by (45), it follows that

$\displaystyle W^{\varepsilon}[\partial_{t}\bm{u},\bm{u}]$
$\displaystyle=-W^{\varepsilon}\left[\operatorname{op^{w}_{\varepsilon}}\left[\frac{i}{\varepsilon}H(x,\xi)\operatorname{Id}+A(x,\xi)\right]\bm{u},\bm{u}\right],$

so that, using the property (44), we arrive at

$W^{\varepsilon}[\partial_{t}\bm{u},\bm{u}]=-\frac{i}{\varepsilon}H\mathcal{W}^{\varepsilon}-\frac{1}{2}\\{H,\mathcal{W}^{\varepsilon}\\}-A\mathcal{W}^{\varepsilon}+O_{(S(1))_{\gamma
d+2}^{\prime}}(\varepsilon).$

Substituting into (59) and using that $H$ is a real scalar symbol, we obtain

$\partial_{t}\mathcal{W}^{\varepsilon}+\\{H\operatorname{Id},\mathcal{W}^{\varepsilon}\\}+A\mathcal{W}^{\varepsilon}+\mathcal{W}^{\varepsilon}A^{*}=O_{(S(1))_{\gamma
d+2}^{\prime}}(\varepsilon).$ (60)

Recalling that $v_{H}$ is the divergence free vector field defined by (47) the
conclusion follows. ∎

As in the proof of Theorem 3.1 we rely on a duality argument given by the
following result.

######  Lemma 4.16.

Let $W(t,x,\xi)$ and $M(t,x,\xi)$ two matrix-valued time-dependent symbols
such that

$\left\\{\begin{aligned}
\partial_{t}W+\operatorname{\bf{div}}(Wv_{H})+AW+WA^{*}&=\mathcal{R},\\\
\partial_{t}M+\operatorname{\bf{div}}(Mv_{H})-MA-A^{*}M&=\mathcal{F}.\end{aligned}\right.$

Then

$\displaystyle\Big{(}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Big{(}W:M\Big{)}\,\mathrm{d}x\mathrm{d}\xi\Big{)}(T)=\Big{(}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Big{(}W:M\Big{)}\,\mathrm{d}x\mathrm{d}\xi\Big{)}(0)$
(61)
$\displaystyle+\int_{0}^{T}\Big{(}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Big{(}\mathcal{R}:M+W:\mathcal{F}\Big{)}\,\mathrm{d}x\mathrm{d}\xi\Big{)}(t)\,dt.$

###### Proof.

We observe that the time derivative $E^{\prime}$ of

$E(t):=\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}M:W\,\mathrm{d}x\,\mathrm{d}\xi,$

satisfies

$\displaystyle E^{\prime}(t)=$
$\displaystyle\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\left(-\operatorname{\bf{div}}(Mv_{H})+MA+A^{*}M+\mathcal{F}\right):W\,\mathrm{d}x\,\mathrm{d}\xi$
$\displaystyle+\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}M:\left(-\operatorname{\bf{div}}(Wv_{H})-AW-
WA^{*}+\mathcal{R}\right)\,\mathrm{d}x\,\mathrm{d}\xi.$ (62)

Using that $\operatorname{\bf{div}}(v_{H})=0$, an integration by parts yields

$\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\operatorname{\bf{div}}(Mv_{H}):W\,\mathrm{d}x\,\mathrm{d}\xi=-\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}M:\operatorname{\bf{div}}(Wv_{H})\,\mathrm{d}x\,\mathrm{d}\xi.$

For the other terms, simple computations give

$(MA+A^{*}M):W=M:(AW+WA^{*}).$

Using those two equalities in (62) leads to

$E^{\prime}(t)=\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\mathcal{F}:W+M:\mathcal{R})\,\mathrm{d}x\,\mathrm{d}\xi.$
(63)

Integrating this equality in time, from $0$ to $T$, we conclude the proof of
Lemma 4.16. ∎

We now apply Lemma 4.16 with $W=\mathcal{W}^{\varepsilon}$ and $M$, which
yields, thanks to (57) and Lemma 4.15,

$\displaystyle\Big{(}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Big{(}\mathcal{W}^{\varepsilon}:M\Big{)}\,\mathrm{d}x\mathrm{d}\xi\Big{)}(T)=\int_{0}^{T}\iint_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Big{(}\mathcal{W}^{\varepsilon}:B^{*}B\Big{)}\,\mathrm{d}x\mathrm{d}\xi\Big{)}(t)\,dt+O(\varepsilon).$
(64)

From (55) and (64), we deduce (54).

Now let $x_{0}\in\mathbb{R}^{d},\xi_{0}\in\mathbb{R}^{d}\setminus\\{0\\}$ and
$\mathbf{b}_{0}\in\mathcal{S}^{N-1}$ and assume that
$(\bm{u}_{0}^{\varepsilon})_{\varepsilon\in(0,1]}$ is localized at
$(x_{0},\xi_{0},b_{0})$ in the sense of Definition 4.8. To pass to the limit
in (54) we rely on the following intermediate result.

######  Lemma 4.17.

Let $n,N\in\mathbb{N}^{*}$ and $T>0$. Let $V$ be a regular field of vector
fields from $\mathbb{R}^{d}\times\mathbb{R}^{d}$ to itself, $A,\tilde{A}\in
S(1)$ be two smooth vector fields of matrices in $\mathbb{R}^{N\times N}$, and
$(\mathcal{R}_{\varepsilon})_{\varepsilon\in(0,1]}\subset(S(1))^{\prime}_{\gamma
d+2}$ such that

$\lim_{\varepsilon\to
0}\|\mathcal{R}_{\varepsilon}(t)\|_{\mathcal{S}^{\prime}_{\gamma
d+2}}=0,\text{ uniformly in time.}$ (65)

Let

$\mathfrak{C}:=C([0,T];(S(1))^{\prime}-w*),$

the space of the distributions in the dual space to $S(1)$, depending on time
in a continuous way when $(S(1))^{\prime}$ is endowed by its weak-* topology.
Then

  1. (1)

For any $\mu_{0}$ in $(S(1))^{\prime}$, $\varepsilon$ in $(0,1]$, there exists
a unique solution $\mu\in\mathfrak{C},$ to the PDE:

$\partial_{t}\mu+\operatorname{\bf{div}}(\mu
V)+A\mu+\mu\tilde{A}=\mathcal{R}_{\varepsilon},$ (66)

with initial value $\mu_{0}$.

  2. (2)

If the initial data $\mu_{0}$ is of the form
$\mu_{0}^{x_{0},\xi_{0},P_{0}}=P_{0}\delta_{x_{0},\xi_{0}}$ with
$x_{0},\xi_{0}\in\mathbb{R}^{d}$ and $P_{0}\in\mathbb{R}^{N\times N}$ then the
unique corresponding solution to the homogeneous equation is given by

$\mu^{x_{0},\xi_{0},P_{0}}(t,\cdot)=P(t)\delta_{x(t),\xi(t)},$ (67)

where $(x,\xi)$ solves the nonlinear ODE:

$(x,\xi)^{\prime}=V(x,\xi)\quad\text{ and }\quad(x,\xi)(0)=(x_{0},\xi_{0}),$
(68)

and $P$ the linear ODE:

$P^{\prime}+A(x(\cdot),\xi(\cdot))P+P\tilde{A}(x(\cdot),\xi(\cdot))=0\quad\text{
and }\quad P(0)=P_{0}.$ (69)

  3. (3)

Finally, if $(\mu_{0}^{\varepsilon})_{\varepsilon\in(0,1]}$ converges weakly-*
to $\mu_{0}^{x_{0},\xi_{0},P_{0}}$ in $(S(1))^{\prime}$ as $\varepsilon$ goes
to 0, then the corresponding solution $(\mu^{\varepsilon})_{(0,1]}$ converges
to $\mu^{x_{0},\xi_{0},M_{0}}$ in $\mathfrak{C}$.

