# Revisiting Video Saliency: A Large-scale Benchmark and a New Model

Wenguan Wang 1 Jianbing Shen 1 Corresponding author: Jianbing Shen. This work
was supported in part by the Beijing Natural Science Foundation under Grant
4182056, the National Basic Research Program of China under Grant
2013CB328805, the Fok Ying Tung Education Foundation under Grant 141067, and
the Specialized Fund for Joint Building Program of Beijing Municipal Education
Commission.  Fang Guo 1 Ming-Ming Cheng 2 Ali Borji 3
1Beijing Lab of Intelligent Information Technology School of Computer Science
Beijing Institute of Technology  China
2CCCE Nankai University  China 3Department of Computer Science University of
Central Florida  USA
<EMAIL_ADDRESS>{shenjianbing<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
https://github.com/wenguanwang/DHF1K

###### Abstract

In this work, we contribute to video saliency research in two ways. First, we
introduce a new benchmark for predicting human eye movements during dynamic
scene free-viewing, which is long-time urged in this field. Our dataset, named
DHF1K (Dynamic Human Fixation), consists of 1K high-quality, elaborately
selected video sequences spanning a large range of scenes, motions, object
types and background complexity. Existing video saliency datasets lack variety
and generality of common dynamic scenes and fall short in covering challenging
situations in unconstrained environments. In contrast, DHF1K makes a
significant leap in terms of scalability, diversity and difficulty, and is
expected to boost video saliency modeling. Second, we propose a novel video
saliency model that augments the CNN-LSTM network architecture with an
attention mechanism to enable fast, end-to-end saliency learning. The
attention mechanism explicitly encodes static saliency information, thus
allowing LSTM to focus on learning more flexible temporal saliency
representation across successive frames. Such a design fully leverages
existing large-scale static fixation datasets, avoids overfitting, and
significantly improves training efficiency and testing performance. We
thoroughly examine the performance of our model, with respect to state-of-the-
art saliency models, on three large-scale datasets (i.e., DHF1K, Hollywood2,
UCF sports). Experimental results over more than 1.2K testing videos
containing 400K frames demonstrate that our model outperforms other
competitors.

##  1 Introduction

Human visual system (HVS) has an astonishing ability to quickly select
visually important regions in its visual field. This cognitive process enables
humans to easily interpret complex scenes in real time. Over the last few
decades, several computational models have been proposed for imitating
attentional mechanisms of HVS during static scene viewing. Significant
advances have been achieved recently with the rapid spread of deep learning
techniques and the availability of large-scale static gaze datasets (e.g.,
SALICON [31]). In stark contrast, predicting observers’ fixations during
dynamic scene free-viewing has less been explored. This task, referred to as
dynamic fixation prediction or video saliency detection is very useful for
understanding human attentional behaviors and has several practical real-word
applications (e.g., video captioning, compression, question answering, object
segmentation, etc). It is thus highly desired to have a standard, high-quality
dataset composed of diverse and representative video stimuli. Exiting datasets
are severely limited in their coverage and scalability, and they only include
special scenarios such as limited human activities in constrained situations.
None of them includes general, representative, and diverse instances in
unconstrained, task-independent scenarios. As a consequence, existing datasets
often fail to offer a rich set of fixations for learning video saliency and to
assess models. Moreover, the existing datasets did not provide an evaluation
server with standalone held out test set to avoid potential dataset over-
fitting, which hinders further development on this topic.

While saliency benchmarks (e.g., MIT300 [32] and LSUN [68]) have been very
instrumental in progressing the static saliency field, such standard
widespread benchmarks are missing for video saliency modeling. We believe such
benchmarks are highly needed to move the field forward. To this end, we
propose a new benchmark “DHF1K (Dynamic Human Fixation 1K)” with a public
server for reporting evaluation results on a preserved test set. Our benchmark
contains a dataset that is unique in terms of generality, diversity and
difficulty. It includes 1K videos with more than 600K frames and per-frame
fixation annotations from 17 observers. The sequences have been carefully
collected to include diverse scenes, motion patterns, object categories, and
activities. DHF1K is accompanied with a comprehensive evaluation of several
state-of-the-art approaches [16, 52, 50, 23, 13, 20, 37, 30, 2, 28, 18, 24,
57, 47]. Moreover, each video is annotated with a main category label (e.g.,
daily activities, animals) and rich attributes (e.g., camera/content movement,
scene lighting, presence of humans), which would enable a deeper understanding
of gaze guidance in free viewing of dynamic scenes.

Further, we propose a novel CNN-LSTM architecture [12, 46] based video
saliency model with a supervised attention mechanism. CNN layers are utilized
for extracting static features within input frames, while convolutional LSTM
(convLSTM) [66] is utilized for sequential fixation prediction over successive
frames. An attention module, learned from existing large-scale image saliency
datasets, is used to enhance spatially informative features of the CNN. Such a
design helps disentangle underlying spatial and temporal factors of dynamic
attention and allows convLSTM to learn temporal saliency representations
efficiently.

Our contributions are three-fold. First, we introduce a standard benchmark of
1K videos covering a wide range of scenes, motions, activities, etc. To the
best of our knowledge, the proposed dataset is the largest eye-tracking
dataset for dynamic, free-viewing fixation prediction. Second, we present a
novel attentive CNN-LSTM architecture for predicting human gaze in dynamic
scenes, which explicitly encodes static attention into dynamic saliency
representation learning by leveraging both static and dynamic fixation data.
Third, we present a comprehensive analysis of video saliency models (the first
one, to the best of our knowledge) on existing datasets (Hollywood-2, UCF
sports), and our new DHF1K dataset. Results show that our model significantly
outperforms previous methods.

##  2 Related Work

###  2.1 Video Eye-Tracking Datasets

There exist several datasets [43, 44, 25, 17] for dynamic visual saliency
prediction, but they are limited and often lack variety, generality and
scalability of instances. Some statistics of these datasets are summarized in
Table 1.

The Hollywood-2 dataset [43] comprises all the $1,707$ videos from Hollywood-2
action recognition dataset [42]. The videos are collected from $69$ Hollywood
movies with $12$ action categories, such as eating, kissing and running. The
human fixation data were tracked from $19$ observers belonging to $3$ groups
for free viewing ($3$ observers), action recognition ($12$ observers), and
context recognition ($4$ observers). Although this dataset is large, its
content is limited to human actions and movie scenes. It mainly focuses on
task-driven viewing mode, rather than free viewing. With $1,000$ frames
randomly sampled from Hollywood-2, we found $84.5$% fixations are located
around the faces.

