# LAD: Language Augmented Diffusion for Reinforcement Learning

Edwin Zhang University of California, Santa Barbara.
{ete, yujielu<EMAIL_ADDRESS>Yujie Lu University of California, Santa
Barbara.
{ete, yujielu<EMAIL_ADDRESS>William Wang University of California,
Santa Barbara.
{ete, yujielu<EMAIL_ADDRESS>Amy Zhang Meta AI
<EMAIL_ADDRESS>
###### Abstract

Learning skills from language provides a powerful avenue for generalization in
reinforcement learning, although it remains a challenging task as it requires
agents to capture the complex interdependencies between language, actions, and
states. In this paper, we propose leveraging Language Augmented Diffusion
models as a planner conditioned on language (LAD). We demonstrate the
comparable performance of LAD with the state-of-the-art on the CALVIN language
robotics benchmark with a much simpler architecture that contains no inductive
biases specialized to robotics, achieving an average success rate (SR) of 72%
compared to the best performance of 76%. We also conduct an analysis on the
properties of language conditioned diffusion in reinforcement learning.

##  1 Introduction

It has been a longstanding dream of the AI community to be able to create a
household robot that can follow natural language instructions and execute
behaviors such as cleaning dishes or organizing the living room [1, 2, 3, 4,
5, 6, 7, 8, 9]. Incorporating language into Reinforcement Learning (RL) has
great potential for generalization, enabling agents to utilize common sense
priors across tasks and environments. Language provides an expressive
abstraction of the environment and systematic generalization to new
actions[10, 11, 12, 13, 14]. Given the recent progress in Natural Language
Processing, how can one incorporate the powerful capabilities of language
models and utilize them for downstream decision-making?

Language-conditioned policies (LCPs) are one class of policies in RL [15, 16,
13] that can be used to formulate this task, through the conditioning of
behavior on natural language instruction. In this paper, we consider a novel
approach for constructing LCPs by viewing the image as a sequence of state-
actions rather than pixels, reformulating language conditioned decision-making
into text-to-image generation.

Diffusion models such as DALL-E 2 [17] and GLIDE [18] have recently shown
promise as generative models, with state-of-the-art text-to-image generation
results demonstrating a surprisingly deep understanding of semantic
relationships and generation of novel scenes. A key driver of the recent
success in generative modeling is the usage of classifier-free guidance, which
is amenable to the RL framework through the usage of language as a reward
function. Inspired by the recent success of diffusion models in generative
modeling [17, 18], we propose a new algorithm for LCPs (LAD) via latent
diffusion models [19]. We demonstrate comparable performance to the state of
the art on the CALVIN benchmark [20], with average success rate (SR) of 72%
compared to the best performance of 76%.

Figure 1: An overview of our training pipeline. The VAE is used to encode a
sampled horizon of RGB observations into a lower dimensional latent space. We
concatenate this sequence of states with the corresponding sequence of actions
to construct the latent plan. We then noise the plan according to a uniformly
sampled timestep from the diffusion process’ variance schedule. We train a
Temporal U-Net to reverse this process when conditioned on an encoded natural
language instruction from an upstream language model, effectively learning how
to conditionally denoise the latent plan. To train the U-Net, one can simply
use the p-norm between the predicted latent plan and the ground truth latent
plan as the loss. We set $p=1$ in practice.

##  2 Language Augmented Diffusion

###  2.1 Method Overview

We consider a language-conditioned RL setting where we assume that the true
reward function $\pazocal{R}$ is unknown, and must be inferred from a natural
language instruction $\textit{L}\in\mathscr{L}$. Formally, let $\pazocal{F}$
be the function space of $\pazocal{R}$. Then the goal becomes learning a
$\phi:\mathscr{L}\mapsto\pazocal{F}$, and maximizing the policy objective
conditioned on the reward function $\phi(\textit{L})$, $J(\pi(\cdot\mid
s,\pazocal{R}))=\displaystyle\mathbb{E}_{a\sim\pi,s\sim p}$
$\sum_{t=0}^{\infty}\gamma^{t}r_{t}$. Note that the space of tasks that can be
specified by language is much larger than that of reward, due to the Markov
restriction of the latter. For example, ‘‘pour the milk’’ and ‘‘pour the milk
after five o’clock’’ are both valid instructions, but are indistinguishable
from a reward function if the state does not contain temporal information. We
assume access to a prior collected dataset $\pazocal{D}$ of $N$ annotated
trajectories $\tau_{i}$ =
$\langle(s_{0},a_{0},...s_{T}),\textit{L}_{i}\rangle$. The language
conditioned policy $\pi_{\beta}$, or the behavior policy. is defined to be the
policy that generates the aforementioned dataset. In this paper, we assume
access to a dataset of expert trajectories, such that $\pi_{\beta}=$ optimal
policy $\pi^{\star}$. In this case, the policy objective reduces to imitation
learning, or

$\min_{\pi}\mathbb{E}_{s,\pazocal{R}\sim\pazocal{D}}\left[D_{\text{KL}}\left(\pi_{\beta}(\cdot\mid
s,\pazocal{R}),\pi(\cdot\mid s,\pazocal{R})\right)\right].$ (1)

As we tackle the problem from a planning perspective, we define a trajectory
generator as $\pazocal{P}$ and switch the atomic object from actions to
trajectories $\bm{\tau}$. Thus we aim to

$\displaystyle\min_{\pazocal{P}}D_{\text{KL}}\left(\pazocal{P}_{\beta}(\bm{\tau}\mid\pazocal{R}),\pazocal{P}(\bm{\tau}\mid\pazocal{R})\right)$
(2) $\displaystyle=$
$\displaystyle\min_{\pazocal{P}}\mathbb{E}_{\bm{\tau},\pazocal{R}\sim\pazocal{D}}\left[\log\pazocal{P}_{\beta}(\bm{\tau}\mid\pazocal{R})-\log\pazocal{P}(\bm{\tau}\mid\pazocal{R})\right]$

To model this, we turn to diffusion models [21]. Inspired by non-equilibrium
thermodynamics, the common forms of diffusion models [22, 23, 24] propose
modeling the data distribution $p(\bm{\tau})$ as a random process that
steadily adds increasingly varied amounts of Gaussian noise to samples from
$p(\bm{\tau})$ until the distribution converges to the standard normal. We
denote the forward process as $f(\bm{\tau}_{t}|\bm{\tau}_{t-1})$, with a
sequence of variances $(\beta_{0},\beta_{1}...\beta_{T})$. We define
$\alpha_{t}:=1-\beta_{t}$ and $\bar{\alpha}_{t}:=\prod_{s=1}^{t}\alpha_{s}$.

$f(\bm{\tau}_{1:T}|\bm{\tau}_{0})=\prod^{T}_{t=1}f(\bm{\tau}_{t}|\bm{\tau}_{t-1}),\quad\text{where
}f(\bm{\tau}_{t}|\bm{\tau}_{t-1})=\pazocal{N}(\bm{\tau}_{t};\sqrt{1-\beta_{t}}\bm{\tau}_{t-1},\beta_{t}\mathbf{I}).$
(3)

One can tractably reverse this process when conditioned on
$\mathbf{\tau_{0}}$, which allows for the construction of a sum of the typical
variational lower bounds for learning the backward process’ density function
[22]. Since the backwards density also follows a Gaussian, it suffices to
predict $\mu_{\theta}$ and $\Sigma_{\theta}$ which parameterize the backwards
distribution:

$p_{\theta}\left(\bm{\tau}_{t-1}\mid\bm{\tau}_{t}\right)=\pazocal{N}\left(\bm{\tau}_{t-1};\bm{\mu}_{\theta}\left(\bm{\tau}_{t},t\right),\mathbf{\Sigma}_{\theta}\left(\bm{\tau}_{t},t\right)\right).$
(4)

In practice, $\Sigma_{\theta}$ is often fixed to constants, but can also be
learned through reparameterization. Following [23] we consider learning only
$\mu_{\theta}$, which can be computed just as a function of $\tau_{t}$ and
$\epsilon_{\theta}(\tau_{t},t)$. One can derive that
$\bm{\tau}_{t}=\sqrt{\bar{\alpha}_{t}}\bm{\tau}_{0}+\sqrt{1-\bar{\alpha}_{t}}\epsilon$
for $\epsilon\sim\pazocal{N}(\mathbf{0},\mathbf{I})$, through a successive
reparameterization of (3) until arriving at $f(\bm{\tau}_{t}|\bm{\tau}_{0})$.
To sample from $p(\tau)$, we need only to learn $\epsilon_{\theta}$, which is
done by regressing to the ground truth $\epsilon$ given by the tractable
backwards density. Assuming we have $\epsilon_{\theta}$, we can then follow a
Markov chain of updates that eventually converges to the original data
distribution, in a procedure reminiscent of Stochastic Gradient Langevin
Dynamics [25]:

$\bm{\tau}_{t-1}=\frac{1}{\sqrt{1-\beta_{t}}}\left(\bm{\tau}_{t}-\frac{\beta_{t}}{\sqrt{1-\bar{\alpha}_{t}}}\bm{\epsilon}_{\theta}\left(\bm{\tau}_{t},t\right)\right)+\sigma_{t}\mathbf{z},\quad\text{where
}\mathbf{z}\sim\pazocal{N}(\mathbf{0},\mathbf{I}).$ (5)

Thus, by using a variant of $\epsilon_{\theta}$ conditioned on language to
denoise our latent plans, we can effectively model
$-\nabla_{\tau}\pazocal{P}_{\beta}(\tau\mid\pazocal{R})$ with our diffusion
model, iteratively guiding our generated trajectory towards the optimal
trajectories conditioned on language.