###### Proof.

The existence of the solution, as well as the form of the solution associated
with the initial datum $M_{0}\delta_{x_{0},\xi_{0}}$, follows from the
characteristics method. The uniqueness and stability can be proved by a
standard duality argument. ∎

By the second item of Lemma 4.17, for any
$(x_{0},\xi_{0},\mathbf{b}_{0})\in\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}^{N}$,
the unique solution $\mu^{x_{0},\xi_{0},b_{0}}$ to

$\partial_{t}\mathcal{\mu}^{x_{0},\xi_{0},b_{0}}+\operatorname{\bf{div}}(\mathcal{\mu}^{x_{0},\xi_{0},b_{0}}v_{H})+A\mathcal{\mu}^{x_{0},\xi_{0},b_{0}}+\mathcal{\mu}\mu^{x_{0},\xi_{0},b_{0}}A^{*}=0.$
(70)

associated with the initial data
$\mu_{0}^{x_{0},\xi_{0},b_{0}}:=(\mathbf{b}_{0}\otimes\mathbf{b}_{0})\delta_{x_{0},\xi_{0}}$
is

$\mu^{x_{0},\xi_{0},b_{0}}=(\mathbf{b}_{x_{0},\xi_{0},b_{0}}(t)\otimes\mathbf{b}_{x_{0},\xi_{0},b_{0}}(t))\delta_{X_{x_{0},\xi_{0}}(t),\Xi_{x_{0},\xi_{0}}(t)},$

where we recall that the Hamiltonian flow
$(X_{x_{0},\xi_{0}}(t),\Xi_{x_{0},\xi_{0}}(t))$ is defined by the equations
(48) and (49). Moreover, by the third item of Lemma 4.17, since, as
$\varepsilon\rightarrow 0$,
$\mathcal{W}^{\varepsilon}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
w*}}}{\to}}(b_{0}\otimes b_{0})\delta_{x_{0},\xi_{0}}$ in
$\mathcal{(}S(1))^{\prime}$, we have that, as $\varepsilon\rightarrow 0$,

$\mathcal{W}^{\varepsilon}(T)\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
w*}}}{\to}}(\mathbf{b}_{x_{0},\xi_{0},\mathbf{b}_{0}}(t)\otimes\mathbf{b}_{x_{0},\xi_{0},\mathbf{b}_{0}}(T))\delta_{X_{x_{0},\xi_{0}}(T),\Xi_{x_{0},\xi_{0}}(T)}.$
(71)

Passing to the limit in (54) and recalling (52) we conclude the proof of
Theorem 4.14.

##  5\. Application to the incompressible Euler equations linearized at a
Couette flow

An interesting example of application of our method is represented by the
incompressible Euler equations in $\mathbb{R}^{2}$, in velocity form, which we
linearize at a steady state $\overline{\bm{u}}$ satisfying
$\operatorname{div}(\overline{\bm{u}})=0$, so that the velocity field
$\bm{u}=\bm{u}(t,x)\in\mathbb{R}^{N}$ and the pressure $p=p(t,x)\in\mathbb{R}$
satisfy

$\left\\{\begin{aligned}
\partial_{t}\bm{u}+(\overline{\bm{u}}\cdot\nabla)\bm{u}+(\bm{u}\cdot\nabla)\overline{\bm{u}}+\nabla
p&=0\\\ \operatorname{div}(\bm{u})&=0.\end{aligned}\right.$ (72)

Using the Leray projection onto divergence-free vector fields, the
semiclassical approximation of (72) then reads

$\varepsilon\partial_{t}\bm{u}+\operatorname{op^{w}_{\varepsilon}}[iH(x,\xi)\operatorname{Id}+\varepsilon
A(x,\xi)]\bm{u}=0,$ (73)

where

$H(x,\xi)=\overline{\bm{u}}(x)\cdot\xi,\quad
A(x,\xi)=\left(\operatorname{Id}-2\frac{\xi\otimes\xi}{|\xi|^{2}}\right)D\overline{\bm{u}}.$
(74)

Since we want to study the behavior of solutions that are localized, and in
high-frequency, we may multiply the symbols $A$ and $H$ by a cut-off function
that is equal to $0$ near $\xi=0$ to avoid the singularity of $A$ at $\xi=0$,
so that one may consider a symbol $A$ in $S(1)$ which is given by (74) for
$\xi=0$ outside of a ball of radius say $1/2$.

We consider the case where the steady state $\overline{\bm{u}}$ is the Couette
flow $\overline{\bm{u}}=(x_{2},0)$, and where the observation matrix is

$B=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}.$

Then for any $x_{0},\xi_{0}$ in $\mathbb{R}^{2}$, the Gramian matrix
$G_{x_{0},\xi_{0}}$ microlocalized at $(x_{0},\xi_{0})$ is given by

$G_{x_{0},\xi_{0}}(T)=\int_{0}^{T}(R_{x_{0},\xi_{0}}(t,T)^{*}B^{*}BR_{x_{0},\xi_{0}}(t,T))\,\mathrm{d}t,$

where $R_{x_{0},\xi_{0}}(t,T)$ is the resolvent matrix, given by solving the
linear system

$\frac{\mathrm{d}}{\mathrm{d}t}R_{x_{0},\xi_{0}}(t,T)=-A(X_{x_{0},\xi_{0}}(t),\Xi_{x_{0},\xi_{0}}(t))R_{x_{0},\xi_{0}}(t,T),\quad
R(T,T)=\operatorname{Id}.$

A tedious computation yields, with the notation
$\xi_{0}=\left(\xi_{1,0},\xi_{2,0}\right)$,

$R_{x_{0},\xi_{0}}(t,T)=\begin{pmatrix}1&f(t,\xi_{0})\\\
0&g(t,\xi_{0})\end{pmatrix},$

where

$\displaystyle f(t,\xi_{0})=\begin{cases}T-t&\text{if }\xi_{1,0}=0,\\\
-\frac{(\xi_{2,0}-t\xi_{1,0})(\xi_{1,0}^{2}+(\xi_{2,0}-T\xi_{1,0})^{2})}{\xi_{1,0}(\xi_{1,0}^{2}+(\xi_{2,0}-t\xi_{1,0})^{2})}+\frac{\xi_{2,0}-T\xi_{1,0}}{\xi_{1,0}}&\text{otherwise,}\end{cases}$
(75)

and

$g(t,\xi_{0})=\begin{cases}1&\text{if }\xi_{1,0}=0,\\\
\frac{\xi_{1,0}^{2}+(\xi_{2,0}-T\xi_{1,0})^{2}}{\xi_{1,0}^{2}+(\xi_{2,0}-t\xi_{1,0})^{2}}&\text{otherwise.}\end{cases}.$

Moreover

$G_{x_{0},\xi_{0}}(T)=\int_{0}^{T}\begin{pmatrix}1&f(t,\xi_{0})\\\
f(t,\xi_{0})&f^{2}(t,\xi_{0})\end{pmatrix}\mathrm{d}t.$

We now consider
$(x_{0},\xi_{0},\mathbf{b}_{0})\in\mathbb{R}^{2}\times(\mathbb{R}^{2}\setminus\\{0\\})\times\mathcal{S}^{1}$
and $(\bm{u}_{0}^{\varepsilon})_{\varepsilon\in(0,1]}\subset
L^{2}(\mathbb{R}^{2};\mathbb{R}^{2})$ a family of initial data localized in
$(x_{0},\xi_{0},\mathbf{b}_{0}).$ By Theorem 4.14, the observability cost
$\mathcal{C}[\bm{u}_{0}^{\varepsilon}]$ satisfies, as $\varepsilon$ goes to
$0$,

$\lim_{\varepsilon\to
0}(\mathcal{C}[\bm{u}_{0}^{\varepsilon}])^{-1}=(\mathbf{b}_{T}^{\mathcal{N}})^{*}G_{x_{0},\xi_{0}}(T)\mathbf{b}_{T}^{\mathcal{N}},$

with

$\mathbf{b}_{T}^{\mathcal{N}}=\frac{\mathbf{b}_{T}}{|\mathbf{b}_{T}|},\quad\mathbf{b}_{T}=R_{x_{0},\xi_{0}}(T,0)\mathbf{b}_{0}=\begin{pmatrix}1&-\frac{f(0,\xi_{0})}{g(0,\xi_{0})}\\\
0&\frac{1}{g(0,\xi_{0})}\end{pmatrix}\mathbf{b}_{0}.$