The UCF sports fixation dataset [43] contains $150$ videos taken from the UCF
sports action dataset [49]. The videos cover $9$ common sports action classes,
such as diving, swinging and walking. Similar to Hollywood-2, the viewers have
been biased towards task-aware observation by being instructed to “identify
the actions occurring in the video sequence”. From the statistics of $1,000$
frames randomly selected from UCF sports, we found $82.3$% fixations fall
inside the human body area.

The DIEM dataset [44] is a public video eye-tracking dataset that has $84$
videos collected from publicly accessible video resources (e.g.,
advertisements, documentaries, sport events, and movie trailers, etc). For
each video, free-viewing fixations of around $50$ observers were collected.
This dataset is mainly limited in its coverage and scale.

Other datasets are either limited in terms of variety and scale of video
stimuli [25, 17], or collected for a special purpose (e.g., salient objects in
videos [59]). More importantly, none of the aforementioned datasets includes a
preserved test set for avoiding potential data overfitting, which has
seriously hampered the research process.

Dataset | Year | Videos | Resolution | Duration(s) | Viewers | Task
---|---|---|---|---|---|---
CRCNS [25] | 2004 | 50 | $640\times 480$ | 6-94 | 15 | task-goal
Hollywood-2 [43] | 2012 | 1,707 | $720\times 480$ | 2-120 | 19 | task-goal
UCF sports [43] | 2012 | 150 | $720\times 480$ | 2-14 | 19 | task-goal
DIEM [44] | 2011 | 84 | $1280\times 720$ | 27-217 |  $\sim$50 | free-view
SFU [17] | 2012 | 12 | $352\times 288$ | 3-10 | 15 | free-view
DHF1K(Ours) | 2017 | 1,000 | $640\times 360$ | 17-42 | 17 | free-view

Table 1: Statistics of typical dynamic eye-tracking datasets.

###  2.2 Computational Models for Fixation Prediction

The study of human gaze patterns in static scenes has received significant
interests, which can be dated back to [28, 27]. Early static saliency models
[36, 69, 15, 7, 18, 22, 33, 63] are mostly based on the contrast assumption
that conspicuous visual features “pop-out” and involuntarily capture attention
(see [5, 6] for review). Computational models compute multiple visual features
such as color, edge, and orientation at multiple spatial scales to produce a
“saliency map”: an image distribution predicting the conspicuity of specific
locations and their likelihood in attracting attention [27, 44]. The locations
with more distinct feature responses over surroundings usually gain higher
saliency values. Deep learning based static saliency models [54, 35, 24, 39,
47, 29, 56, 57] have achieved astonishing improvements, relying on the
powerful end-to-end learning ability of neural network and the availability of
large-scale static saliency datasets [31].

Previous investigations of dynamic human fixation [14, 16, 41, 50, 52, 23, 13,
20, 37] leveraged both static stimulus features and temporal information
(e.g., optical flow, difference-over-time, etc). Some of those studies [14,
41, 52] can be viewed as extensions of exiting static saliency models with
additional motion features. Those models are mainly bound to significant
feature engineering and limited representation ability of hand-crafted
features. To date, only a few deep learning based video saliency models [2,
30] exist in this field. They are mainly based on two-stream network
architecture [2] that accounts for color images and motion fields separately,
or two-layer LSTM with object information [30]. These works show a better
performance and demonstrate the potential advantages in applying neural
networks to this problem. However, they do not 1) consider attentive
mechanisms; 2) utilize existing large-scale static fixation datasets; and 3)
exhaustively assess their performance over large amount of data.

There are some salient object detection models [40, 1, 11, 61, 58, 60, 4, 62,
21] that attempt to uniformly highlight salient object regions in images or
videos. Those models are often task-driven and focus on inferring the main
object, instead of investigating the behavior of the HVS during scene free
viewing.

###  2.3 Attention Mechanisms in Neural Networks

Recently, incorporating attention mechanisms into network architectures has
shown great success in several computer vision [67, 9, 55] and natural
language processing tasks [51, 48]. In such studies, attention is learned in
an automatic, top-down, and task-specific manner, allowing the network to
focus on the most relevant parts in images or sentences. In this paper, we use
attention for enhancing intra-frame salient features, thus allowing the LSTM
to model dynamic representations more easily. In contrast to previous models
learning attentions implicitly, our attention module encodes strong static
saliency information and can be learned from existing static saliency dataset
in a supervised manner. This design leads to improved generality and
prediction performance. It is the first attempt to incorporate a supervised
attention mechanism into the network structure to achieve state-of-art results
in dynamic fixation prediction.

##  3 DHF1K Dataset

(a)  (b)  (c)

Figure 1: Average annotation maps of three datasets used in benchmarking: (a)
Hollywood-2, (b) UCF sports, (c) DHF1K. Figure 2: Example frames from DHF1K
with fixations (red dots) and corresponding categories.

DHF1K | Human | Animal | Artifact | Scenery
---|---|---|---|---
Daily ac. | Sports | Social ac. | Art
#sub-classes* | 20 | 29 | 13 | 10 | 36 | 21 | 21
#videos | 134 | 185 | 116 | 101 | 192 | 162 | 110

  * •

∗Number of sub-classes in each category is reported. For example,
Sports has sub-classes like swimming, jumping, etc.

Table 2: Statistics for video categories in DHF1K dataset.

We introduce DHF1K, a large-scale dataset of gaze in free-viewing of videos.
Our dataset includes 1K videos with diverse content and length, with eye-
tracking annotations from $17$ observers. Fig. 1 shows the center bias of
DHF1K, compared to Hollywood-2, and UCF sports datasets.

Stimuli. The collection of dynamic stimuli mainly follows the following 4
principles.

• Large scale and high quality. Both scale and quality are necessary to ensure
the content diversity of a dataset and is crucial to guarantee a longer
lifespan for a benchmark. To this end, we searched the Youtube engine with
about 200 key terms (e.g., dog, walking, car, etc) and carefully selected
$1,000$ video sequences from the retrieval results. All videos were converted
from their original sources to a 30 fps Xvid MPEG-4 video file in an AVI
container and were resized uniformly into $640\times 360$ spatial resolution.
Thus, DHF1K comprises a total $1,000$ video sequences with $582,605$ frames
with total duration of $19,420$ seconds.