###  2.2 Model Architecture

It is computationally infeasible to operate directly on the pixel space.
Instead, we do planning in latent state space by first compressing the visual
input with a $\beta$-TCVAE [26], similar to latent diffusion models [19]. We
choose $\beta$-TCVAE for its efficiency in compression. The disentangled
representation is a property that is also worth taking into consideration, and
its effects on the denoising training process. It is still unclear whether
disentanglement is beneficial or not. We adopt CLIP [27] as our textual
encoder. CLIP is trained on large-scale image-text pairs and is able to align
visual and textual input in its embedding space. Specifically, we use the
Transformer variant [28] as the text encoder with the architecture
modifications described in [29]. We use a modified temporal U-Net[30], which
performs 1D convolution only across the time dimension, rather than the 2D
convolution typical in text-to-image generation. This is motivated by our wish
to preserve equivariance along the time dimension but not the state-action
dimension. We modify the architecture in [30] by adding conditioning via cross
attention in a fashion that resembles the latent diffusion model, but also
uses a temporal convolution for projection to the token embeddings rather than
the traditional 2D convolution. We use DDIM [24] during inference for
increased computational efficiency and faster planning. DDIM uses strided
sampling and is able to capture almost the same level of fidelity as typical
DDPM sampling [23] with an order of magnitude speedup. For rolling out the
latent plans generated by the denoiser, we resample a new plan with the
frequency of $H$, until either the task is completed successfully or the
maximum timestep is reached. In between samples, we roll out the open-loop
plan without replanning. We set the total time horizon equal to $3H$ in our
experiments, which means at maximum we will sample three latent plans from the
model.

[capbesideposition=left,top,capbesidewidth=.32]figure[]

Figure 2: An overview of our inference pipeline. We pass sampled noise to our
denoising autoencoder along with an encoded language prompt. The autoencoder
is instantiated as a temporal U-Net. By repeating this process iteratively, we
are able to generate high-fidelity latent plans conditioned on language. We
are also able to decode the latent states into pixel space to analyze and
interpret the plans generated by the denoising autoencoder.

##  3 Experiments

###  3.1 Dataset and Metric

We evaluate a subset of the CALVIN benchmark [20] due to time, selecting the
five most common subtasks (i.e tasks with the most pre-collected data) that do
not belong to the same subtask category. This metric allows for fast
experimentation since there is significantly less data to converge on, while
still covering a diverse and broad range of tasks as we enforce that each task
will belong to a different category. After pretraining our VAE on all data, we
freeze the autoencoder and train the U-Net just with data from the five tasks.
This setup is motivated by the assumption that we only wish to evaluate the
performance of the downstream decision-making, so we assume that there already
exists a robust pre-trained autoencoder. In order to provide a fair
comparison, we obtain results from our comparisons by freezing the
autoencoders of their final checkpoint which was trained on all data and
training the rest of the model on just the five tasks. We roll out all
evaluated policies for $5$ trajectories per task, for a total of $25$ rollouts
per policy. We compare the two strongest models released on this benchmark so
far, HULC and MCIL [9, 31] All comparisons are trained in their official
repository111https://github.com/lukashermann/hulc.

Table 1: Comparison of success rates between our diffusion model and prior
benchmarks. Although our absolute performance does not beat the prior SOTA, we
note that HULC is a significantly more complex model containing many inductive
biases for robotics that may not be transferable to other RL environments,
such as using a separate logistic loss for modeling the gripper action.

Task | MCIL | HULC | LAD
---|---|---|---
Place in Slider | $1.0$ | $0.8$ | $1.0$
Open Drawer | $1.0$ | $1.0$ | $1.0$
Move Slider Right | $0.4$ | $1.0$ | $1.0$
Stack Block | $0.2$ | $0.4$ | $0.2$
Lift Blue Block Table | $0.4$ | $0.6$ | $0.4$
$\mathbf{Total~{}(avg)}$ | $\mathbf{0.60}$ | $\mathbf{0.76}$ | $\mathbf{0.72}$

###  3.2 Effects of Diffusion

An interesting phenomenon we observe when rolling out with the diffusion model
is its ability to robustly model arbitrary starting points of trajectories,
leading to an ability to recover from failures in rollouts and attempt tasks
again. For this reason, we find that simply by replanning more times instead
of just rolling out a single plan improves the performance of LAD. There is
much more to explore in this direction, and future rollout strategies that
include conditioning on more than the last state or replanning more often than
just once every H timesteps will likely lead to substantial gains in
performance.

##  4 Conclusion

Learning the atomic sub-skills is critical to solving the multi-task planning
problem and enabling the solving of more complex and open environments through
state and temporal abstraction. We explore reformulating the language-
conditioned planning process as the text-to-plan generation to better learn
the alignment between language and state-action pairs. Experiments and
qualitative analysis demonstrate the simplicity and effectiveness of our
model. Future work looks to extend to the long horizon setting and further
probing of the generalization and compositional capabilities of the model
through classifier-free guidance or improved sampling methods, as well as
incorporating value functions for guiding towards optimality.

#### Acknowledgments

We would like to thank Ashley Jiang for creating the main figures, Michael
Janner and Weixi Feng for fruitful discussions, and Oier Mees and Luka
Shermann for clarifications and support on the CALVIN benchmark.

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##  Appendix A Qualitative Analysis

Figure 4: Comparison of a ground truth trajectory for the "place in slider"
task (left), the reconstructed trajectory (right), and the generated
trajectory through denoising (bottom).

##  Appendix B Background and Related Work

Reinforcement Learning

We formulate the RL framework as Markov decision process (MDP) $\pazocal{M}$ =
($\pazocal{S},\pazocal{A},\pazocal{R},\gamma,p$), with state space
$\pazocal{S}$, action space $\pazocal{A}$, reward function $\pazocal{R}$,
discount factor $\gamma$, and transition dynamics $p$. At each time step $t$,
agents observe a state $s\in\pazocal{S}\subseteq\mathbb{R}^{n}$, take an
action $a\in\pazocal{A}\subseteq\mathbb{R}^{m}$, and transition to a new state
$s^{\prime}$ with reward $r$ following $s^{\prime},r\sim p(\cdot,\cdot|s,a)$.
The goal of RL is then to learn either a deterministic policy
$\pi:\pazocal{S}\mapsto\pazocal{A}$ or a stochastic policy where
$a\sim\pi(\cdot|s)$ with $a\in\pazocal{A}$ that maximises the policy
objective, $J(\pi)=\displaystyle\mathbb{E}_{a\sim\pi,s^{\prime}\sim p}$
$\sum_{t=0}^{\infty}\gamma^{t}r_{t}$. Notice this is nothing but the expected
discounted cumulative reward or expected return.

Language-Conditioned Policy Language-conditioned policies [15, 16, 13] have
been explored in the reinforcement learning community to improve the ability
to abstract the goal and the generalization to a new environment [10, 11, 12,
13]. However, these LCPs still struggle with long-horizon language commands.
We are the first to leverage the advantage of the diffusion model in
compositionality and long-horizon decision-making to address such challenges.

Diffusion in Offline Reinforcement Learning

Given the success of the denoising diffusion probabilistic models [23] (DPM)
applied in text-to-image synthesis [32], the DPM has been further explored in
both discrete and continuous data domains, including image and video synthesis
[33, 34], text generation [35], and time series [36]. Diffusion planning [30]
first proposed to transform the planning problem into inpainting and utilize
diffusion models to solve the problem. Specifically, they diffuse the state
and actions jointly to implement imitation learning and goal-conditioned
reinforcement learning. This leverage the diffusion to solve long-horizon and
compositionality issue in planning. Instead of predicting the whole trajectory
for each state, [37] apply the diffusion model to sample a single action at a
time conditioned with states. Analogous to the Diffuser [30] in trajectory-
planning perspective and [37] in offline policy-optimization perspective, we
are in the language-grounding perspective.