We can notice that, for any value of $\xi_{0}\neq 0$, the function
$f(\cdot,\xi_{0})$ is continuous and not constant, so that, by the Cauchy-
Schwarz inequality,

$\displaystyle\det
G_{x_{0},\xi_{0}}(T)=T\int_{0}^{T}f^{2}(t,\xi_{0})\,\mathrm{d}t-\left(\int_{0}^{T}f(t,\xi_{0})\,\mathrm{d}t\right)^{2}>0.$

Thus the system is observable in high frequencies for initial data localized
in any $(x_{0},\xi_{0},\mathbf{b}_{0})$, and for any positive time $T>0$. The
limit of the observability cost as $\varepsilon$ goes to $0$ is finite, and
can be explicitly computed in terms of $\xi_{0}$ and $\mathbf{b}_{0}.$ The
short-time and long-time behaviors of the observability cost can be given by
further computations. First, in the case where $\xi_{1,0}=0$, the Gramian
matrix has the simple expression:

$G_{x_{0},\xi_{0}}(T)=\begin{pmatrix}T&\frac{T^{2}}{2}\\\
\frac{T^{2}}{2}&\frac{T^{3}}{3}\end{pmatrix}.$

Immediate computations then yield

$\lambda_{\rm{min}}(G_{x_{0},\xi_{0}}(T))=\frac{T}{2}+\frac{1}{6}(T^{3}-\sqrt{T^{6}+3T^{4}+9T^{2}}),$

so that in particular
$\lambda_{\rm{min}}(G_{x_{0},\xi_{0}}(T))^{-1}\sim\frac{12}{T^{3}}$ for short
times and $\lambda_{\rm{min}}(G_{x_{0},\xi_{0}}(T))^{-1}\sim\frac{4}{T}$ for
long times. Note that $\lambda_{\rm{min}}(G_{x_{0},\xi_{0}}(T))^{-1}$ is the
”worst-case” cost for initial data localized in $(x_{0},\xi_{0})$, namely

$\lambda_{\rm{min}}(G_{x_{0},\xi_{0}}(T))^{-1}=\sup_{\mathbf{b}_{0}\in\mathcal{S}^{N-1}}\\{\lim_{\varepsilon\to
0}\mathcal{C}(\bm{u}_{0}^{\varepsilon}):(\bm{u}_{0}^{\varepsilon})\text{ is
localized in }(x_{0},\xi_{0},\mathbf{b}_{0})\\}.$

The asymptotic behaviour of the cost at short times can be extended to the
other values of $\xi_{0}$. If $\xi_{1,0}\neq 0$ and
$\xi_{1,0}^{2}\neq\xi_{2,0}^{2}$, then by using the Taylor expansion of $f$ at
first order, we obtain

$\lambda_{\rm{min}}(G_{x_{0},\xi_{0}}(T))\sim\frac{T^{3}(\xi_{1,0}^{2}-\xi_{2,0}^{2})^{2}}{12|\xi_{0}|^{4}}.$
(76)

Otherwise, that is if $\xi_{1,0}\neq 0$ and $\xi_{1,0}^{2}=\xi_{2,0}^{2}$,
then by using the Taylor expansion of $f$ at second order, we obtain

$\lambda_{\rm{min}}(G_{x_{0},\xi_{0}}(T))\sim\frac{T^{5}}{45}.$ (77)

Those estimates are consistent with the results of [21]. Indeed, in the case
where $\xi_{1,0}^{2}\neq\xi_{2,0}^{2}$, the rank condition is fulfilled on the
second bracket, namely

$\text{rank}[B|A_{x_{0},\xi_{0}}(0)B]=2,\text{ while rank}(B)<2.$

Then the analysis of [21] gives a cost of order $T^{-3}$, as given by (76).
When $\xi_{1,0}^{2}=\xi_{2,0}^{2}$, we need to take one more bracket to verify
the rank condition, namely

$\text{rank}[B|A_{x_{0},\xi_{0}}(0)B|(A_{x_{0},\xi_{0}}^{2}(0)+A^{\prime}_{x_{0},\xi_{0}}(0))B]=2,\text{
while rank}[B|A_{x_{0},\xi_{0}}(0)B]<2.$

The results in [21] then yield a cost of order $T^{-5}$, as given by (77). Our
approach furthermore gives an explicit constant.

##  6\. Case of systems without crossing modes

This section is devoted to an extension of the results of Section 4 to more
general systems.

###  6.1. Statement of the result

In this section we extend the previous analysis to the case of pseudo-
differential hyperbolic systems of the form:

$\displaystyle\partial_{t}\bm{u}+\operatorname{op^{w}_{\varepsilon}}\left[\frac{i}{\varepsilon}H+A\right]\bm{u}=0,\quad(x,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d}.$
(78)

Above, $A$ and $H$ are two matrices of symbols which are supposed to fulfill
the following assumption.

######  Assumption 6.1.

  * We assume that $A$ and $H$ satisfy the following.

  * (H1):

$H=H(x,\xi)$ is a $N\times N$ field of Hermitian matrices in $S(1)$, and there
exist $1\leq m\leq N$ and $C>0$ such that for all
$(x,\xi)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$, the matrix $H(x,\xi)$ has $m$
real semi-simple eigenvalues $\lambda_{k}(x,\xi)$, for $1\leqslant k\leqslant
m$, with

$|\lambda_{j}(x,\xi)-\lambda_{k}(x,\xi)|>C\quad\text{for all}\;1\leq j\neq
k\leq m\leq N.$

  * (H2):

$A=A(x,\xi)$ is a $N\times N$ field of matrices in $S(1).$

As a consequence of (H1) the matrix
$\mathcal{U}(x,\xi):=(\textbf{r}_{\ell})_{\ell=1,\cdots N}$ of the right
normalized eigenvectors $(\textbf{r}_{\ell})_{\ell=1,\cdots N}$ of $H(x,\xi)$
is in $S(1)$, takes values which are unitary matrices, and

$\mathcal{U}^{*}H=\mathcal{D}\mathcal{U}^{*}=(\lambda_{\ell}\textbf{r}_{\ell})^{*}_{\ell=1,\cdots
N},$ (79)

and

$\mathcal{D}(x,\xi):=(\mathcal{U}^{*}H\mathcal{U})(x,\xi)=\sum_{k=1}^{m}\lambda_{k}(x,\xi)\Pi_{k}\in
S(1),$ (80)

where the $\left(\Pi_{k}\right)_{1\leq k\leq m}$ are the constant orthogonal
projectors onto
$\ker(\mathcal{D}(x,\xi)-\lambda_{k}(x,\xi)\operatorname{Id}),$ which are
independent of $(x,\xi)\in\mathbb{R}^{d}$.

Our aim is to answer to Question 4.9 in the case of the system (78), that is
to determine, as $\varepsilon\rightarrow 0$, the limit observability cost
$\mathcal{C}[\bm{u}_{0}^{\varepsilon}]$, as given by Definition 4.7, of a
family of initial data $(\bm{u}_{0}^{\varepsilon})_{\varepsilon\in(0,1]}$ in
$L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})$ which is localized at
$(x_{0},\xi_{0},b_{0})\in\mathbb{R}^{d}\times(\mathbb{R}^{d}\setminus\\{0\\})\times\mathbb{R}^{N}$,
for the corresponding solution $\bm{u}$ of the system (78) and an
observability operator $B\in S(1)$. The main result of this section provides
an answer to this question which generalizes the earlier case where a single
characteristic was considered. Again the result involves a few ingredients,
which we now recall in this more general setting. First it makes use of the
Hamiltonian flows associated with each eigenvalue $\lambda_{k}$, with $1\leq
k\leq m$, as given by Definition 4.13. For sake of clarity we explicitely
recall this definition below, which involves nonlinear ODEs.