DHF1K | Content motion | Camera motion | #Objects
---|---|---|---
stable | slow | fast | stable | slow | fast | 0 | 1 | 2 |  $\geq$3
#videos | 126 | 505 | 369 | 343 | 386 | 271 | 56 | 335 | 254 | 355

Table 3: Statistics regarding motion patterns and number of main objects in
DHF1K dataset.

DHF1K | Scene illumination | #People
---|---|---
day | night | indoor | 0 | 1 | 2 |  $\geq$3
#videos | 577 | 37 | 386 | 345 | 307 | 236 | 112

Table 4: Statistics regarding scene illumination and number of people in DHF1K
dataset.

• Diverse content. Stimulus diversity is essential for avoiding overfitting
and to delay performance saturation. It offers evenly distributed exogenous
control for studying person-external stimulus factors during scene free-
viewing. In DHF1K, each video is manually annotated with a category label
(totally $150$ classes). Those labels are further classified into $7$ main
categories (see Table 2). Those semantic annotations would enable a deeper
understanding of the high-level stimuli factors guiding human gaze in dynamic
scenes and be indicative for potential research. In Fig. 2, we show example
frames from each category.

• Varied motion patterns. Previous investigations [26, 14, 44] suggested that
motion is one of the key factors that directs attention allocation in dynamic
viewing. For this, DHF1K is designed to span varied motion patterns
(stable-/slow-/fast-motion of content and camera). Please see Table 3 for the
information regarding motion patterns.

• Various objects. Previous studies [65, 38, 3] in cognitive and computer
vision confirmed that object information is indicative to human fixations. The
objects in the dataset vary in their type (e.g., human, animal, in Table 2)
and frequency (see Table 3). For each video, five subjects were instructed to
count the number of the main objects. The majority vote of their counts was
considered as the final count.

For completeness, in Table 4, we offer the information of the scene
illumination and the amount of humans in the dataset. As demonstrated in [45],
luminance is an important exogenous factor for attentive selection. Further,
human beings are important high-level stimuli [10, 8] in free-viewing.

Apparatus and technical specifications. Participants’ eye movements were
monitored binocularly using a Senso Motoric Instruments (SMI) RED 250 system
at a sampling rate of 250 Hz. The dynamic stimuli were displayed on a 19”
display (resolution $1440\times 900$). A headrest was used to help
participants’ heads still at a distance of around $68$ cm, as advised by the
product manual.

Participants. $17$ participants ($10$ males and $7$ females, aging between
$20$ and $28$) who passed the calibration of the eye tracker and had less than
$10$% fixation dropping rate, were quantified for our eye tracking experiment.
All participants had normal or corrected-to-normal vision. All subjects had
not seen the stimuli in DHF1K before. All provided informed consent and were
naïve to the underlying purposes of the experiment.

Data capturing. The stimuli were equally partitioned into 10 non-overlapping
sessions. Participants were required to freeview 10 sessions of videos in
random order. In each session, the videos were also displayed at random.
Before the experiments, every participant was calibrated using the standard
routine in product manual with recommended settings for the best results. To
avoid eye fatigue, each video presentation was followed by a 5-second waiting
interval with black screen. After undergoing a session of videos, the
participant can take a rest until she was ready for viewing the next session.
Finally, $51,038,600$ fixations were recorded from 17 subjects on $1,000$
videos.

Training/testing split. We split $1,000$ dynamic stimuli into separate
training, validation and test sets. Following random selection, we arrive at a
unique split consisting of $600$ training and $100$ validation videos with
publicly available fixation records, as well as $300$ test videos with
annotations held-out for benchmarking purpose.

##  4 Our Approach

Overview. Fig. 3 presents the overall architecture of our video saliency
model. It is based on a CNN-LSTM architecture that combines convolutional
network and recurrent model to exploit both spatial and temporal information
for predicting video saliency. The CNN-LSTM network is extended with a
supervised attention mechanism, which explicitly captures static saliency
information and allows the LSTM to focus on learning dynamic information. The
attention module is trained from rich static eye-tracking data. Thus our model
is able to produce accurate, spatiotemporal saliency with improved
generalization ability. Next, we explain each component of our model in
detail.

Figure 3: Network architecture of the proposed video saliency model. (a)
Attentive CNN-LSTM architecture. (b) CNN layers with attention module are used
for learning intra-frame static features, where the attention module is
learned with the supervision from static saliency data. (c) ConvLSTM used for
learning sequential saliency representations.

CNN-LSTM architecture. Formally, given an input video $\\{I_{t}\\}_{t}$, we
first obtain a sequence of convolutional features $\\{\mathcal{X}_{t}\\}_{t}$
from CNN. Then the features $\\{\mathcal{X}_{t}\\}_{t}$ are fed into a
convLSTM [66] as input. Here, the convLSTM is used for modeling the temporal
dynamic nature of this sequential problem, which is achieved by incorporating
memory units with gated operations. Additionally, through replacing dot
products with convolutional operations, the convLSTM is able to preserve
spatial information, which is essential for making spatially-variant pixel-
level prediction.

More specifically, the convLSTM utilizes three convolution gates (input,
output and forget) to control the flow of signal within the cell. With the
input feature $\mathcal{X}_{t}$ at time step $t$, the convLSTM outputs a
hidden state $\mathcal{H}_{t}$ and maintains a memory cell $\mathcal{C}_{t}$
for controlling state update and output:

$\displaystyle\\!\\!\\!\\!i_{t}\\!$
$\displaystyle=\sigma(W^{\mathcal{X}}_{i}\\!\ast\\!\mathcal{X}_{t}\\!+\\!W^{\mathcal{H}}_{i}\\!\ast\\!\mathcal{H}_{t-1}\\!+\\!W^{\mathcal{C}}_{i}\circ\mathcal{C}_{t-1}\\!+\\!b_{i}),$
(1) $\displaystyle\\!\\!\\!\\!f_{t}\\!$
$\displaystyle=\sigma(W^{\mathcal{X}}_{f}\\!\ast\\!\mathcal{X}_{t}\\!+\\!W^{\mathcal{H}}_{f}\\!\ast\\!\mathcal{H}_{t-1}\\!+\\!W^{\mathcal{C}}_{f}\circ\mathcal{C}_{t-1}\\!+\\!b_{f}),$
(2) $\displaystyle\\!\\!\\!\\!o_{t}\\!$
$\displaystyle=\sigma(W^{\mathcal{X}}_{o}\\!\ast\\!\mathcal{X}_{t}\\!+\\!W^{\mathcal{H}}_{o}\\!\ast\\!\mathcal{H}_{t-1}\\!+\\!W^{\mathcal{C}}_{o}\circ\mathcal{C}_{t}\\!+\\!b_{o}),$
(3) $\displaystyle\\!\\!\\!\\!\mathcal{C}_{t}\\!$
$\displaystyle=f_{t}\circ\mathcal{C}_{t-1}\\!+\\!i_{t}\\!\circ\\!\tanh(W^{\mathcal{X}}_{c}\\!\ast\\!\mathcal{X}_{t}\\!+\\!W^{\mathcal{H}}_{c}\\!\ast\\!\mathcal{H}_{t-1}\\!+\\!b_{c}),$
(4) $\displaystyle\\!\\!\\!\\!\mathcal{H}_{t}\\!$
$\displaystyle=o_{t}\circ\tanh(\mathcal{C}_{t}),$ (5)