# Uniform synchronization of an abstract linear second order evolution system

###### Abstract.

Although the mathematical study on the synchronization of wave equations at
finite horizon has been well developed, there was few results on the
synchronization of wave equations for long-time horizon. The aim of the paper
is to investigate the uniform synchronization at the infinite horizon for one
abstract linear second order evolution system in a Hilbert space.

First, using the classical compact perturbation theory on the uniform
stability of semigroups of contractions, we will establish a lower bound on
the number of damping, necessary for the uniform synchronization of the
considered system. Then, under the minimum number of damping, we clarify the
algebraic structure of the system as well as the necessity of the conditions
of compatibility on the coupling matrices. We then establish the uniform
synchronization by the compact perturbation method and then give the dynamics
of the asymptotic orbit. Various applications are given for the system of wave
equations with boundary feedback or (and) locally distributed feedback, and
for the system of Kirchhoff plate with distributed feedback. Some open
questions are raised at the end of the paper for future development.

The study is based on the synchronization theory and the compact perturbation
of semigroups.

Tatsien Li111School of Mathematical Sciences, Fudan University, Shanghai
200433, China; Shanghai Key Laboratory for Contemporary Applied Mathematic;
Nonlinear Mathematical Modeling and Methods Laboratory<EMAIL_ADDRESS>Bopeng Rao222Corresponding author. Institut de Recherche Mathématique Avancée,
Université de Strasbourg, 67084 Strasbourg, France<EMAIL_ADDRESS>

Keywords: uniform synchronization, condition of compatibility, second order
evolution system.

MS Classification 2010 93B05, 93C20, 35L53

##  1\. Introduction

Synchronization is a widespread natural phenomenon. It was first observed by
Huygens in 1665 [7]. The theoretical research on synchronization from the
mathematical point of view dates back to Wiener in 1950s in [29] (Chapter 10).
Since 2012, Li and Rao started the research on the synchronization in a finite
time for a coupled system of wave equations with Dirichele boundary controls
[13, 14]). Later, the synchronization has been carried out for a coupled
system of wave equations with various boundary controls, the most part of
their results was recently collected in the monograph [16]. The optimal
control for the exact synchronization of parabolic system was recently
investigated by Wang and Yan in [28]. Consequently, the study of
synchronization becomes a part of research in control theory.

In a recent work [17, 18], we showed that under Kalman’s rank condition, the
observability of a scalar equation implies the uniqueness of solution to a
system of elliptic operators. Using this result, we have established the
asymptotic synchronization by groups for second order evolution systems.

The objective of this work is to investigate the uniform synchronization for
second order evolution systems. Let us briefly describe the formulation and
the main ideas.

Let $H$ and $V$ be two separated Hilbert spaces such that $V\subset H\subset
V^{\prime},$ $V^{\prime}$ being the dual of $V$, with dense and compact
imbeddings. Let $L$ be the duality mapping from $V$ onto $V^{\prime}$, and $g$
be a linear continuous symmetric operator from $V$ into $V^{\prime}$. Let $I$
denote the identity of $\mathbb{R}^{N}$. We define the diagonal operators

(1.1) $\mathcal{L}=LI\quad\hbox{and}\quad\mathcal{G}=gI.$

Let $A$ and $D$ be symmetric and semi-positive definite matrices with constant
elements. Consider the following second order evolution system for the state
variable $U=(u^{(1)},\cdots,u^{(N)})^{T}$:

(1.2) $U^{\prime\prime}+\mathcal{L}U+AU+D\mathcal{G}U^{\prime}=0,$

where $``\ ^{\prime}\ "$ stands for the time derivative.

We first show that if system (1.2) is uniformly stable in the space $(V\times
H)^{N}$, then rank$(D)=N$ (see Corollary 2.5 below). When rank$(D)<N$, system
(1.2) is not uniformly stable, we then turn to consider the synchronization.

For any given integer $p\geqslant 1$, let

(1.3) $0=n_{0}<n_{1}<n_{2}<\cdots<n_{p}=N$

be integers such that $n_{r}-n_{r-1}\geqslant 2$ for all $r$ with $1\leqslant
r\leqslant p.$ We re-arrange the components of the state variable $U$ into $p$
groups

(1.4) $(u^{(1)},\cdots,u^{(n_{1})}),\
(u^{(n_{1}+1)},\cdots,u^{(n_{2})}),\cdots\cdots,(u^{(n_{p-1}+1)},\cdots,u^{(n_{p})}).$

######  Definition 1.1.

System (1.2) is uniformly (exponentially) synchronizable by $p$-groups, if
there exist constants $M\geqslant 1$ and $\omega>0$, such that for any given
initial data $(U_{0},U_{1})\in(V\times H)^{N}$, the corresponding solution $U$
to system (1.2) satisfies

(1.5)
$\displaystyle\|(u^{(k)}(t)-u^{(l)}(t),{u^{(k)}}^{\prime}(t)-{u^{(l)}}^{\prime}(t))\|_{V\times
H}$ $\displaystyle\leqslant$ $\displaystyle Me^{-\omega
t}\|(u^{(k)}_{0}-u^{(l)}_{0},u_{1}^{(k)}-u_{1}^{(l)})\|_{V\times H},\quad
t\geqslant 0$

for all $k,l$ with $n_{r-1}+1\leqslant k,l\leqslant n_{r}$ and all $r$ with
$1\leqslant r\leqslant p.$

Now let us outline the main ideas in the study of the uniform synchronization
by $p$-groups.

Let $C_{p}$ be the matrix given by (3.2) below. Then (1.5) can be equivalently
rewritten as

(1.6) $\|C_{p}(U(t),U^{\prime}(t))\|_{(V\times H)^{N-p}}\leqslant Me^{-\omega
t}\|C_{p}(U_{0},U_{1})\|_{(V\times H)^{N-p}}\quad t\geqslant 0.$

The matrix $A$ satisfies the condition of $C_{p}$-compatibility, if there
exists a symmetric and semi-positive definite matrix $\overline{A}_{p}$ such
that

(1.7)
$\displaystyle(C_{p}C_{p}^{T})^{-1/2}C_{p}A=\overline{A}_{p}(C_{p}C_{p}^{T})^{-1/2}C_{p}.$

Correspondingly, the reduced matrix $\overline{D}_{p}$ can be introduced for
$D$ (see Proposition 3.2). Applying $(C_{p}C_{p}^{T})^{-1/2}C_{p}$ to (1.2)
and setting $W=(C_{p}C_{p}^{T})^{-1/2}C_{p}U$, we get a self-closed reduced
system

(1.8)
$W^{\prime\prime}+\mathcal{L}W+\overline{A}_{p}W+\overline{D}_{p}\mathcal{G}W^{\prime}=0.$

It is clear that the uniform synchronization by $p$-groups of system (1.2) is
equivalent to the uniform stability of the reduced system (1.8).

In Theorem 3.7, we will show that under the condition $\hbox{rank}(D)=N-p$, if
the scalar equation

(1.9) $u^{\prime\prime}+Lu+gu^{\prime}=0$

is uniformly stable in the space $V\times H$, then system (1.2) is uniformly
synchronizable by $p$-groups.

Furthermore (see Theorem 3.9), there exist some functions
$u_{1},\cdots,u_{p}$, such that

(1.10)
$\displaystyle\|(u^{(k)}(t)-u_{r}(t),{u^{(k)}}^{\prime}(t)-u_{r}^{\prime}(t))\|_{V\times
H}$ $\displaystyle\leqslant$ $\displaystyle Me^{-\omega
t}\|(u^{(k)}_{0}-u^{(l)}_{0},u_{1}^{(k)}-u_{1}^{(l)})\|_{V\times H},\quad
t\geqslant 0$

for all $k,l$ with $n_{r-1}+1\leqslant k,l\leqslant n_{r}$ and all $r$ with
$1\leqslant r\leqslant p.$

Moreover, the functions $u_{1},\cdots,u_{p}$ satisfy a homogeneous system,
then, the solution $U$ to system (1.2) follows a conservative orbit. This is
quite different from the approximate boundary synchronization by $p$-groups,
since the approximate boundary synchronization by $p$-groups in the consensus
sense does not imply that in the pinning sense in general (see Chapter 11 in
[16]).

The above approach is direct and efficient. The difficult part of the problem
is to show the necessity of the conditions of $C_{p}$-compatibility which are
imposed as physically reasonable hypotheses even for the systems of ordinary
differential equations. So, we have to first justify the necessity of the
conditions of compatibility, then, the uniform synchronization will be studied
by means of a serious mathematical consideration.