######  Definition 6.1.

For $1\leq k\leq m$, for any $(x,\xi)$, we define the Hamiltonian flow
$\big{(}X_{k,x,\xi},\Xi_{k,x,\xi}\big{)}$ induced by the $k$-th eigenvalue
$\lambda_{k}(x,\xi)$ as the unique smooth solution of the following Cauchy
problem:

$\big{(}X_{k,x,\xi},\Xi_{k,x,\xi}\big{)}^{\prime}=v_{\lambda_{k}}\big{(}X_{k,x,\xi},\Xi_{k,x,\xi}\big{)},\quad\left(X_{k,x,\xi},\Xi_{k,x,\xi}\right)(0)=(x,\xi).$
(81)

where $v_{\lambda_{k}}$ is the divergence free vector field from
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ to itself defined by

$v_{\lambda_{k}}=\left(\nabla_{\xi}\lambda_{k},-\nabla_{x}\lambda_{k}\right)$
(82)

We also rely on some Gramian matrices associated with each Hamiltonian flow
and with $H$ and $A$, similarly to Definition 4.12, but with some surprising
extra terms, which encode the influence between the various characteristic
fields of the system. More precisely, for $1\leq k\leq m$, let

$\displaystyle A^{\rm{eff}}_{k}$
$\displaystyle:=\Pi_{k}\mathcal{U}^{*}A\mathcal{U}\Pi_{k}+\frac{1}{2}\Pi_{k}\Big{(}\\{\mathcal{U}^{*},H\\}\mathcal{U}-\\{\mathcal{D},\mathcal{U}^{*}\\}\mathcal{U}\Big{)}\Pi_{k},$
(83) $\displaystyle B^{\rm{eff}}_{k}$ $\displaystyle:=B\mathcal{U}\Pi_{k},$
(84)

which both belong to $S(1)$. We may now give the definition of the Gramian
matrices, which involves linear ODEs.

######  Definition 6.2.

Let $x_{0},\xi_{0}\in\mathbb{R}^{d}$ and $1\leq k\leq m$. We define the $k$-th
Gramian matrix microlocalized in $(x_{0},\xi_{0})$ as the unique solution
$G_{k,x_{0},\xi_{0}}$ to the differential Lyapunov equation

$Lyap(A_{k,x_{0},\xi_{0}},B_{k,x_{0},\xi_{0}}),$

where

$A_{k,x_{0},\xi_{0}}(t):=A_{k}^{\rm{eff}}(X_{k,x_{0},\xi_{0}}(t),\Xi_{k,x_{0},\xi_{0}}(t))\,\text{
and
}\,B_{k,x_{0},\xi_{0}}(t):=B_{k}^{\rm{eff}}(X_{k,x_{0},\xi_{0}}(t),\Xi_{k,x_{0},\xi_{0}}(t))).$

We also define the bicharacteristic amplitudes, which are given as solutions
to linear ODEs.

######  Definition 6.3.

For any $x_{0},\xi_{0}\in\mathbb{R}^{d}$, for any $\mathbf{b}_{0}$ in
$\mathbb{R}^{N}$ and $1\leq k\leq m$, we define the $k$-th bicharacteristic
amplitude $\mathbf{b}_{k,x_{0},\xi_{0},\mathbf{b}_{0}}$ as the unique solution
to the linear ODE:

$\mathbf{b}_{k,x_{0},\xi_{0},\mathbf{b}_{0}}^{\prime}=-A_{k,x_{0},\xi_{0}}\,\mathbf{b}_{k,x_{0},\xi_{0},\mathbf{b}_{0}},\quad\mathbf{b}_{k,x_{0},\xi_{0},\mathbf{b}_{0}}(0)=\mathbf{b}_{0}.$

The main result of this section is that the inverse of observability cost
$\mathcal{C}[\bm{u}_{0}^{\varepsilon}]$ for a family of initial data
$(\bm{u}_{0}^{\varepsilon})_{\varepsilon\in(0,1]}$ in
$L^{2}(\mathbb{R}^{d};\mathbb{C}^{N})$ which is localized at
$(x_{0},\xi_{0},\mathbf{b}_{0})$, in the sense of Definition 4.8, converges to
a weighted mean of Rayleigh quotients of the $m$ Gramian matrices with the
bicharacteristic amplitudes.

######  Theorem 6.4.

Let
$(x_{0},\xi_{0},b_{0})\in\mathbb{R}^{d}\times(\mathbb{R}^{d}\setminus\\{0\\})\times\mathbb{R}^{N}$.
For $1\leq k\leq m$, let $G_{T,k}$ be the Gramian matrix given at time $T$,
that is $G_{T,k}:=G_{k,x_{0},\xi_{0}}(T),$ and $\mathbf{b}_{T,k}$ the
characteristic amplitude at time $T$, that is
$\mathbf{b}_{T,k}:=\mathbf{b}_{k,x_{0},\xi_{0},\mathbf{b}_{0}}(T).$ Let
$(\bm{u}_{0}^{\varepsilon})_{\varepsilon\in(0,1]}$ in
$L^{2}(\mathbb{R}^{d};\mathbb{R}^{N})$ be localized at $(x_{0},\xi_{0},b_{0})$
in the sense of Definition 4.8. Then the observability cost
$\mathcal{C}[\bm{u}_{0}^{\varepsilon}]$ verifies, as $\varepsilon\to 0$,

$\mathcal{C}[\bm{u}_{0}^{\varepsilon}]^{-1}\to\frac{\sum_{k=1}^{m}\mathbf{b}_{T,k}^{*}G_{T,k}\mathbf{b}_{T,k}}{\sum_{k=1}^{m}|\mathbf{b}_{T,k}|^{2}}.$
(85)

Theorem 6.4 extends to the general case of hyperbolic systems, without
characteristics crossing, some earlier results by [7, 8, 17] in the case of
the wave equation and of the Klein-Gordon equation. In these works, the
authors use some Egorov type methods, for scalar equations. Despite some
counterparts of the Egorov theorem for systems have been established in [1, 3,
5], in the proof below we prefer to extend the approach performed in the
previous sections, an approach which we believe is more flexible, direct and
general.

###  6.2. Proof of Theorem 6.4

The first step of the proof is to rewrite the system in normal form. To this
end, we need the two preliminary results, that we will prove later. Recall
that $\mathcal{D}$ is defined in (80), and set

$2A^{\flat}(x,\xi):=\\{\mathcal{U}^{*},H\\}\mathcal{U}-\\{\mathcal{D},\mathcal{U}^{*}\\}\mathcal{U},$
(86)

which is in $S(1)$.

######  Lemma 6.5.

Let $\bm{u}$ be the solution to (78) with initial datum $\bm{u}_{0}$, and let
$\widetilde{\bm{u}}:=\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\bm{u}$.
Then $\widetilde{\bm{u}}$ solves the system

$\left\\{\begin{aligned}
\partial_{t}\widetilde{\bm{u}}+\frac{i}{\varepsilon}\operatorname{op^{w}_{\varepsilon}}[\mathcal{D}]\widetilde{\bm{u}}+\operatorname{op^{w}_{\varepsilon}}[\Gamma]\widetilde{\bm{u}}&=\varepsilon\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}]\widetilde{\bm{u}},\\\
\widetilde{\bm{u}}(0)&=\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\bm{u}_{0},\end{aligned}\right.$
(87)

where $\Gamma=\mathcal{U}^{*}A\mathcal{U}+A^{\flat}$ and the remainder
$r_{\varepsilon}\in S(1)$ is such that $\mathcal{N}_{\gamma
d}(r_{\varepsilon})=O(1)$.

For all $(x,\xi)\in\mathbb{R}^{2d}$, we set

$Q(x,\xi):=-\sum_{(j,k)/\,j\neq
k}\frac{i\Pi_{j}\Gamma(x,\xi)\Pi_{k}}{\lambda_{j}(x,\xi)-\lambda_{k}(x,\xi)}.$
(88)

Then $Q\in S(1)$. Recall that $A^{\rm{eff}}_{k}$ is given by (83).