$i_{t}$, $f_{t}$, $o_{t}$ are the gates. $\sigma$ and $\tanh$ are the
activation functions of logistic sigmoid and hyperbolic tangent, respectively.
‘$\ast$’ denotes the convolution operator and ‘$\circ$’ represents Hadamard
product. The dynamic fixation map can be obtained via convolving the hidden
states $\mathcal{H}$ with a $1\times 1$ kernel (see Fig. 3 (c)).

In our implementation, the first five conv blocks of VGG-16 [53] are used. For
preserving more spatial details, we remove pool4 and pool5 layers, which
results in $\times 8$ instead of $\times 32$ downsampling. At time step $t$,
with an input frame $I_{t}$ with 224$\times$224 resolution, we have
$\mathcal{X}_{t}\in\mathbb{R}^{28\times 28\times 512}$ and a 28$\times$28
dynamic saliency map from the convLSTM. The kernel size of the conv layer in
convLSTM is set as 3.

Attention module. We extend above CNN-LSTM architecture with an attention
mechanism, which is learned from existing static fixation data in a supervised
manner. Such design is mainly driven by the following three motivations:

• Previous studies [26, 64] shown that human attention is guided by both
static and dynamic factors. Through the additional attention module, the CNN
is enforced to generate a more explicit spatial saliency representation. This
helps disentangle underlying spatial and temporal factors of dynamic
attention, and allows convLSTM better capture temporal dynamics.

• CNN-LSTM architecture introduces a large number of parameters for modeling
spatial and temporal patterns. However, for sequential data such as videos,
obtaining labelled data is costly. Even though there are large-scale datasets
like DHF1K that have 1K videos, the amount of training data is still
insufficient, considering the high correlation among frames within same video.
The supervised attentive module is able to leverage existing rich static
fixation data to improve the generalization power of our model.

• In VGG-16, we remove the last two pooling layers to obtain a large feature
map. This dramatically decreases the receptive field
($212$$\times$$212$$\rightarrow$$140$$\times$$140$), which cannot cover the
whole frame ($224$$\times$$224$). To remedy this, we insert a set of down-
and up-sampling operations into the attention module, which would enhance the
intra-frame saliency information with an enlarged receptive field. By this,
our model is able to make more accurate predictions from a global view.

As demonstrated in Fig. 3 (b), our attentive module is built upon the conv5-3
layer, as an additional branch of several conv layers interleaved with
pooling, and upsampling operations. Given the input feature $\mathcal{X}$,
with pooling layers, the attention module generates a downsampled attention
map (7$\times$7) with an enlarged receptive field (260$\times$260). Then the
small attention map is $\times 4$ upsampled as the same spatial dimensions of
$\mathcal{X}$. Let $M\in[0,1]^{28\times 28}$ be the upsampled attention map,
the feature $\mathcal{X}\in\mathbb{R}^{28\times 28\times 512}$ from conv5-3
layer can be further enhanced by:

$\displaystyle\hat{\mathcal{X}}^{c}=M\circ\mathcal{X}^{c},$ (6)

where $c\in\\{1,\ldots,512\\}$ is the index of the channel. Here, the
attention module work as a feature selector to enhance the feature
representation.

The above attention module may lose useful information for learning a dynamic
saliency representation, as the attention module only considers static
saliency information in still video frames. For this, inspired by the recent
advances of attention mechanism and residual connection [19, 55], we improve
Eq. 6 in residual form:

$\displaystyle\hat{\mathcal{X}}^{c}=(1+M)\circ\mathcal{X}^{c}.$ (7)

With the residual connection, both the original CNN features and the enhanced
features are combined and fed to the LSTM model. In §5.2 and §5.4, more
detailed explorations for the attention module are offered.

Different from previous attention mechanisms that learn task-related attention
in an implicit way, our attention module can learn from existing large-scale
static fixation data in an explicit and supervised manner (detailed in next
part).

Loss function. We use the following loss function [24] that considers three
different saliency evaluation metrics instead of one. The rationale here is
that no single metric can fully capture how satisfactory a saliency map is.

We denote the predicted saliency map as $Y\\!\\!\in\\![0,1]^{28\times 28}$,
the map of fixation locations as $P\\!\\!\in\\!\\!\\{0,1\\}^{28\times 28}$ and
the continuous saliency map (distribution) as $Q\\!\\!\in\\!\\![0,1]^{28\times
28}$. Here the fixation map $P$ is discrete, that records whether a pixel
receives human fixation. The continuous saliency map is obtained via blurring
each fixation location with a small Gaussian kernel. Our loss functions is
defined as follows:

$\displaystyle\\!\\!\\!\mathcal{L}(Y\\!,\\!P\\!,\\!Q)\\!=$
$\displaystyle\mathcal{L}_{K\\!L}(Y,\\!Q)\\!+\\!\alpha_{1}\mathcal{L}_{CC}(Y,\\!Q)\\!+\\!\alpha_{2}\mathcal{L}_{N\\!S\\!S}(Y,\\!P),$
(8)

where $\mathcal{L}_{KL}$, $\mathcal{L}_{CC}$ and $\mathcal{L}_{NSS}$ are the
Kullback-Leibler (KL) divergence, the Linear Correlation Coefficient (CC), and
the Normalized Scanpath Saliency (NSS), respectively, which are derived from
commonly used metrics to evaluate saliency prediction models. $\alpha$s are
balance parameters and are empirically set to $\alpha_{1}=\alpha_{2}=0.1$.