The necessity of the condition of $C_{p}$-compatibility for $A$, respectively
$D$ is intrinsically linked with the rank of the matrix $D$. We will show (see
Proposition 3.5) that $\hbox{rank}(D)\geqslant N-p$ is a necessary condition
for the uniform synchronization by $p$-groups. Then under the minimum rank
condition $\hbox{rank}(D)=N-p$, we establish the necessity of the condition of
$C_{p}$-compatibility for the matrix $A$, respectively $D$ (see Theorem 3.7).

Now we give some related literatures. One of the motivation of studying the
synchronization consists of establishing the controllability for fewer
boundary controls. When the number of boundary controls is fewer than the
number of state variables, the non-exact boundary controllability for a
coupled system of wave equations with various boundary controls in the usual
energy space was established in Li and Rao [16]. However, if the components of
initial data are allowed to have different levels of energy, then the exact
boundary controllability for a system of two wave equations was established by
means of only one boundary control in Alabau-Boussouira [1, 2], Liu and Rao
[20], Rosier and de Teresa [25]. In [4], Dehman established the
controllability of two coupled wave equations on a compact manifold with only
one local distributed control. In [21, 31], Zuazua proposed the average
controllability as another way to deal with the controllability with fewer
controls. The observability inequality is particularly interesting for a trial
on the decay rate of approximate controllability.

The paper is organized as follows. In §2, we consider the uniform stability
and establish a lower bound on the rank of the control matrix, which is
necessary for the study of the uniform synchronization. §3 is devoted to the
uniform synchronization by $p$-groups. Under the minimum rank condition, we
show the necessity of the conditions of $C_{p}$-compatibility for the coupling
matrices in the considered system. In §4, we give some examples of
applications such as the system of wave equations with boundary feedback or
(and) locally distributed feedback, and the system of Kirchhoff plate with
distributed feedback. In §5, we give some comments on the obtained results and
propose some open questions for future development.

##  2\. Uniform stability

We first recall the following well-posedness result (see Proposition 3.1 in
[18]).

######  Proposition 2.1.

System (1.2) generates a semi-group of contractions with a compact resolvent
in the space $(V\times H)^{N}$. More precisely, for any given initial data
$(U_{0},U_{1})\in(V\times H)^{N}$, the corresponding weak solution $U$ to
system (1.2) satisfies

(2.1) $U\in C^{0}(\mathbb{R}^{+},V^{N})\cap C^{1}(\mathbb{R}^{+},H^{N})$

and

(2.2) $\|(U(t),U^{\prime}(t))\|_{(V\times
H)^{N}}\leqslant\|(U_{0},U_{1})\|_{(V\times H)^{N}},\quad t\geqslant 0.$

######  Definition 2.2.

System (1.2) is uniformly (exponentially) stable in the space $(V\times
H)^{N}$, if there exist constants $M\geqslant 1$ and $\omega>0$, such that for
any given initial data $(U_{0},U_{1})\in(V\times H)^{N}$, the corresponding
solution $U$ to system (1.2) satisfies

(2.3) $\|(U(t),U^{\prime}(t))\|_{(V\times H)^{N}}\leqslant Me^{-\omega
t}\|(U_{0},U_{1})\|_{(V\times H)^{N}},\quad t\geqslant 0.$

######  Proposition 2.3.

Let $\mathcal{R}$ be a linear compact mapping from $V$ to $L^{2}(0,T;H)$. Then
we can not find positive constants $M\geqslant 1$ and $\omega>0$, such that
for all $\theta\in V$, the solution to the following problem

(2.4) $\begin{cases}u^{\prime\prime}+Lu=\mathcal{R}\theta,\\\ t=0:\quad
u=\theta,\ u^{\prime}=0\end{cases}$

satisfies

(2.5) $\|(u(t),u^{\prime}(t))\|_{V\times H}\leqslant Me^{-\omega
t}\|\theta\|_{V},\quad t\geqslant 0.$

###### Proof.

Noting that problem (2.4) is time invertible, by well-posedness we have

(2.6)
$\|\theta\|_{V}\leqslant\|u(T)\|_{V}+\|u^{\prime}(T)\|_{H}+\int_{0}^{T}\|\mathcal{R}\theta\|_{H}dt.$

Assume by contradiction that (2.5) holds for all $\theta\in V$, then we have

(2.7) $\|\theta\|_{V}\leqslant Me^{-\omega
T}\|\theta\|_{V}+\int_{0}^{T}\|\mathcal{R}\theta\|_{H}dt.$

When $T$ is large enough, it follows that for all $\theta\in V$, we have

(2.8) $\|\theta\|_{V}\leqslant\frac{\sqrt{T}}{1-Me^{-\omega
T}}\|\mathcal{R}\theta\|_{L^{2}(0,T;H)}.$

This contradicts the compactness of $\mathcal{R}$. The proof is complete. ∎

######  Theorem 2.4.

Let $\widetilde{C}_{q}$ be a full row-rank matrix of order $(N-q)\times N$
with $0\leqslant q<N$. Assume that there exist constants $M\geqslant 1$ and
$\omega>0$, such that for any given initial data $(U_{0},U_{1})\in(V\times
H)^{N}$, the corresponding solution $U$ to system (1.2) satisfies

(2.9) $\|\widetilde{C}_{q}(U(t),U^{\prime}(t))\|_{(V\times H)^{N-q}}\leqslant
Me^{-\omega t}\|(U_{0},U_{1})\|_{(V\times H)^{N}},\quad t\geqslant 0.$

Then

(2.10) $rank(\widetilde{C}_{q}D)\geqslant N-q.$

###### Proof.

Assume by contradiction that the rank condition (2.10) fails. Then, we have

(2.11)
$rank(\widetilde{C}_{q}D)=rank(D\widetilde{C}_{q}^{T})<N-q=rank(\widetilde{C}_{q}^{T}).$

By Proposition 2.11 in [16], we have

(2.12) $Im(\widetilde{C}_{q}^{T})\cap Ker(D)\not=\\{0\\}.$

Let $E\in\hbox{Im}(\widetilde{C}_{q}^{T})$ be a unit vector such that $DE=0.$
Applying $E$ to system (1.2) associated with the initial data

(2.13) $t=0:\quad U=\theta E,\quad U^{\prime}=0$

with $\theta\in V$, and setting $u=(\\!(E,U)\\!)$, we get

(2.14) $\begin{cases}u^{\prime\prime}+Lu=-(\\!(E,AU)\\!),\\\ t=0:\quad
u=\theta,\ u^{\prime}=0,\end{cases}$

here and hereafter $(\\!(\cdot,\cdot)\\!)$ denotes the inner product with the
associated norm $\|\cdot\|$ in the euclidian space $\mathbb{R}^{N}$.

Now, we define the linear mapping

(2.15) $\mathcal{R}:\quad\theta\rightarrow(\\!(E,AU)\\!).$

Since the matrices $A$ and $D$ are symmetric and semi-positive definite, by
the dissipation of system (1.2) with the initial data (2.13), we have

(2.16)
$\|\mathcal{R}\theta\|_{L^{2}(0,T;V)}+\|\mathcal{R}\theta\|_{H^{1}(0,T;H)}\leqslant
c_{T}\|\theta\|_{V},$

where $c_{T}$ is a positive constant depending only on $T$.

Noting that the imbedding from $L^{2}(0,T;V)\cap H^{1}(0,T;H)$ into
$L^{2}(0,T;H)$ is compact (see Theorem 5.1 in [19]), the mapping $\mathcal{R}$
is compact from $V$ into $L^{2}(0,T;H).$

On the other hand, noting $E=\widetilde{C}_{q}^{T}x$, we have

(2.17) $u=(\\!(E,U)\\!)=(\\!(x,\widetilde{C}_{q}U)\\!).$

Then, it follows from (2.9) that

(2.18) $\|(u(t),u^{\prime}(t))\|_{V\times H}\leqslant
c\|\widetilde{C}_{q}(U(t),U^{\prime}(t))\|_{V\times H}\leqslant cMe^{-\omega
t}\|\theta\|_{V},\quad t\geqslant 0$

for all $\theta\in V$. This contradicts Proposition 2.3. ∎

In particular, taking $\widetilde{C}_{q}=I$ in Theorem 2.4, we get immediately

######  Corollary 2.5.

If system (1.2) is uniformly stable, then we have rank$(D)=N.$

Conversely, we have

######  Theorem 2.6.

Assume that the scalar equation

(2.19) $u^{\prime\prime}+Lu+gu^{\prime}=0$

is uniformly stable in the space $V\times H$. If rank$(D)=N$, then system
(1.2) is uniformly stable in the space $(V\times H)^{N}$.

###### Proof.

Following the classical theory (see [26, 27]), the uniform stability of a
semi-group is robust by compact perturbation. This property was served in [8,
22] for obtaining the uniform stability.