######  Lemma 6.6.

Let $\widetilde{\bm{u}}$ be the solution to (87). We set
$\bm{v}:=(\operatorname{Id}+\varepsilon\operatorname{op^{w}_{\varepsilon}}[Q])\widetilde{\bm{u}}$
and $\bm{v}_{k}:=\Pi_{k}\bm{v}$ for $1\leq k\leq m$. Then

$\left\\{\begin{aligned}
\partial_{t}\bm{v}_{k}+\frac{i}{\varepsilon}\operatorname{op^{w}_{\varepsilon}}[\lambda_{k}\operatorname{Id}]\bm{v}_{k}+\operatorname{op^{w}_{\varepsilon}}[A^{\rm{eff}}_{k}]\bm{v}_{k}&=\varepsilon\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}]\bm{v},\\\
\bm{v}_{k}(0)&=(\operatorname{Id}+\varepsilon\operatorname{op^{w}_{\varepsilon}}[Q])\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\bm{u}_{0},\end{aligned}\right.$
(89)

where $r_{\varepsilon}$ is again a remainder in $S(1)$ such that
$\mathcal{N}_{\gamma d}(r_{\varepsilon})=O(1).$

Let us take these two results as granted and finish the proof of Theorem 6.4.
Rewriting both energy terms involved in the observability cost in terms of
$\bm{v}$ gives, using that
$\mathcal{U}\mathcal{U}^{*}=\mathcal{U}^{*}\mathcal{U}=\operatorname{Id}$,

$\displaystyle\|\bm{u}(T)\|_{L^{2}}^{2}$
$\displaystyle=\sum_{k=1}^{m}\|\bm{v}_{k}(T)\|_{L^{2}}^{2}(1+O(\varepsilon)),$
(90)
$\displaystyle\|\operatorname{op^{w}_{\varepsilon}}[B]\bm{u}\|^{2}_{L^{2}}$
$\displaystyle=\sum_{k=1}^{m}\|\operatorname{op^{w}_{\varepsilon}}[B^{\rm{eff}}_{k}]\bm{v}_{k}\|_{L^{2}}^{2}+\sum_{j\neq
k}\left\langle\operatorname{op^{w}_{\varepsilon}}[B\mathcal{U}]\bm{v}_{j},\operatorname{op^{w}_{\varepsilon}}[B\mathcal{U}]\bm{v}_{k}\right\rangle_{L^{2}}+O(\varepsilon)\|\bm{v}\|_{L^{2}}^{2}.$
(91)

Note that we split the diagonal and off-diagonal terms in the observed energy
because they exhibit very distinct properties.

For the off-diagonal terms, we rely on the following lemma.

######  Lemma 6.7.

Let $\Lambda(x,\xi)\in S(1)$ be a matrix-valued symbol. Let $1\leq j,k\leq m$
such that $j\neq k$. Then it holds

$\int_{0}^{T}\langle\operatorname{op^{w}_{\varepsilon}}[\Pi_{k}\Lambda\Pi_{j}]\bm{v}(t),\bm{v}(t)\rangle_{L^{2}}\,\mathrm{d}t=O(\varepsilon)\|\bm{v}(T)\|^{2}_{L^{2}}.$
(92)

Applying this lemma with $\Lambda=\mathcal{U}^{*}B^{*}B\mathcal{U}$ yields,
for $1\leq j\neq k\leq m$,

$\left\langle\operatorname{op^{w}_{\varepsilon}}[B\mathcal{U}]\bm{v}_{j},\operatorname{op^{w}_{\varepsilon}}[B\mathcal{U}]\bm{v}_{k}\right\rangle_{L^{2}}=O(\varepsilon)\|\bm{v}(T)\|_{L^{2}}^{2}.$
(93)

For the diagonal terms, for each $1\leq k\leq m$, we apply Theorem 4.14 to the
system (89), for $1\leq k\leq m$, to obtain that

$\displaystyle\|\bm{v}(T)\|_{L^{2}}^{2}$
$\displaystyle=|\mathbf{b}_{T,k}|^{2}+o(1),$ (94)
$\displaystyle\int_{0}^{T}\|\operatorname{op^{w}_{\varepsilon}}[B\mathcal{U}]\bm{v}_{k}\|_{L^{2}}^{2}$
$\displaystyle=\mathbf{b}_{T,k}^{*}G_{T,k}(T)\mathbf{b}_{T,k}+o(1).$ (95)

Combining (93), (94) and (95), and substituting them into (90) and (91) yields

$\displaystyle\|\bm{u}(T)\|_{L^{2}}^{2}$
$\displaystyle=\sum_{k=1}^{m}|\mathbf{b}_{k,x_{0},\xi_{0},\mathbf{b}_{0}}(T)|^{2}+o(1),$
$\displaystyle\int_{0}^{T}\|\operatorname{op^{w}_{\varepsilon}}[B]\bm{u}\|_{L^{2}}^{2}$
$\displaystyle=\sum_{k=1}^{m}\mathbf{b}_{k,x_{0},\xi_{0},\mathbf{b}_{0}}(T)^{*}G_{k,x_{0},\xi_{0}}(T)\mathbf{b}_{k,x_{0},\xi_{0},\mathbf{b}_{0}}(T)+o(1).$

Plugging those estimates into the definition of the cost as given by (46)
concludes the proof.

###  6.3. Diagonalisation of the propagation term. Proof of Lemma 6.5

We consider the system (78), and apply to it the operator
$\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]$, giving

$\partial_{t}\widetilde{\bm{u}}+\frac{i}{\varepsilon}\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\operatorname{op^{w}_{\varepsilon}}[H]\bm{u}+\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\operatorname{op^{w}_{\varepsilon}}[A]\bm{u}=0.$
(96)

Applying Lemma 4.6 for the identity (79), we obtain

$\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\operatorname{op^{w}_{\varepsilon}}[H]+\frac{\varepsilon}{2i}\operatorname{op^{w}_{\varepsilon}}[\\{\mathcal{U}^{*},H\\}]=\operatorname{op^{w}_{\varepsilon}}[\mathcal{D}]\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]+\frac{\varepsilon}{2i}\operatorname{op^{w}_{\varepsilon}}[\\{\mathcal{D},\mathcal{U}^{*}\\}]+\varepsilon^{2}\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}],$
(97)

where $r_{\varepsilon}\in S(1)$ is a remainder symbol such that
$\mathcal{N}_{\gamma d}(r_{\varepsilon})$ is bounded uniformly in
$\varepsilon$. By a slight abuse of notation, we will denote all such
remainders $r_{\varepsilon}$, despite them not being the same in all the
equalities. Rearranging, using that
$\mathcal{U}\mathcal{U}^{*}=\operatorname{Id}$ and recalling the definition of
$A^{\flat}$ in (86), it follows that

$\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\operatorname{op^{w}_{\varepsilon}}[H]=\operatorname{op^{w}_{\varepsilon}}[\mathcal{D}]\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]+\frac{\varepsilon}{i}\operatorname{op^{w}_{\varepsilon}}[A^{\flat}]\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]+\varepsilon^{2}\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}].$
(98)

Meanwhile, for the term of order 0, we have

$\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\operatorname{op^{w}_{\varepsilon}}[A]=\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}A\mathcal{U}]\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]+\varepsilon\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}].$
(99)

Substituting (98) and (99) into (96), it finally yields

$\partial_{t}\widetilde{\bm{u}}+\left(\frac{i}{\varepsilon}\operatorname{op^{w}_{\varepsilon}}[\mathcal{D}]+\operatorname{op^{w}_{\varepsilon}}[A^{\flat}]+\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}A\mathcal{U}]\right)\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\bm{u}=\varepsilon\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}]\widetilde{\bm{u}},$
(100)

and the claim immediately follows.

######  Remark 6.8.

For any $1\leq k\leq m$, multiplying equality (98) on the left by $\Pi_{k}$
and on the right by $\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}]\Pi_{k}$
gives, denoting by $\sigma_{p}$ the principal symbol of a pseudo-differential
operator,

$\sigma_{p}\left(\Pi_{k}(\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\operatorname{op^{w}_{\varepsilon}}[H]\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}]-\operatorname{op^{w}_{\varepsilon}}[\mathcal{D}]\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}^{*}]\operatorname{op^{w}_{\varepsilon}}[\mathcal{U}])\Pi_{k}\right)=-i\varepsilon\Pi_{k}A^{\flat}\Pi_{k}.$
(101)

By property (41) of the Weyl quantization, the first operator in the left-hand
side of (101) is self-adjoint, and the second one is also self-adjoint, as a
consequence of the symbolic identity
$(\Pi_{k}\mathcal{D})(x,\xi)=(\lambda_{k}\Pi_{k})(x,\xi)$. Therefore the
symbolic matrix $i(\Pi_{k}A^{\flat}\Pi_{k})(x,\xi)$ is Hermitian for $1\leq
k\leq m$.