$\mathcal{L}_{KL}$ is widely adopted for training saliency models and is
chosen as the primary loss in our work:

$\displaystyle\mathcal{L}_{KL}(Y,Q)=\sum\nolimits_{x}Q(x)\log\big{(}\frac{Q(x)}{Y(x)}\big{)}.$
(9)

$\mathcal{L}_{CC}$ measures the linear relationship between $Y$ and $Q$:

$\displaystyle\mathcal{L}_{CC}(Y,Q)=-\frac{cov(Y,Q)}{\rho(Y)\rho(Q)},$ (10)

where $cov(Y,Q)$ is the covariance of $Y$ and $Q$, and $\rho(\cdot)$ stands
for standard deviation.

$\mathcal{L}_{NSS}$ is derived from NSS metric:

$\displaystyle\mathcal{L}_{NSS}(Y,P)=-\frac{1}{N}\sum\nolimits_{x}\overline{Y}(x)\times
P(x),$ (11)

where $\overline{Y}=\frac{Y-\mu(Y)}{\rho(Y)}$ and $N=\sum_{x}P(x)$. It is
calculated by taking the mean of scores from the normalized saliency map
$\overline{Y}$ (with zero mean and unit standard deviation) at human eye
fixations $P$. Since $CC$ and $NSS$ are similarity metrics, their negatives
are adopted for minimization.

Training protocol. Our model is iteratively trained with sequential fixation
and image data. In training, a video training batch is cascaded with an image
training batch. More specifically, in a video training batch, we apply a loss
defined over the final dynamic saliency prediction from LSTM. Let
$\\{Y^{d}_{t}\\}_{t=1}^{T}$, $\\{P^{d}_{t}\\}_{t=1}^{T}$, and
$\\{Q^{d}_{t}\\}_{t=1}^{T}$ denote the dynamic saliency predictions, the
dynamic fixation sequence and the continuous ground-truth saliency maps, we
minimize the following loss:

$\displaystyle\mathcal{L}^{d}=\sum\nolimits_{t=1}^{T}\mathcal{L}(Y^{d}_{t},P^{d}_{t},Q^{d}_{t}).$
(12)

In this process, the attention module is trained in an implicit way, since we
do not have the groundtruth fixation of each frame in static scene.

In an image training batch, we only train our attention module via minimizing:

$\displaystyle\mathcal{L}^{s}=\mathcal{L}(M,P^{s},Q^{s}),$ (13)

where the $M$, $P^{s}$, $Q^{s}$ indicate the attention map for our static
attention module, the ground-truth static fixation map, and the ground-truth
static saliency map. In this process, the training of attention module is
supervised by the ground-truth static fixation. Note that, in image training
batch, we do not train our LSTM module, as it is used for learning the dynamic
representation.

For each video training batch, 20 consecutive frames from the same video are
used. Both the video and the start frame are randomly selected. For each image
training batch, we set the batch size as 20, and the images are randomly
sampled from existing static fixation dataset. More implementation details can
be found in § 5.1.

##  5 Experiments

###  5.1 Experimental Setup

|  Method Dataset | DHF1K | Hollywood-2 | UCF sports
---|---|---|---|---
AUC-J$\uparrow$ |  SIM$\uparrow$ | s-AUC$\uparrow$ |  CC$\uparrow$ |  NSS$\uparrow$ | AUC-J$\uparrow$ |  SIM$\uparrow$ | s-AUC$\uparrow$ |  CC$\uparrow$ |  NSS$\uparrow$ | AUC-J$\uparrow$ |  SIM$\uparrow$ | s-AUC$\uparrow$ |  CC$\uparrow$ |  NSS$\uparrow$
|  ∗PQFT [16] | 0.699 | 0.139 | 0.562 | 0.137 | 0.749 | 0.723 | 0.201 | 0.621 | 0.153 | 0.755 | 0.825 | 0.250 | 0.722 | 0.338 | 1.780
|  ∗Seo et al. [52] | 0.635 | 0.142 | 0.499 | 0.070 | 0.334 | 0.652 | 0.155 | 0.530 | 0.076 | 0.346 | 0.831 | 0.308 | 0.666 | 0.336 | 1.690
Dynamic |  ∗Rudoy et al. [50] | 0.769 | 0.214 | 0.501 | 0.285 | 1.498 | 0.783 | 0.315 | 0.536 | 0.302 | 1.570 | 0.763 | 0.271 | 0.637 | 0.344 | 1.619
models |  ∗Hou et al. [23] | 0.726 | 0.167 | 0.545 | 0.150 | 0.847 | 0.731 | 0.202 | 0.580 | 0.146 | 0.684 | 0.819 | 0.276 | 0.674 | 0.292 | 1.399
|  ∗Fang et al. [13] | 0.819 | 0.198 | 0.537 | 0.273 | 1.539 | 0.859 | 0.272 | 0.659 | 0.358 | 1.667 | 0.845 | 0.307 | 0.674 | 0.395 | 1.787
|  ∗OBDL [20] | 0.638 | 0.171 | 0.500 | 0.117 | 0.495 | 0.640 | 0.170 | 0.541 | 0.106 | 0.462 | 0.759 | 0.193 | 0.634 | 0.234 | 1.382
|  ∗AWS-D [37] | 0.703 | 0.157 | 0.513 | 0.174 | 0.940 | 0.694 | 0.175 | 0.637 | 0.146 | 0.742 | 0.823 | 0.228 | 0.750 | 0.306 | 1.631
| OM-CNN [30] | 0.856 | 0.256 | 0.583 | 0.344 | 1.911 | 0.887 | 0.356 | 0.693 | 0.446 | 2.313 | 0.870 | 0.321 | 0.691 | 0.405 | 2.089
| Two-stream [2] | 0.834 | 0.197 | 0.581 | 0.325 | 1.632 | 0.863 | 0.276 | 0.710 | 0.382 | 1.748 | 0.832 | 0.264 | 0.685 | 0.343 | 1.753
|  ∗ITTI [28] | 0.774 | 0.162 | 0.553 | 0.233 | 1.207 | 0.788 | 0.221 | 0.607 | 0.257 | 1.076 | 0.847 | 0.251 | 0.725 | 0.356 | 1.640
|  ∗GBVS [18] | 0.828 | 0.186 | 0.554 | 0.283 | 1.474 | 0.837 | 0.257 | 0.633 | 0.308 | 1.336 | 0.859 | 0.274 | 0.697 | 0.396 | 1.818
Static | SALICON [24] | 0.857 | 0.232 | 0.590 | 0.327 | 1.901 | 0.856 | 0.321 | 0.711 | 0.425 | 2.013 | 0.848 | 0.304 | 0.738 | 0.375 | 1.838
models | Shallow-Net [47] | 0.833 | 0.182 | 0.529 | 0.295 | 1.509 | 0.851 | 0.276 | 0.694 | 0.423 | 1.680 | 0.846 | 0.276 | 0.691 | 0.382 | 1.789
| Deep-Net [47] | 0.855 | 0.201 | 0.592 | 0.331 | 1.775 | 0.884 | 0.300 | 0.736 | 0.451 | 2.066 | 0.861 | 0.282 | 0.719 | 0.414 | 1.903
| DVA [57] | 0.860 | 0.262 | 0.595 | 0.358 | 2.013 | 0.886 | 0.372 | 0.727 | 0.482 | 2.459 | 0.872 | 0.339 | 0.725 | 0.439 | 2.311
Training | Ours | 0.885 | 0.311 | 0.553 | 0.415 | 2.259 | 0.905 | 0.471 | 0.757 | 0.577 | 2.517 | 0.894 | 0.403 | 0.742 | 0.517 | 2.559
setting (i) | Attention module | 0.854 | 0.251 | 0.545 | 0.332 | 1.755 | 0.880 | 0.415 | 0.748 | 0.529 | 2.283 | 0.853 | 0.333 | 0.719 | 0.435 | 1.946
Training | Ours | 0.878 | 0.297 | 0.543 | 0.388 | 2.125 | 0.912 | 0.519 | 0.754 | 0.609 | 3.049 | 0.874 | 0.364 | 0.727 | 0.452 | 2.186
setting (ii) | Attention module | 0.855 | 0.250 | 0.541 | 0.318 | 1.703 | 0.885 | 0.416 | 0.690 | 0.490 | 2.113 | 0.860 | 0.322 | 0.656 | 0.367 | 1.667
Training | Ours | 0.866 | 0.277 | 0.596 | 0.362 | 1.951 | 0.884 | 0.449 | 0.749 | 0.534 | 2.647 | 0.905 | 0.496 | 0.767 | 0.603 | 3.200
setting (iii) | Attention module | 0.852 | 0.260 | 0.582 | 0.350 | 1.945 | 0.898 | 0.429 | 0.763 | 0.543 | 2.409 | 0.884 | 0.354 | 0.743 | 0.500 | 2.339
Training | Ours | 0.890 | 0.315 | 0.601 | 0.434 | 2.354 | 0.913 | 0.542 | 0.757 | 0.623 | 3.086 | 0.897 | 0.406 | 0.744 | 0.510 | 2.567
setting (iv) | Attention module | 0.870 | 0.273 | 0.577 | 0.380 | 2.077 | 0.878 | 0.479 | 0.686 | 0.478 | 2.060 | 0.877 | 0.379 | 0.685 | 0.411 | 1.899