More precisely, since the mapping $U\rightarrow AU$ is compact from $V$ into
$H$, the asymptotic stability of the coupled system (1.2) and the uniform
stability of the following decoupled system

(2.20) $U^{\prime\prime}+\mathcal{L}U+D\mathcal{G}U^{\prime}=0$

yield the uniform stability of the coupled system (1.2).

Since rank$(D)=N$, system (2.20) can be decomposed into $N$ scalar equations
of the same type as those in (2.19), therefore, it is uniformly stable.

On the other hand, by Proposition 2.1, the resolvent of system (1.2) is
compact. Then by the classical theory of semigroups, the asymptotic stability
of system (1.2) is equivalent to the uniqueness of the following over-
determined system:

(2.21)
$(\mathcal{L}+A)\Phi=\beta^{2}\Phi\quad\hbox{and}\quad\mathcal{G}\Phi=0.$

Let $AE=\lambda E$ with $\lambda\geqslant 0$. Then, setting $\phi=(E,\Phi)$,
it follows that

(2.22) $L\phi=(\beta^{2}-\lambda)\phi\quad\hbox{and}\quad g\phi=0.$

By the definition of the dual mapping, we have

$\langle L\phi,\phi\rangle_{V^{\prime},V}=(L\phi,\phi)_{H}=\|\phi\|^{2}_{V}.$

It follows that $\beta^{2}-\lambda>0$. Then, we check easily that

(2.23) $U=e^{it\sqrt{\beta^{2}-\lambda}}\phi E$

satisfies system (2.20), which is uniformly stable. We get
$(\\!(E,\Phi)\\!)=\phi=0$ for each eigenvector $E$ of $A$, then $\Phi=0$. ∎

######  Remark 2.7.

Roughly speaking, Theorem 2.6 indicates that the uniform stability of system
(1.2) can be obtained by means of the scalar equation (2.19). It provides thus
a direct and efficient approach to solve a seemingly difficult problem of
uniform stability of a complex system.

##  3\. Uniform synchronization by $p$-groups

By Corollary 2.5, when rank$(D)<N$, system (1.2) is not uniformly stable.
Instead of the stability, we turn to consider its synchronization by
$p$-groups.

Let $S_{r}$ be the full row-rank matrix of order
$(n_{r}-n_{r-1}-1)\times(n_{r}-n_{r-1})$:

(3.1) $S_{r}=\begin{pmatrix}1&-1\\\ &1&-1&\cr&&\ddots&\ddots\\\
&&&1&-1\end{pmatrix},\quad 1\leqslant r\leqslant p.$

Define the $(N-p)\times N$ matrix $C_{p}$ of synchronization by $p$-groups as

(3.2)
$C_{p}=\begin{pmatrix}S_{1}\cr&S_{2}\cr&&\ddots\cr&&&S_{p}\end{pmatrix}.$

The uniform synchronization by $p$-groups (1.5) can be equivalently rewritten
by (1.6), which is easy to be analyzed.

Let $\epsilon_{1},\cdots,\epsilon_{N}$ be the vectors of the canonical basis
of $\mathbb{R}^{N}$. Defining

(3.3) $e_{r}=\sum_{i=n_{r-1}+1}^{n_{r}}\epsilon_{i},\quad 1\leqslant
r\leqslant p,$

we have

(3.4) $Ker(C_{p})=Span\\{e_{1},\cdots,e_{p}\\}.$

######  Proposition 3.1.

(see Proposition 4.2 in [18]) The matrix $A$ satisfies the condition of
$C_{p}$-compatibility:

(3.5) $AKer(C_{p})\subseteq Ker(C_{p})$

if and only if there exists a symmetric and semi-positive definite matrix
$\overline{A}_{p}$ of order $(N-p)$, such that (1.7) holds.

######  Proposition 3.2.

(see Proposition 4.4 in [18]) The matrix $D$ satisfies the condition of strong
$C_{p}$-compatibility:

(3.6) $Ker(C_{p})\subseteq Ker(D)$

if and only if there exists a symmetric and semi-positive definite matrix $R$
of order $(N-p)$, such that

(3.7) $D=C_{p}^{T}RC_{p}.$

Moreover, setting

(3.8) $\overline{D}_{p}=(C_{p}C_{p}^{T})^{1/2}R(C_{p}C_{p}^{T})^{1/2},$

we have

(3.9)
$(C_{p}C_{p}^{T})^{-1/2}C_{p}D=\overline{D}_{p}(C_{p}C_{p}^{T})^{-1/2}C_{p}.$

######  Remark 3.3.

By the expression (3.3), it is easy to check that the condition of
$C_{p}$-compatibility (3.5) is equivalent to the row-sum condition by blocks

(3.10) $\displaystyle\sum_{j=n_{s-1}+1}^{n_{s}}a_{ij}=\alpha_{rs},\quad
n_{r-1}+1\leqslant i\leqslant n_{r},\quad 1\leqslant r,s\leqslant p,$

where $\alpha_{rs}$ are some constants. In particular, when $p=1$, $A$
satisfies the row-sum condition:

(3.11) $\displaystyle\sum_{p=1}^{N}a_{kp}=\alpha,\quad k=1,\cdots,N.$

The condition of strong $C_{p}$-compatibility (3.6) is equivalen to

(3.12) $DKer(C_{p})=\\{0\\}.$

That means that $D=(d_{ij})$ satisfies the null row-sum condition by blocks

(3.13) $\displaystyle\sum_{j=n_{s-1}+1}^{n_{s}}d_{ij}=0,\quad
n_{r-1}+1\leqslant i\leqslant n_{r},\quad 1\leqslant r,s\leqslant p.$

Now applying $C_{p}$ to system (1.2), and setting $W=C_{p}U$, we get a self-
closed reduced system

(3.14)
$W^{\prime\prime}+\mathcal{L}W+\overline{A}_{p}W+\overline{D}_{p}GW^{\prime}=0.$

Moreover, it is easy to check the following basic result.

######  Proposition 3.4.

Assume that the matrices $A$ and $D$ satisfy the condition of
$C_{p}$-compatibility (3.5) and the condition of strong $C_{p}$-compatibility
(3.6), respectively. The uniform synchronization by $p$-groups of system (1.2)
in the space $(V\times H)^{N}$ is equivalent to the uniform stability of the
reduced system (3.14) in the space $(V\times H)^{N-p}$.

Since the reduced matrices $\overline{A}_{p}$ and $\overline{D}_{p}$ are still
symmetric and semi-positive definite, the uniform stability of the reduced
system (3.14) can be treated by Theorem 2.6. So, the uniform synchronization
by $p$-groups is reduced to the uniform stability. However, the necessity of
the condition of $C_{p}$-compatibility for $A$ and that of the condition of
strong $C_{p}$-compatibility for $D$ are intrinsically linked with the rank of
the matrix $D$.

######  Proposition 3.5.

If system (1.2) is uniformly synchronizable by $p$-groups, then we necessarily
have

(3.15) $\hbox{rank}(C_{p}D)\geqslant N-p.$

###### Proof.

It is sufficient to take $\widetilde{C}_{q}=C_{p}$ in Theorem 2.4. ∎

######  Proposition 3.6.

The following rank condition

(3.16) $\hbox{rank}(D)=\hbox{rank}(C_{p}D)=N-p$

holds, if and only if $Ker(D)$ and $Ker(C_{p})$ are bi-orthonormal.

###### Proof.

By Proposition 2.11 in [16], the rank condition (3.16) is equivalent to

(3.17) $Ker(D)\cap Im(C_{p}^{T})=Ker(C_{p})\cap Im(D)=\\{0\\},$

namely,

(3.18)
$Ker(D)\cap\\{Ker(C_{p})\\}^{\perp}=Ker(C_{p})\cap\\{Ker(D^{T})\\}^{\perp}=\\{0\\}.$

Hence by Proposition 2.5 in [16], Ker$(D)$ and Ker$(C_{p})$ are bi-orthogonal.
∎

######  Theorem 3.7.

Assume that system (1.2) is uniformly synchronizable by $p$-groups under the
minimal rank conditions (3.16). Then $A$ satisfies the condition of
$C_{p}$-compatibility (3.5) and $D$ satisfies the condition of strong
$C_{p}$-compatibility (3.6).