###  6.4. Block-diagonalisation of the amplification term. Proof of Lemma 6.6

We follow a method initially used by Poincaré in the setting of ODEs to
further diagonalize the system, into a pseudo-differential normal form where
the term of order $\frac{1}{\varepsilon}$ is diagonal and the term of order
$1$ is block-diagonal. Since
$(\operatorname{Id}+\varepsilon\operatorname{op^{w}_{\varepsilon}}[Q])^{-1}=\operatorname{Id}-\varepsilon\operatorname{op^{w}_{\varepsilon}}[Q]+\varepsilon^{2}\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}]$,
we get

$\displaystyle(\operatorname{Id}+\varepsilon\operatorname{op^{w}_{\varepsilon}}[Q])\left(\operatorname{op^{w}_{\varepsilon}}\left[\frac{i}{\varepsilon}\mathcal{D}+\Gamma\right]\right)(\operatorname{Id}+\varepsilon\operatorname{op^{w}_{\varepsilon}}[Q])^{-1}$
$\displaystyle=\operatorname{op^{w}_{\varepsilon}}\left[\frac{i}{\varepsilon}\mathcal{D}-i[\mathcal{D},Q]+\Gamma\right]$
$\displaystyle\quad+\varepsilon\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}].$
(102)

Now, we observe from the definition of $Q$ in (88) that

$\displaystyle-i[\mathcal{D},Q]+\Gamma=\sum_{k=1}^{m}\Pi_{k}\Gamma\Pi_{k},$
(103)

Substituting (103) into this equality then yields

$\displaystyle(\operatorname{Id}+\varepsilon\operatorname{op^{w}_{\varepsilon}}[Q])\left(\operatorname{op^{w}_{\varepsilon}}\left[\frac{i}{\varepsilon}\mathcal{D}+\Gamma\right]\right)(\operatorname{Id}+\varepsilon\operatorname{op^{w}_{\varepsilon}}[Q])^{-1}$
$\displaystyle=\operatorname{op^{w}_{\varepsilon}}\left[\frac{i}{\varepsilon}\mathcal{D}+\sum_{k=1}^{m}\Pi_{k}\Gamma\Pi_{k}\right]$
$\displaystyle\quad+\varepsilon\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}]$
(104)

Performing the change of variable
$\bm{v}=(\operatorname{Id}+\varepsilon\operatorname{op^{w}_{\varepsilon}}[Q])\widetilde{\bm{u}}$
in the system (87) then gives

$\partial_{t}\bm{v}+\operatorname{op^{w}_{\varepsilon}}\left[\frac{i}{\varepsilon}\mathcal{D}+\sum_{k=1}^{m}\Pi_{k}\Gamma\Pi_{k}\right]\bm{v}=\varepsilon\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}]\bm{v}.$
(105)

For $1\leq k\leq m$, multiplying this equation on the left by $\Pi_{k}$ then
leads to the system (89) and the claim is proven.

###  6.5. Estimate of the off-diagonal energy terms. Proof of Lemma 6.7

By (105) and standard pseudo-differential calculations, for any $1\leq j\neq
k\leq m$, recalling that $\lambda_{j}$ denotes the eigenvalues of
$\mathcal{D}$,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\langle\operatorname{op^{w}_{\varepsilon}}\left[\frac{\Pi_{k}\Lambda\Pi_{j}}{\lambda_{j}-\lambda_{k}}\right]\bm{v},\bm{v}\right\rangle$
$\displaystyle=-\frac{i}{\varepsilon}\left\langle\operatorname{op^{w}_{\varepsilon}}\left[\left[\frac{\Pi_{k}\Lambda\Pi_{j}}{\lambda_{j}-\lambda_{k}},\mathcal{D}\right]\right]\bm{v},\bm{v}\right\rangle+O(1)\|\bm{v}\|_{L^{2}}^{2}$

Now, we observe that, for any $1\leq j\neq k\leq m$,

$\left[\frac{\Pi_{k}\Lambda\Pi_{j}}{\lambda_{j}-\lambda_{k}},\mathcal{D}\right]=\Pi_{k}\Lambda\Pi_{j},$

and we set

$E_{j,k}(t):=\langle\operatorname{op^{w}_{\varepsilon}}[\Pi_{k}\Lambda\Pi_{j}]\bm{v},\bm{v}\rangle_{L^{2}}.$

This immediately yields

$E_{j,k}(t)=i\varepsilon\frac{\mathrm{d}}{\mathrm{d}t}\left\langle\operatorname{op^{w}_{\varepsilon}}\left[\frac{\Pi_{k}\Lambda\Pi_{j}}{\lambda_{j}-\lambda_{k}}\right]\bm{v},\bm{v}\right\rangle+O(\varepsilon)\|\bm{v}(t)\|^{2}_{L^{2}}$
(106)

Then, integrating in time yields

$\displaystyle\int_{0}^{T}E_{j,k}(t)\,\mathrm{d}t$
$\displaystyle=i\varepsilon\left\langle\operatorname{op^{w}_{\varepsilon}}\left[\frac{\Pi_{k}\mathcal{U}^{*}B^{*}B\mathcal{U}\Pi_{j}}{\lambda_{j}-\lambda_{k}}\right]\bm{v}(T),\bm{v}(T)\right\rangle_{L^{2}}$
$\displaystyle\quad-i\varepsilon\left\langle\operatorname{op^{w}_{\varepsilon}}\left[\frac{\Pi_{k}\mathcal{U}^{*}B^{*}B\mathcal{U}\Pi_{j}}{\lambda_{j}-\lambda_{k}}\right]\bm{v}(0),\bm{v}(0)\right\rangle_{L^{2}}+O(\varepsilon)\|\bm{v}(T)\|_{L^{2}}^{2}.$

By the non-crossing condition,
$\operatorname{op^{w}_{\varepsilon}}\left[\frac{\Pi_{k}\mathcal{U}^{*}B^{*}B\mathcal{U}\Pi_{j}}{\lambda_{j}-\lambda_{k}}\right]$
is a bounded operator on $L^{2}$, and the claim immediately follows.

##  7\. Compressible gas-dynamics in a rapidly rotating frame

As an illustration we consider the non-isentropic compressible Euler equations
for perfect gases in a rotating frame (see for instance [2, 10]) in $2$D:

$\left\\{\begin{aligned}
\partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}+\rho^{-1}\nabla p&=0,\\\
\partial_{t}p+\bm{u}\cdot\nabla p+\rho
c^{2}\nabla\cdot\bm{u}+\frac{b(x_{2})}{\rm{Ro}}\bm{u}^{\perp}&=0,\\\
\partial_{t}s+\bm{u}\cdot\nabla s&=0,\end{aligned}\right.$ (107)

where $\rho$ is the density, $\bm{u}$ the velocity, $p$ the pressure, $s$ the
entropy, $\rm{Ro}\in\mathbb{R}^{+}$ the Rossby number,
$\bm{u}^{\perp}=(-u_{2},u_{1})$, $b$ is a smooth function of $x_{2}$ and $c$
is the sound speed. We consider the the case of polytropic gases, for which
the pressure law is $p=(\gamma-1)\rho^{\gamma}e^{\frac{s}{c}},$ for
$\gamma>1$, and the sound speed $c$ is given by $c=\sqrt{\gamma p/\rho}$.
Then, setting $\pi:=(p/(\gamma-1))^{2\gamma},$ the system (107) is recast as