  * ∗ Non-deep learning model.

Table 5: Quantitative results on DHF1K, Hollywood2, and UCF sports datasets.
The best scores are marked in bold. Training settings (§5.1) for video
saliency datasets: (i) DHF1K, (ii) Hollywood-2, (iii) UCF sports, and (iv)
DHF1K+Hollywood-2+UCF sports.

Training/testing protocols. We use the static stimuli ($10,000$ images) from
the training set of SALICON [31] dataset for training our attention module.
For dynamic stimuli, we consider 4 settings: using the training set(s) from
(i) DHF1K, (ii) Hollywood-2, (iii) UCF sports, and (iv) DHF1K+Hollywood-2+UCF
sports. For DHF1K, we use the original training/validation/testing splitting
($600/100/300$). For Hollywood-2, following [42], $823$ videos for training
and $884$ videos for testing. For UCF sports, the training and testing sets
include $103$ and $47$ videos, respectively, as suggested by [49]. We randomly
sample 10% videos from the training sets of Hollywood-2, and UCF sports as
their validation sets. We evaluate our model on the testing sets of DHF1K,
Hollywood-2, and UCF sports dataset, in total $1,231$ video sequences with
more than $400,000$ frames.

Implementation details. Our model is implemented in Python on Keras, and
trained with the Adam optimizer [34]. During the training phase, the learning
rate was set to $0.0001$ and was decreased by a factor of 10 every 2 epochs.
The network was trained for 10 epochs. We perform early-stopping on the
validation set.

Competitors. We compare our model with nine dynamic saliency models: PQFT
[16], Seo et al. [52], Rudoy et al. [50], Hou et al. [23], Fang et al. [13],
OBDL [20], AWS-D [37], OM-CNN [30], and Two-stream [2]111We re-implemented [2]
since the official codes cannot run correctly.. For the sake of complementary,
we further compare with six state-of-the-art static attention models: ITTI
[28], GBVS [18], SALICON [24], DVA [57], Shallow-Net [47], and Deep-Net [47].
OM-CNN, Two-stream, SALICON, DVA, Shallow-Net, and Deep-Net are deep learning
models, and others are classical saliency models. Those models are selected
due to: 1) their representability of the diversity of the state-of-the-art; or
2) publicly available implementations.

Baselines. We further derive $8$ baselines. For each training setting, we
derive two baselines: Our and Attention module, refer to our final dynamic
saliency prediction and the intermediate output of our attention module,
respectively.

Evaluation metrics. Here, we employ five classic metrics, namely Normalized
Scanpath Saliency (NSS), Similarity Metric (SIM), Linear Correlation
Coefficient (CC), AUC-Judd (AUC-J), and shuffled AUC (s-AUC). Please refer to
[5, 57] for detailed descriptions of these metrics.

Computation load. The whole model is trained in an end-to-end manner. The
entire training procedure takes about $30$ hours with a single NVIDIA TITAN X
GPU and a 4.0GHz Intel processor (in training setting (iv)). Since our model
does not need any pre- or post-processing, it takes only about $0.08$s to
process an frame image of size $224\times 224$.