###### Proof.

Let $U$ be the solution to system (1.2) with the following initial data:

(3.19) $U_{0}=\sum_{r=1}^{p}u_{0r}e_{r},\quad
U_{1}=\sum_{r=1}^{p}u_{1r}e_{r},$

where $u_{0r}\in V$ and $u_{1r}\in H$ for $r=1,\cdots,p$. Then by (1.6) we
have

(3.20) $t\geqslant 0:\quad\|C_{p}(U(t),U^{\prime}(t))\|_{(V\times
H)^{N-p}}\leqslant Me^{-\omega t}\|C_{p}(U_{0},U_{1})\|_{(V\times
H)^{N-p}}=0.$

There exist some functions $u_{1},\cdots,u_{p}$ in
$C^{0}(\mathbb{R}^{+},V)\cap C^{1}(\mathbb{R}^{+},H)$, such that

(3.21) $U=\sum_{s=1}^{p}u_{s}e_{s}.$

Then

(3.22)
$\sum_{s=1}^{p}u_{s}^{\prime\prime}e_{s}+\sum_{s=1}^{p}Lu_{s}e_{s}+\sum_{s=1}^{p}gu_{s}^{\prime}De_{s}+\sum_{s=1}^{p}u_{s}Ae_{s}=0.$

Applying $C_{p}$ to both sides of the above system, it follows that

(3.23)
$\sum_{s=1}^{p}gu_{s}^{\prime}C_{p}De_{s}+\sum_{s=1}^{p}u_{s}C_{p}Ae_{s}=0.$

In particular, by the continuity at $t=0$, we have

(3.24) $\sum_{s=1}^{p}gu_{1s}C_{p}De_{s}+\sum_{s=1}^{p}u_{0s}C_{p}Ae_{s}=0,$

then

(3.25) $C_{p}Ae_{s}=0,\quad C_{p}De_{s}=0,\quad s=1,\cdots,p.$

Thus $A$ satisfies the condition of $C_{p}$-compatibility (3.5), and $D$
satisfies a similar condition of $C_{p}$-compatibility as in (3.5).

We next show that $D$ satisfies the condition of strong $C_{p}$-compatibility
(3.6). In fact, for $s=1,\cdots,p$, we have

(3.26) $(\\!(De_{s},d)\\!)=(\\!(e_{s},Dd)\\!)=0,\quad d\in Ker(D),$

then $De_{s}\in\hbox{Ker}(D)^{\perp}\cap\hbox{Ker}(C_{p})$. By Proposition
3.6, Ker$(D)$ is bi-orthogonal to Ker$(C_{p})$, so
$\hbox{Ker}(D)^{\perp}\cap\hbox{Ker}(C_{p})=\\{0\\}$. Then

(3.27) $De_{s}=0,\quad s=1,\cdots,p.$

We get thus the condition of strong $C_{p}$-compatibility (3.6) for the matrix
$D$. ∎

######  Theorem 3.8.

Assume that $A$ satisfies the condition of $C_{p}$-compatibility (3.5) and $D$
satisfies the condition of strong $C_{p}$-compatibility (3.7) with
rank$(R)=N-p.$ Assume furthermore that the scalar equation (2.19) is uniformly
stable in the space $V\times H$. Then system (1.2) is uniformly synchronizable
by $p$-groups in $(V\times H)^{N}$.

###### Proof.

By Proposition 3.4, it is sufficient to show the uniform stability of the
reduced system (3.14). By (3.8),
$\hbox{rank}(\overline{D}_{p})=\hbox{rank}(R)=N-p$. Then by Theorem 2.6, the
reduced system (3.14) is uniformly stable. ∎

######  Theorem 3.9.

Assume that system (1.2) is uniformly synchronizable by $p$-groups in
$(V\times H)^{N}$, then there exist some functions $u_{1},\cdots,u_{p}$ in
$C^{0}(\mathbb{R}^{+},V)\cap C^{1}(\mathbb{R}^{+},H)$ and some positive
constants $M\geqslant 1$ and $\omega>0$, such that setting

(3.28) $u=\sum_{r=1}^{p}u_{r}e_{r}/\|e_{r}\|,$

we have for all $t\geqslant 0$,

(3.29) $\displaystyle\|(U(t)-u(t),U^{\prime}(t)-u^{\prime}(t))\|_{(V\times
H)^{N}}\leqslant Me^{-\omega t}\|C_{p}(U_{0},U_{1})\|_{(V\times H)^{N-p}}.$

Assume furthermore that $A$ satisfies the condition of $C_{p}$-compatibility
(3.5) and $D$ satisfies the condition of strong $C_{p}$-compatibility (3.6).
Then $u$ obeys a conservative system.

###### Proof.

Let $U$ be the solution to system (1.2) with any given initial data
$(U_{0},U_{1})\in(V\times H)^{N}$. For $r=1,\cdots,p$, let
$u_{r}=(\\!(U,e_{r})\\!)/\|e_{r}\|.$ Noting that
$\mathbb{R}^{N}=\hbox{Ker}(C_{p})\oplus\hbox{Im}(C_{p}^{T})$, we have

(3.30)
$U=\sum_{r=1}^{p}u_{r}e_{r}/\|e_{r}\|+C_{p}^{T}(C_{p}C_{p}^{T})^{-1}C_{p}U=u+C_{p}^{T}(C_{p}C_{p}^{T})^{-1}C_{p}U.$

By (1.6), we get

(3.31) $\displaystyle\|(U(t)-u(t),U^{\prime}(t)-u^{\prime}(t))\|_{(V\times
H)^{N}}$ $\displaystyle\leqslant$
$\displaystyle\|C_{p}^{T}(C_{p}C_{p}^{T})^{-1}\|\|C_{p}(U(t),U^{\prime}(t))\|_{(V\times
H)^{N-p}}$ $\displaystyle\leqslant$ $\displaystyle M^{\prime}e^{-\omega
t}\|C_{p}(U_{0},U_{1})\|_{(V\times H)^{N-p}},\quad t\geqslant 0$

for some constant $M^{\prime}\geqslant 1.$

Now we will precisely show the dynamics of the functions $u_{1},\cdots,u_{p}$.
First, recall that the condition of $C_{p}$-compatibility (3.5) implies

(3.32) $Ae_{r}=\sum_{s=1}^{p}\beta_{rs}{\|e_{r}\|\over\|e_{s}\|}e_{s},\quad
r=1,\cdots,p.$

Moreover, since $A$ is symmetric, a straightforward computation shows that

(3.33)
$(Ae_{r},e_{s})=\sum_{q=1}^{p}\beta_{rq}{\|e_{r}\|\over\|e_{q}\|}(\\!(e_{q},e_{s})\\!)=\beta_{rs}\|e_{r}\|\|e_{s}\|$

and

(3.34)
$(\\!(e_{r},Ae_{s})\\!)=\sum_{q=1}^{p}\beta_{sq}{\|e_{s}\|\over\|e_{q}\|}(\\!(e_{r},e_{q})\\!)=\beta_{sr}\|e_{s}\|\|e_{r}\|.$

It follows that

$\beta_{rs}=\beta_{sr},\quad 1\leqslant r,s\leqslant p.$

On the other hand, the condition of strong $C_{p}$-compatibility (3.6) implies

(3.35) $De_{r}=0,\quad r=1,\cdots,p.$

Then, applying $e_{r}$ to system (1.2), we get the following conservative
system

(3.36)
$\begin{cases}u_{r}^{\prime\prime}+Lu_{r}+\sum_{s=1}^{p}\beta_{rs}u_{s}=0,\\\
t=0:\quad u_{r}=(\\!(U_{0},e_{r})\\!)/\|e_{r}\|,\quad
u_{r}^{\prime}=(\\!(U_{1},e_{r})\\!)/\|e_{r}\|\end{cases}$

for $r=1,\cdots,p$. ∎

######  Remark 3.10.

Classically, the convergence (1.5) or equivalently (1.6) is called uniform
synchronization by $p$-groups in the consensus sense, while the convergence
(1.5) is in the pinning sense. Moreover, the $p$-tuple
$u=(u_{1},\cdots,u_{p})$ is called the uniformly synchronizable state by
$p$-groups. Theorem 3.9 indicates that two notions are simply the same.

Moreover, setting the matrix $B=(\beta_{rs})$, we define the energy by

$E(t)=\|u(t)\|_{V^{p}}^{2}+(Bu(t),u(t))_{H^{p}}+\|u^{\prime}(t)\|^{2}_{H^{p}}.$

Since $B$ is symmetric, we have

(3.37) $E(t)=E(0),\quad t\geqslant 0.$

Then the orbit of $u$ is lacalized on the sphere (3.37) which is uniquely
determined by the projection of the initial data $(U_{0},U_{1})$ to
Ker$(C_{p})$.

######  Remark 3.11.

The condition of strong $C_{p}$-compatibility (3.6) implies that (see
Proposition 2.13 in [16])

$\hbox{rank}(D,AD,\cdots,A^{N-1}D)=N-p.$

Following Theorem 4.7 in [18], there does not exist an extended matrix
$\widetilde{C}_{q}\ (q<p)$, such that

$\widetilde{C}_{q}(U(t),U^{\prime}(t))\rightarrow(0,0)\quad\hbox{in }(V\times
H)^{N}\hbox{ as }t\rightarrow+\infty.$

Unlike in the case of the approximate boundary synchronization by $p$-groups,
there is no possibility to get any induced synchronization in the present
situation (see Chapter 11 in [16]).