$\left\\{\begin{aligned} \partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}+2\gamma
e^{\frac{s}{\gamma
c}}\pi\nabla\pi+\frac{b(x_{2})}{\rm{Ro}}\bm{u}^{\perp}&=0,\\\
\partial_{t}\pi+\bm{u}\cdot\nabla\pi+\frac{\gamma-1}{2}\pi\nabla\cdot\bm{u}&=0,\\\
\partial_{t}s+\bm{u}\cdot\nabla s&=0.\end{aligned}\right.$

The linearization of the system at a steady solution
$(\overline{\bm{u}},\overline{\pi},\overline{s})$, where
$\overline{\bm{u}}=(\overline{u}_{1}(x_{2}),0)$ is a shear flow,
$\overline{s}\in\mathbb{R}^{+}$ is a constant value and
$\overline{\pi}=\overline{\pi}(x_{2})$ satisfies the relation
$2\gamma\text{Ro}\exp(\overline{s}/c\gamma)(\overline{\pi}\overline{\pi}^{\prime})(x_{2})=-b(x_{2})\overline{u}_{1}(x_{2})$.
In the case where $\text{Ro}=\varepsilon$, the semi-classical approximation
yields the following system:

$\partial_{t}\textbf{U}+\operatorname{op^{w}_{\varepsilon}}\left[\frac{i}{\varepsilon}H+A\right]\textbf{U}=0,$
(108)

where

$\textbf{U}=\begin{pmatrix}u_{1}\\\ u_{2}\\\ \pi\\\ s\end{pmatrix},\quad
H=\begin{pmatrix}\overline{u}_{1}(x_{2})\xi_{1}&-ib(x_{2})&2\gamma
e^{\frac{\overline{s}}{\gamma c}}\overline{\pi}\xi_{1}&0\\\
ib&\overline{u}_{1}(x_{2})\xi_{1}&2\gamma
e^{\frac{\overline{s}}{\gamma}}\overline{\pi}\xi_{2}&0\\\
\frac{\gamma-1}{2}\overline{\pi}\xi_{1}&\frac{\gamma-1}{2}\overline{\pi}\xi_{2}&\overline{u}_{1}(x_{2})\xi_{1}&0\\\
0&0&0&\overline{u}_{1}(x_{2})\xi_{1}\end{pmatrix},$

and

$A=\begin{pmatrix}0&\overline{u}_{1}^{\prime}(x_{2})&0&0\\\ 0&0&2\gamma
e^{\frac{\overline{s}}{\gamma}}\overline{\pi}^{\prime}(x_{2})&2\gamma
e^{\frac{\overline{s}}{\gamma}}\overline{\pi}\overline{\pi}^{\prime}(x_{2})\\\
0&\overline{\pi}^{\prime}(x_{2})&0&0\\\ 0&0&0&0\end{pmatrix}.$

The principal symbol $H$ is not symmetric; however, setting

$S:=\operatorname{diag}(1,1,4\frac{\gamma}{\gamma-1}\exp({\overline{s}}/{c\gamma}),1),$

which only depends on $x$, and computing

$\widetilde{H}=\sqrt{S}H\sqrt{S}^{-1}=\begin{pmatrix}\overline{u}_{1}\xi_{1}&-ib&\eta\xi_{1}&0\\\
ib&\overline{u}_{1}\xi_{1}&\eta\xi_{2}&0\\\
\eta\xi_{1}&\eta\xi_{2}&\overline{u}_{1}\xi_{1}&0\\\
0&0&0&\overline{u}_{1}\xi_{1}\end{pmatrix},$

where
$\eta:=\sqrt{\gamma(\gamma-1)}\exp(\frac{\overline{s}}{2c\gamma})\overline{\pi}$
and $\sqrt{S}$ is the positive square root of $S$, we observe that
$\widetilde{H}$ is symmetric. We are then led to a system of the form:

$\partial_{t}\textbf{W}+\operatorname{op^{w}_{\varepsilon}}\left[\frac{i}{\varepsilon}\widetilde{H}+\widetilde{A}\right]\textbf{W}=\varepsilon\operatorname{op^{w}_{\varepsilon}}[r_{\varepsilon}]\textbf{W}.$
(109)

The eigenvalues of $\widetilde{H}$ are
$\lambda_{1}=\overline{u}_{1}(x_{2})\xi_{1}$ with multiplicity $2$,

$\lambda_{2}=\overline{u}_{1}\xi_{2}+\sqrt{b^{2}+C_{1}^{2}\overline{\pi}^{2}\exp({\overline{s}}/{c\gamma})|\xi|^{2}},$

with multiplicity $1$, and

$\lambda_{3}=\overline{u}_{1}\xi_{2}-\sqrt{b^{2}+C_{1}^{2}\overline{\pi}^{2}\exp({\overline{s}}/{c\gamma})|\xi|^{2}},$

with multiplicity $1$. The non-crossing assumption is verified for
$(x_{1},x_{2})$ such that

$b^{2}(x_{2})+C_{1}^{2}\overline{\pi}^{2}(x_{2})\exp({\overline{s}}/{c\gamma})|\xi|^{2}\neq
0.$

Two orthogonal eigenvectors of $\lambda_{1}$ are given by

$(0,0,0,1)^{T},\quad\frac{1}{\langle\xi\rangle_{b}}(-i\xi_{2},i\xi_{1},\widetilde{b},0)^{T},\quad\text{where}\quad\langle\xi\rangle_{b}=\sqrt{|\xi^{2}|+\widetilde{b}^{2}},\quad\widetilde{b}=\frac{b}{C_{1}\overline{\pi}}\exp(-{\overline{s}}/{2c\gamma}).$

We can then compute the block of the (block-diagonal) matrix
$A^{\flat}(x,\xi)$ in (86) associated to $\lambda_{1}$

$\displaystyle A^{\flat}_{\lambda_{1}}(x,\xi)=\begin{pmatrix}0&0\\\
0&-\frac{i\xi_{1}C_{1}\exp(\overline{s}/2\gamma)\overline{\pi}\partial_{2}\widetilde{b}(x_{2})}{\langle\xi\rangle^{2}_{b}}\end{pmatrix}.$

Acknowledgements. F. S. was partially supported by the Agence Nationale de la
Recherche, Project SINGFLOWS, grant ANR-18-CE40-0027-01, Project BOURGEONS,
grant ANR-23-CE40-0014-01, and Institut Universitaire de France. R.B.
acknowledges financial support by the Italian Ministry for Universities and
Research (MUR), Project PRIN 2022 TESEO, grant 2022HSSYPN.

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$\frac{1}{3}D^{\rho}_{h}-\frac{2\lambda_{h}}{3K}I\preceq\widehat{D}^{\rho}_{h}\preceq\frac{5}{3}D^{\rho}_{h}+\frac{2\lambda_{h}}{3K}I~{}.$
(31)

Equations (30) and (31) imply that, with probability at least $1-\delta$, we
have

$\widehat{D}^{\rho}_{h}-\left(\frac{1}{c}D^{\pi,\nu}_{h}-\frac{2\lambda_{h}}{3K}I\right)\succeq
0~{}.$

Since the left-hand side is a diagonal matrix, the above is equivalent to

$\min_{(s,a,b)}\left(\widehat{d}^{\rho}_{h}(s,a,b)-\left(\frac{1}{c}-\frac{2\lambda_{h}}{3Kd^{\pi,\nu}_{h}(s,a,b)}\right)d^{\pi,\nu}_{h}(s,a,b)\right)\geq
0~{}.$

Now let us define constant $c_{1}$ as

$\displaystyle
c_{1}=\max\\{\frac{1}{c}-\frac{2d\log(KH/\delta)}{3Kd^{\pi,\nu}_{h}(s,a,b)}:d^{\pi,\nu}_{h}(s,a,b)>0\\}~{}.$

Then we obtain

$\min_{(s,a,b)}\left(\widehat{d}^{\rho}_{h}(s,a,b)-c_{1}d^{\pi,\nu}_{h}(s,a,b)\right)\geq
0,$

which means that we have

$\widehat{D}^{\rho}_{h}-c_{1}D^{\pi,\nu}_{h}\succeq 0~{}.$

Now, since $\Phi$ has full rank, its null space is $0$. Thus, given non-zero
$x\in\mathbb{R}^{d}$, we have $\Phi x\neq 0$ and thus