###  5.2 Performance comparison

Performance on DHF1K. Table 5 reports the comparative results with the
aforementioned saliency models, on the testing set ($300$ video sequences) of
DHF1K dataset. In can be observed that the proposed model consistently and
significantly outperforms other competitors, across all the metrics. This can
be contributed to our specially designed attention module, which makes our
model explicitly learn static and dynamic saliency representations in CNN and
LSTM separately. Our model even does not use any optical flow algorithm
(different with Fang et al. [13], Two-stream [2]). This significantly improves
the applicability of our model and demonstrates the effectiveness of our
training protocol that leveraging both static and dynamic stimuli.

DHF1K  Hollywood-2  UCF sports

Figure 4: Qualitative results of our video saliency model on three datasets.
Best viewed in color.

Performance on Hollywood-2. We further test our model on Hollywood-2 dataset,
where the testing sets comprises $884$ video sequences. The results are
summarized in Table 5. Again, our model consistently significantly higher than
other methods across various metrics. Besides, when we go insight into the
performance with training settings, the performance would increase with
increasing amount of training data. This suggests that the large-scale
training data volume is important for the performance of neural network.

Performance on UCF sports. With the test set ($47$ video sequences) of UCF
sports dataset, we again observe the proposed model provides consistently good
results, compared to related state-of-the-art (see Table 5). Interestingly, we
find that, with small amount of training data (training setting (iii), $103$
video stimuli from UCF sports dataset), the proposed model achieves a very
high performance, even better than the model (Our, training setting (iv))
trained with large-scale data (1.5K video stimuli). This could be explained by
lack of diversity in the video training data, as the videos in UCF sports
dataset are highly related (with similar scenes and actors) and small scale.
This is also consistent with our research for UCF sports which shows that
$82.3$% fixations are located on the human body area (see § 2.1).

###  5.3 Analysis

Based on our extensive experiments, we provide more detailed analyses, which
would give deeper insights of previous studies and suggest some hints for
future research.

Dynamic saliency models: deep vs non-deep learning. In dynamic scenes,
previous deep learning based dynamic saliency models (i.e., OM-CNN, Two-
stream) show significant improvements over classic dynamic models (_e.g_.,
PQFT, Seo _et al_. et al., Rudoy et al., Hou et al., Fang et al.). This
demonstrates the strong learning ability of neural network and the promise of
developing neural network in this challenging area.

Non-deep learning models: static vs dynamic. An interesting finding is classic
dynamic methods (i.e., PQFT, Seo et al., Rudoy et al., Hou et al., Fang et
al.) did not perform better than their static counterparts: ITTI, GBVS. This
is probably due to two reasons. First, the perceptual cues and underlying
mechanisms of visual attention allocation during dynamic viewing are more
complex and still not clear. Second, previous studies are more focused on
computational models of static saliency, while less efforts were paid for
modeling dynamic saliency.

Deep learning models: static vs dynamic. Compared with state-of-the-art deep
learning based static models (i.e., DVA, Deep-Net), previous deep learning
based dynamic models (i.e., OM-CNN, Two-stream) only obtain slightly better
performance (or only competitive). Although strong motion information (i.e.,
optical flow, motion network) have been encoded into OM-CNN and Two-stream,
their performance are still limited. We attribute this into the inherent
difficulties of video saliency prediction and previous models’ neglect of
utilizing existing rich static saliency data.

###  5.4 Ablation study

In this section, we offer a more detailed exploration of our proposed approach
in several aspects with DHF1K dataset. We verify the effectiveness of the
proposed mechanism, and examine the influence of different training protocols.
The results are summarized in Table 6.

Aspects | Variants | AUC-J$\uparrow$ |  SIM$\uparrow$ | s-AUC$\uparrow$ |  CC$\uparrow$ |  NSS$\uparrow$
---|---|---|---|---|---|---
Baseline |  | training setting (iv)
---
(1.5K videos+10K images)
0.890 | 0.315 | 0.601 | 0.434 | 2.354
Attention |  | w/o attention
---
(1.5K videos)
0.847 | 0.236 | 0.579 | 0.306 | 1.685
module |  | w/o residual connection
---
(1.5K videos+10K images)
0.874 | 0.303 | 0.594 | 0.401 | 2.174
|  | w/o downsampling
---
(1.5K videos+10K images)
0.870 | 0.298 | 0.583 | 0.389 | 2.085
Training |  | reduced training samples
---
(1.5K videos+5K images)
0.877 | 0.297 | 0.588 | 0.372 | 2.098
convLSTM |  | w/o convLSTM
---
(1.5K videos+10K images)
0.867 | 0.269 | 0.573 | 0.382 | 2.034

Table 6: Ablation study on DHF1K. See §5.4 for details.

Effect of attention mechanism. By disabling the attention module, and only
training with video stimuli we observe a performance drop (e.g., AUC-J:
$0.890$$\rightarrow$$0.847$), verifying the effectiveness of attention module
and showing that the leverage of static stimuli indeed improves the
predication accuracy in dynamic scenes. For exploring the effect of the
residual connection in attention module (Eq. 8), we train the model based on
Eq. 5 (without residual connection). We observe a minor decrease; showing that
employing residual connection could avoid distorting spatial features in
frames. In our attention module, we apply down-sampling for enlarging the
receptive field. We also study the influence of such design. We find that the
attention module with enlarged receptive field would gain better performance,
since the model could make prediction in global view.

Training. We assess different training protocols. By reducing the amount of
static training stimuli from 10K to 5K, we observe a performance drop (e.g.,
AUC-J: $0.890$$\rightarrow$$0.877$). The baseline (w/o attention) can also be
viewed as the model without any static training stimuli, which gains worse
performance (e.g., AUC-J: $0.890$$\rightarrow$$0.847$).

Effect of convLSTM. To study the influence of convLSTM, we re-train our model
without convLSTM (using training setting (iv)) and get a baseline: w/o
convLSTM. We observe a drop of performance; showing that the dynamic
information learnt in convLSTM could boost the performance.

##  6 Discussion and Conclusion

In this work, we presented “Dynamic Human Fixation (DHF1K)”, a large-scale
carefully designed and systematically collected benchmark dataset to
facilitate research in video saliency modeling. To the best of our knowledge,
our work is the most comprehensive performance evaluation of video saliency
models. DHF1K contains 1K videos, which capture representative instances,
diverse contents and various motions, with human eye-tracking annotations.

Further, we proposed a novel deep learning based video saliency model, which
encodes a supervised attention mechanism to explicitly capture static saliency
information and help LSTM better capture dynamic saliency representations over
successive frames. We performed extensive experiments on DHF1K, Hollywood-2,
and UCF-sports datasets, and analyzed the performance of our model compared to
previous attention models in dynamic scenes. Our experimental results
demonstrate that our proposed model outperforms other competitors and is quite
efficient.