##  4\. Applications

###  4.1. Wave equations with boundary feedback

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with a smooth boundary
$\Gamma=\Gamma_{1}\cup\Gamma_{0}$ such that
$\overline{\Gamma}_{1}\cap\overline{\Gamma}_{0}=\emptyset$ and
mes$(\Gamma_{1})>0$. For fixing idea, we assume that mes$(\Gamma_{0})>0$.

Consider the following wave equation

(4.1) $\left\\{\begin{array}[]{ll}u^{\prime\prime}-\Delta u=0&\mbox{in
}\mathbb{R}^{+}\times\Omega,\\\ u=0&\mbox{on
}\mathbb{R}^{+}\times\Gamma_{0},\\\ \partial_{\nu}u+u^{\prime}=0&\mbox{on
}\mathbb{R}^{+}\times\Gamma_{1},\end{array}\right.$

where $\partial_{\nu}$ denotes the outward normal derivative on the boundary.
The uniform stability of (4.1) was abundantly studied by different approaches
in the literature, we only quote [3, 10, 11] and the references therein.

Now, let $A$ and $D$ be symmetric and semi-positive definite matrices of order
$N$. We consider the following system of wave equations:

(4.2) $\left\\{\begin{array}[]{ll}U^{\prime\prime}-\Delta U+AU=0&\mbox{in
}\mathbb{R}^{+}\times\Omega,\\\ U=0&\mbox{on
}\mathbb{R}^{+}\times\Gamma_{0},\\\ \partial_{\nu}U+DU^{\prime}=0&\mbox{on
}\mathbb{R}^{+}\times\Gamma_{1}.\end{array}\right.$

Let $H^{1}_{\Gamma_{0}}(\Omega)$ denote the subspace of $H^{1}(\Omega)$,
composed of functions with vanishing trace on $\Gamma_{0}$. Multiplying (4.2)
by $\Phi\in H^{1}_{\Gamma_{0}}(\Omega)$ and integrating by parts, we get the
following variational formulation:

(4.3) $\int_{\Omega}(\\!(U^{\prime\prime},\Phi)\\!)dx+\int_{\Omega}(\\!(\nabla
U,\nabla\Phi)\\!)dx+\int_{\Omega}(\\!(AU,\Phi)\\!)dx+\int_{\Gamma_{1}}(\\!(DU^{\prime},\Phi)\\!)d\Gamma=0.$

Define

(4.4) $\langle Lu,\phi\rangle=\int_{\Omega}\nabla u\cdot\nabla\phi
dx,\quad\langle gv,\phi\rangle=\int_{\Gamma_{1}}v\phi d\Gamma.$

Then (4.3) can be rewritten as

(4.5) $U^{\prime\prime}+\mathcal{L}U+AU+D\mathcal{G}U^{\prime}=0.$

Moreover, since the scalar equation (4.1) is uniformly stable in
$H^{1}_{\Gamma_{0}}(\Omega)\times L^{2}(\Omega)$, applying Theorem 3.8 and
Theorem 3.9, we immediately obtain the following

######  Theorem 4.1.

Assume that $A$ satisfies the condition of $C_{p}$-compatibility (3.5) and $D$
the condition of strong $C_{p}$-compatibility (3.7) with rank$(R)=N-p.$ Then
the system of wave equations (4.2) is uniformly synchronizable by $p$-groups
in the space $(H^{1}_{\Gamma_{0}}(\Omega)\times L^{2}(\Omega))^{N}$.

Moreover, for any given initial data
$(U_{0},U_{1})\in(H^{1}_{\Gamma_{0}}(\Omega)\times L^{2}(\Omega))^{N}$,
consider the problem

(4.6) $\left\\{\begin{array}[]{ll}u_{r}^{\prime\prime}-\Delta
u_{r}+\sum_{s=1}^{p}\beta_{rs}u_{s}=0&\mbox{in }\mathbb{R}^{+}\times\Omega,\\\
u_{r}=0&\mbox{on }\mathbb{R}^{+}\times\Gamma_{0},\\\
\partial_{\nu}u_{r}=0&\mbox{on }\mathbb{R}^{+}\times\Gamma_{1},\\\ t=0:\quad
u_{r}=(\\!(U_{0},e_{r})\\!)/\|e_{r}\|,\quad
u_{r}^{\prime}=(\\!(U_{0},e_{r})\\!)/\|e_{r}\|&\mbox{in
}\Omega\end{array}\right.$

for $r=1,\cdots,p$, and the coefficients $\beta_{rs}$ are given by (3.32).
Then, setting $u=\sum_{r=1}^{p}u_{r}e_{r}/\|e_{r}\|$, the corresponding
solution $U$ to system (4.2) satisfies

(4.7)
$\displaystyle\|(U(t)-u(t),U^{\prime}(t)-u^{\prime}(t))\|_{(H^{1}_{\Gamma_{0}}(\Omega)\times
L^{2}(\Omega))^{N}}$ $\displaystyle\leqslant$ $\displaystyle Me^{-\omega
t}\|C_{p}(U_{0},U_{1})\|_{(H^{1}_{\Gamma_{0}}(\Omega)\times
L^{2}(\Omega))^{N-p}},\quad t\geqslant 0.$

###  4.2. Wave equations with locally distributed feedback

Let $\Omega\subset\mathbb{R}^{n}$ denote a bounded domain with smooth boundary
$\Gamma$. Let $\omega\subset\Omega$ denote the damped domain.

Let $a$ be a smooth function such that

(4.8) $a(x)\geqslant 0,\quad\forall x\in\Omega\quad\hbox{and}\quad
a(x)\geqslant a_{0}>0,\quad\forall x\in\omega.$

Consider the uniform stability of the following locally damped scalar wave
system

(4.9) $\left\\{\begin{array}[]{ll}u^{\prime\prime}-\Delta
u+au^{\prime}=0&\mbox{in }\mathbb{R}^{+}\times\Omega,\\\ u=0&\mbox{on
}\mathbb{R}^{+}\times\Gamma.\end{array}\right.$

This is a very challenge and promising issue. There is a large amount of
literatures that we will comment briefly. The uniform decay was first
established by multipliers in [6] as $\omega$ is a neighbourhood of the
boundary. Later, the result was generalized in [30] to semi-linear case. When
$\Omega$ is a compact Riemann manifold without boundary and $\omega$ satisfies
the geometric optic condition, the uniform stability was established by a
micro-local approach in [24].

Now, consider the following system of locally damped wave equations:

(4.10) $\left\\{\begin{array}[]{ll}U^{\prime\prime}-\Delta
U+AU+aDU^{\prime}=0&\mbox{in }\mathbb{R}^{+}\times\Omega,\\\ U=0&\mbox{on
}\mathbb{R}^{+}\times\Gamma,\end{array}\right.$

where $A$ and $D$ are symmetric and semi-positive definite matrices with
constant elements. Multiplying system (4.10) by $\Phi\in H^{1}_{0}(\Omega)$
and integrating by parts, we get the following variational formulation:

(4.11)
$\int_{\Omega}(\\!(U^{\prime\prime},\Phi)\\!)dx+\int_{\Omega}(\\!(\nabla
U,\nabla\Phi)\\!)dx+\int_{\Omega}(\\!(AU,\Phi)\\!)dx+\int_{\Omega}a(\\!(DU^{\prime},\Phi)\\!)d\Gamma=0.$

Let $L$ and $g$ be the linear continuous mappings from $H^{1}_{0}(\Omega)$
into $H^{-1}(\Omega)$, defined by

(4.12) $\langle Lu,\phi\rangle=\int_{\Omega}\nabla u\cdot\nabla\phi
dx\quad\hbox{and}\quad\langle gv,\phi\rangle=\int_{\Omega}av\phi dx,$

respectively. Then the variational problem (4.11) can be rewritten as

(4.13) $U^{\prime\prime}+\mathcal{L}U+AU+D\mathcal{G}U^{\prime}=0.$

Then, applying Theorem 3.8 and Theorem 3.9, we have

######  Theorem 4.2.

Assume that the damped domain $\omega\subset\Omega$ contains a neighbourhood
of the whole boundary $\Gamma$. Assume furthermore that $A$ satisfies the
condition of $C_{p}$-compatibility (3.5) and $D$ the condition of strong
$C_{p}$-compatibility (3.7) with rank$(R)=N-p.$ Then system (4.13) is
uniformly synchronizable by $p$-groups in the space $(H^{1}_{0}(\Omega)\times
L^{2}(\Omega))^{N}$.