$(\Phi x)^{\top}\left(\widehat{D}^{\rho}_{h}-c_{1}D^{\pi,\nu}_{h}\right)(\Phi
x)\geq 0~{},$

since $\widehat{D}^{\rho}_{h}-c_{1}D^{\pi,\nu}_{h}\succeq 0$. Therefore,

$\Phi^{\top}\left(\widehat{D}^{\rho}_{h}-c_{1}D^{\pi,\nu}_{h}\right)\Phi\succeq
0~{},$

which implies

$\sum^{K}_{\tau=1}\Phi^{\top}\widetilde{D}^{\rho}_{\tau,h}\Phi+I\succeq
I+c_{1}K\Phi^{\top}D^{\pi,\nu}_{h}\Phi~{}.$

Note that the above can be written as

$\Lambda_{h}\succeq
I+c_{1}K\max\left\\{\sup_{\nu}\;\mathbb{E}_{\pi^{*},\nu}\left[\phi_{h}\phi^{\top}_{h}|s_{1}=x\right],\;\sup_{\nu}\;\mathbb{E}_{\pi,\nu^{*}}\left[\phi_{h}\phi^{\top}_{h}|s_{1}=x\right]\right\\}~{}.$

Now assume the MG is tabular. Then $\Phi$ is just the identity matrix. Thus,
if Assumption 3 holds, then there exists a positive constant $c_{1}$ such
that, for any $h\in[H]$, $(\pi,\nu)\in\mathcal{U}(\pi^{*},\nu^{*})$, we have

$\displaystyle\Lambda_{h}\succeq I+c_{1}K\Phi^{\top}D^{\pi,\nu}_{h}\Phi$
$\displaystyle\Rightarrow\Phi^{\top}\left(\widehat{D}^{\rho}_{h}-c_{1}D^{\pi,nu}_{h}\right)\Phi\succeq
0$ $\displaystyle\Rightarrow\widehat{D}^{\rho}_{h}-c_{1}D^{\pi,\nu}_{h}\succeq
0$
$\displaystyle\Rightarrow\frac{5}{3}D^{\rho}_{h}+\frac{2\lambda_{h}}{3K}I-c_{1}D^{\pi,\nu}_{h}\succeq
0$ $\displaystyle\Rightarrow
D^{\rho}_{h}+\frac{2\lambda_{h}}{5K}I-\frac{3}{5}c_{1}D^{\pi,\nu}_{h}\succeq
0$
$\displaystyle\Rightarrow\min_{(s,a,b)}\left(d^{\rho}_{h}(s,a,b)-\left(\frac{3}{5}c_{1}-\frac{2\lambda_{h}}{3Kd^{\pi,\nu}_{h}(s,a,b)}\right)d^{\pi,\nu}_{h}(s,a,b)\right)\geq
0~{}.$

Now let us define

$c^{\prime}=\max\left\\{\frac{3}{5}c_{1}-\frac{2\lambda_{h}}{5Kd^{\pi,\nu}_{h}(s,a,b)}:d^{\pi,\nu}_{h}(s,a,b)>0\right\\}~{}.$

We have

$\min_{(s,a,b)}\left(d^{\rho}_{h}(s,a,b)-c^{\prime}d^{\pi,\nu}_{h}(s,a,b)\right)\geq
0~{}.$

Thus, for $c=1/c^{\prime}$, we obtain

$\frac{d^{\pi,\nu}_{h}(s,a,b)}{d^{\rho}_{h}(s,a,b)}<c<\infty,\forall
h\in[H],(\pi,\nu)\in\mathcal{U}(\pi^{*},\nu^{*}),(s,a,b)\in\\{(s,a,b)\in\mathcal{S}\times\mathcal{A}\times\mathcal{B}:d^{\pi,\nu}_{h}(s,a,b)>0\\}~{}.$

∎

####  C.2 Proof of Proposition 2

First, we show the equivalence of both assumptions in the tabular Markov game
setting. Note that, in the tabular setting, the features are $SAB$ dimensional
and $\phi(s,a,b)$ is the unit vector $e_{s,a,b}$ with coordinate $(s,a,b)$ set
to $1$. First, suppose Assumption 2 is true. Then we have,

$\mathbb{E}_{d^{\rho}_{h}}\left[e_{s,a,b}e_{s,a,b}^{\top}\right]=\textrm{diag}\left[\\{d^{\rho}_{h}(s,a,b)\\}_{s,a,b}\right]\succeq\xi\mathrm{I}$
(32)

This implies that $d^{\rho}_{h}(s,a,b)\geq\xi$ for any $h$ and $s,a,b$ and
Assumption 7 is satisfied.

Now, suppose Assumption 7 is true. Then $d^{\rho}_{h}(s,a,b)>0$ for any $h$
and tuple $(s,a,b)$. Since the number of states is finite, there exists a
constant $C$ such that $d^{\rho}_{h}(s,a,b)\geq C$ for any $h$ and any tuple
$(s,a,b)$. Therefore, Equation (32) is satisfied with $\xi=C$, and thus,
Assumption 2 is satisfied.

Next, we show that Assumption 2 is actually a stronger assumption than
Assumption 7 for the more general linear MDP model. We will write
$\Phi\in\mathbb{R}^{SAB\times d}$ to denote the feature matrix where the row
$(s,a,b)$ corresponds to the $d$-dimensional feature $\phi(s,a,b)$.777For
infinite $\mathcal{S}$, $\Phi$ is interpreted as a function.

######  Lemma 15.

Suppose $\textrm{rank}(\Phi)=d$. Then Assumption 2 implies Assumption 7.

###### Proof.

We will assume $\mathcal{S}$ is finite but possibly very large. The proof can
be easily generalized for infinite $\mathcal{S}$. Since $\Phi$ has rank $d$
let us write $\Phi=U\Lambda V^{\top}$ where $U\in\mathbb{R}^{SAB\times d}$,
$\Lambda$ is a $d$-dimensional diagonal matrix, and $V\in\mathbb{R}^{d\times
d}$. Moreover, we can take $V$ to be orthonormal i.e. $V^{\top}V=\mathrm{I}$.
Let $D^{\rho}_{h}\in\mathbb{R}^{SAB\times SAB}$ be a diagonal matrix with
$D^{\rho}(s,a,b)=d^{\rho}_{h}(s,a,b)$.

$\displaystyle\mathbb{E}_{d^{\rho}_{h}}\left[\phi(s,a,b)\phi(s,a,b)^{\top}\right]=\Phi^{\top}D^{\rho}_{h}\Phi$
$\displaystyle=V\Lambda U^{\top}D^{\rho}_{h}U\Lambda
V^{\top}\succeq\xi\mathrm{I}_{d\times d}$

Since $\Phi$ has rank $d$, both $\Lambda$ and $V$ are invertible. This gives
us

$\displaystyle U^{\top}D^{\rho}_{h}U\succeq\xi(V\Lambda)^{-1}(\Lambda
V^{\top})^{-1}$
$\displaystyle=\xi\Lambda^{-1}(V^{\top}V)^{-1}\Lambda^{-1}\succeq\xi\Lambda^{-2}\mathrm{I}_{d\times
d}$

Since the matrix $U^{\top}D^{\rho}_{h}U$ is already in diagonalized
representation, the minimum eigenvalue of $U^{\top}D^{\rho}_{h}U$ is the
smallest diagonal entry of $D^{\rho}_{h}$. Therefore,

$\lambda_{\min}\left(U^{\top}D^{\rho}_{h}U\right)=\min_{s,a,b}d^{\rho}_{h}(s,a,b)\geq\frac{\xi}{\max_{j\in[d]}\Lambda(j)^{2}}$

Since $\Phi$ has rank $d$ at least one entry of the diagonal matrix $\Lambda$
is non-zero. Therefore, for any $h$, the tuple $(s,a,b)$ is covered with
probability at least $p$ where $p=\xi/\max_{j}\Lambda(j)^{2}$.

On the other hand, note that, for general rank-$d$ feature matrix $\Phi$,
assumption 7 need not imply assumption 2. However, if we put additional
restrictions on the features e.g. diversity, these two assumptions could be
equivalent. ∎