Moreover, for any given initial data $(U_{0},U_{1})\in(H^{1}_{0}(\Omega)\times
L^{2}(\Omega))^{N}$, consider the problem

(4.14) $\left\\{\begin{array}[]{ll}u_{r}^{\prime\prime}-\Delta
u_{r}+\sum_{s=1}^{p}\beta_{rs}u_{s}=0&\mbox{in }\mathbb{R}^{+}\times\Omega,\\\
u_{r}=0&\mbox{on }\mathbb{R}^{+}\times\Gamma,\\\ t=0:\quad
u_{r}=(\\!(U_{0},e_{r})\\!)/\|e_{r}\|,\quad
u_{r}^{\prime}=(\\!(U_{0},e_{r})\\!)/\|e_{r}\|&\mbox{in
}\Omega\end{array}\right.$

for $r=1,\cdots,p$, and the coefficients $\beta_{rs}$ are given by (3.32).
Then, setting $u=\sum_{r=1}^{p}u_{r}e_{r}/\|e_{r}\|$, the corresponding
solution $U$ to system (4.13) satisfies

(4.15)
$\displaystyle\|(U(t)-u(t),U^{\prime}(t)-u^{\prime}(t))\|_{(H^{1}(\Omega)\times
L^{2}(\Omega))^{N}}$ $\displaystyle\leqslant$ $\displaystyle Me^{-\omega
t}\|C_{p}(U_{0},U_{1})\|_{(H^{1}(\Omega)\times L^{2}(\Omega))^{N-p}},\quad
t\geqslant 0.$

###  4.3. Kirchhoff plate equations with locally distributed feedback

In this sub-section $\Omega$ is a bounded domain in $\mathbb{R}^{2}$, occupied
by an elastic thin plate. We refer to [9] for the stabilization of linear
models.

Let $a$ be a smooth and non-negative function such that (4.8) holds. Assume
that $\omega$ contains a neighbourhood of the whole boundary $\Gamma$. Then,
the following system of plate equation

(4.16)
$\left\\{\begin{array}[]{ll}u^{\prime\prime}+\Delta^{2}u+au^{\prime}=0&\mbox{in
}\mathbb{R}^{+}\times\Omega,\\\ u=\partial_{\nu}u=0&\mbox{on
}\mathbb{R}^{+}\times\Gamma\end{array}\right.$

is uniformly stable in $H^{2}_{0}(\Omega)\times L^{2}(\Omega)$ (see [9] for
details).

Consider the following system :

(4.17)
$\left\\{\begin{array}[]{ll}U^{\prime\prime}+\Delta^{2}U+AU+aDU^{\prime}=0&\mbox{in
}\mathbb{R}^{+}\times\Omega,\\\ U=\partial_{\nu}U=0&\mbox{on
}\mathbb{R}^{+}\times\Gamma,\end{array}\right.$

where $A$ and $D$ are symmetric and semi-positive definite matrices with
constant elements. Multiplying system (4.17) by $\Phi\in H^{2}_{0}(\Omega)$
and integrating by parts, we get the following variational formulation:

(4.18)
$\int_{\Omega}(\\!(U^{\prime\prime},\Phi)\\!)dx+\int_{\Omega}(\\!(\Delta
U,\Delta\Phi)\\!)dx+\int_{\Omega}(\\!(AU,\Phi)\\!)dx+\int_{\Omega}a(\\!(DU^{\prime},\Phi)\\!)dx=0.$

Let $L$ and $g$ be defined by

(4.19) $\langle Lu,\phi\rangle=\int_{\Omega}\Delta u\Delta\phi
dx\quad\hbox{and}\quad\langle gv,\phi\rangle=\int_{\Omega}av\phi dx,$

respectively. (4.18) can be interpreted as

(4.20) $U^{\prime\prime}+\mathcal{L}U+AU+D\mathcal{G}U^{\prime}=0.$

Then, applying Theorem 3.8 and Theorem 3.9, we have

######  Theorem 4.3.

Assume that $A$ satisfies the condition of $C_{p}$-compatibility (3.5) and $D$
the condition of strong $C_{p}$-compatibility (3.7) with rank$(R)=N-p.$ Then
system (4.17) is uniformly synchronizable by $p$-groups in
$(H^{2}_{0}(\Omega)\times L^{2}(\Omega))^{N}$.

Moreover, for any given initial data $(U_{0},U_{1})\in(H^{2}_{0}(\Omega)\times
L^{2}(\Omega))^{N}$, consider the problem

(4.21)
$\left\\{\begin{array}[]{ll}u_{r}^{\prime\prime}+\Delta^{2}u_{r}+\sum_{s=1}^{p}\beta_{rs}u_{s}=0&\mbox{in
}\mathbb{R}^{+}\times\Omega,\\\ u_{r}=\partial_{\nu}u_{r}=0&\mbox{on
}\mathbb{R}^{+}\times\Gamma,\\\ t=0:\quad
u_{r}=(\\!(U_{0},e_{r})\\!)/\|e_{r}\|,\quad
u_{r}^{\prime}=(\\!(U_{0},e_{r})\\!)/\|e_{r}\|&\mbox{in
}\Omega\end{array}\right.$

for $r=1,\cdots,p$, and the coefficients $\beta_{rs}$ are given by (3.32).
Then, setting $u=\sum_{r=1}^{p}u_{r}e_{r}/\|e_{r}\|$, the corresponding
solution $U$ to system (4.17) satisfies

(4.22)
$\displaystyle\|(U(t)-u(t),U^{\prime}(t)-u^{\prime}(t))\|_{(H^{2}(\Omega)\times
L^{2}(\Omega))^{N}}$ $\displaystyle\leqslant$ $\displaystyle Me^{-\omega
t}\|C_{p}(U_{0},U_{1})\|_{(H^{2}(\Omega)\times L^{2}(\Omega))^{N-p}},\quad
t\geqslant 0.$

######  Remark 4.4.

The above examples are classic and illustrate the applications of the abstract
theory. In fact, Theorems 3.8 and 3.9 are also applicable for many other
models, such as system of wave equations with viscoelastic (Kelvin-Voigt)
damping, system of Kirchhoff plate equations with boundary shear force and
bending moment damping etc.

##  5\. Perspective comments.

Up to now, we have started the work on a simplified model with only one
damping. Many related problems can be considered later.

(i) By the definition of uniform synchronization by $p$-groups:

(5.1) $\|C_{p}(U(t),U^{\prime}(t))\|_{(V\times H)^{N-p}}\leqslant Me^{-\omega
t}\|C_{p}(U_{0},U_{1})\|_{(V\times H)^{N-p}},\quad t\geqslant 0,$

if $C_{p}(U_{0},U_{1})=(0,0)$, then

(5.2) $C_{p}U(t)\equiv 0,\quad t\geqslant 0.$

Thus, for any given synchronized initial data, the solution is always
synchronized. This simplifies much the study on the necessity of the
conditions of $C_{p}$-compatibility given in Theorem 3.7.

A more natural definition of uniform synchronization by $p$-groups should be
given by

(5.3) $\|C_{p}(U(t),U^{\prime}(t))\|_{(V\times H)^{N-p}}\leqslant Me^{-\omega
t}\|(U_{0},U_{1})\|_{(V\times H)^{N}},\quad t\geqslant 0.$

In this case, the solution is not automatically synchronized even for the
synchronized initial data. The situation will be chaotic and presents
certainly many interesting questions.

(ii) Instead of the uniform decay rate given by (5.1), we can consider the
polynomial decay rate as

(5.4) $\|C_{p}(U(t),U^{\prime}(t))\|_{(V\times
H)^{N-p}}=O((1+t)^{-\delta}),\quad t\geqslant 0,$

with a positive power $\delta$. We refer to [5, 23] and the references therein
for the recent progress on the polynomial stability of indirectly damped wave
equations.

(iii) We may consider a system with several damping of different types:

(5.5)
$U^{\prime\prime}+\mathcal{L}U+AU+D_{1}\mathcal{G}_{1}U^{\prime}+D_{2}\mathcal{G}_{2}U^{\prime}=0,$

where $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ can be internal and boundary
damping for wave equations, and bending moment and shear force damping for
plate equations, respectively. Many related questions can be asked, for
example:

(a) Let $D=(D_{1},D_{2})$ be the composite damping matrix. Is Kalman’s rank
condition on $(A,D)$ still sufficient for the asymptotic stability as what has
been done in [18]?

(b) Is the condition rank$(D)=N$ still sufficient for the uniform stability as
we have done in the present work?

The main difficulty comes from the interaction of the numerous matrices
$A,D_{1},D_{2}$, somewhat like for coupled Robin problem in [12]. The key idea
is to separate them as the coupling terms are compact, so more regularity
seems to be necessary.

We do not have any answer yet for each question, but the first attempt already
shows some interesting results for developing the research in these
directions.

Acknowledgement  This work is supported by National Natural Sciences
Foundation of China under Grant 11831011.

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