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###### Abstract

We investigate the stability and stabilization concepts for infinite
dimensional time fractional differential linear systems in Hilbert spaces with
Caputo derivatives. Firstly, based on a family of operators generated by
strongly continuous semigroups and on a probability density function, we
provide sufficient and necessary conditions for the exponential stability of
the considered class of systems. Then, by assuming that the system dynamics is
symmetric and uniformly elliptic and by using the properties of the
Mittag–Leffler function, we provide sufficient conditions that ensure strong
stability. Finally, we characterize an explicit feedback control that
guarantees the strong stabilization of a controlled Caputo time fractional
linear system through a decomposition approach. Some examples are presented
that illustrate the effectiveness of our results.

###### keywords:

fractional differential equations; fractional diffusion systems; Caputo
derivative; stability and stabilization in Hilbert spaces; decomposition
method.

xx 1 5 Submitted: 31 Dec 2019; Revised: 21 and 28 Feb 2020; Accepted: 02 March
2020. Published in: Mathematics 2020, Volume 8, Issue 3, 353.
DOI:black10.3390/math8030353 The Stability and Stabilization of Infinite
Dimensional Caputo-Time Fractional Differential Linear Systems Hanaa Zitane
1,†, Ali Boutoulout 2,† and Delfim F. M. Torres 3,†* Hanaa Zitane, Ali
Boutoulout and Delfim F. M. Torres Correspondence<EMAIL_ADDRESS>Tel.:
+351-234-370-668 (D.F.M.T.) The authors contributed equally to this work.
26A33; 93D15.

##  1 Introduction

Fractional order calculus is a natural generalization of classical integer
order calculus. It deals with integrals and derivatives of an arbitrary real
or complex order. Fractional order calculus has become very popular, in recent
years, due to its demonstrated applications in many fields of applied sciences
and engineering, such as the spread of contaminants in underground water, the
charge transport in amorphous semiconductors, and diffusion of pollution in
the atmosphere Rahimy (2010); Kilbas et al. (2006); Diethelm (2010). Because
it generalizes and includes in the limit the integer order calculus, the
fractional calculus has the potential to accomplish much more than what
integer order calculus achieves Hilfer (2000). In particular, it has proved to
be a powerful tool to describe long-term memory and hereditary properties of
various dynamical complex processes Sabatier et al. (2007), diffusion
processes, such as those found in batteries Gabano and Poinot (2011) and
electrochemical and control processes Ichise et al. (1971), to model and
control epidemics Rosa and Torres (2019); Silva and Torres (2019) and
mechanical properties of viscoelastic systems and damping materials, such as
stress and strain Bagley and Calico (1991).

One can find in the literature several different fractional calculus. Here we
use the fractional calculus of Caputo, which was introduced by Michele Caputo
in his 1967 paper Caputo (2008). Such calculus has appeared, in a natural way,
for representing observed phenomena in laboratory experiments and field
observations, where the mathematical theory was checked with experimental
data. Indeed, the operator introduced by Caputo in 1967, and used by us in the
present work, represents an observed linear dissipative mechanism phenomena
with a time derivative of order 0.15 entering the stress-strain relation
Caputo (2008). More recently, a variational analysis with Caputo operators has
been developed, which provides further mathematical substance to the use of
Caputo fractional operators Malinowska and Torres (2012); Almeida et al.
(2015).

In the analysis and design of control systems, the stability issue has always
an important role Mahmoud and Karaki (2018); Rocha et al. (2018). For a
dynamical system, an equilibrium state is said to be stable if such system
remains close to this state for small disturbances, and for an unstable system
the question is how to stabilize it, especially by a feedback control law
Sontag (2012). The stabilization concept for integer order systems and related
problems has been considered in several works, see, e.g., Pritchard and
Zabczyk (1981); Curtain and Zwart (1995); Triggiani (1975); Balakrishnan
(1981) and references cited therein. In Pritchard and Zabczyk (1981), the
relationship between the asymptotic behavior of a system, the spectrum
properties of its dynamics, and the existence of a Lyapunov functional is
provided. Several techniques are considered to study different kinds of
stabilization, for example, the exponential stabilization is studied via a
decomposition method Triggiani (1975) while the strong stabilization is
developed using the Riccati approach Balakrishnan (1981).

Similarly as classical dynamical systems, stability analysis is a central task
in the study of fractional dynamical systems, which has attracted increasing
interest of many researchers Silva and Torres (2019); Wojtak et al. (2018).
For finite dimensional systems, the stability concept for fractional
differential systems equipped with the Caputo derivative is investigated in
many works Zhang et al. (2011). In Matignon (1996), Matignon studies the
asymptotic behavior for linear fractional differential systems with the Caputo
derivative, where their dynamics $A$ is a constant coefficient matrix. In this
case the stability is guaranteed if the eigenvalues of the dynamics matrix
$A$, $\lambda\in\sigma(A)$, satisfy $|arg(\lambda)|>\dfrac{\pi\alpha}{2}$
Matignon (1996). Since then, many scholars have carried out further studies on
the stability for different classes of fractional linear systems Qian et al.
(2010); Li et al. (2010). In Qian et al. (2010), stability theorems for
fractional differential systems, which include linear systems, time-delayed
systems, and perturbed systems, are established, while in Li et al. (2010),
Ge, Chen and Kou provide results on the Mittag–Leffler stability and propose a
Lyapunov direct method, which covers the power law stability and the
exponential stability. See also Matar and Abu Skhail (2018), where the
Mittag–Leffler and the class-K function stability of fractional differential
equations of order $\alpha\in(1,2)$ are investigated. In 2018, the notion of
regional stability was introduced for fractional systems in Ge et al. (2018),
where the authors study the Mittag–Leffler stability and the stabilization of
systems with Caputo derivatives, but only on a sub-region of its geometrical
domain. More recently, fractional output stabilization problems for
distributed systems in the Riemann–Liouville sense were studied Zitane et al.
(2020, in press, 2019), where feedback controls, which ensure exponential,
strong, and weak stabilization of the state fractional spatial derivatives,
with real and complex orders, are characterized.

An analysis of the literature shows that existing results on stability of
fractional systems are essentially limited to finite-dimensional fractional
order linear systems, while results on infinite-dimensional spaces are a
rarity. In contrast, here we investigate global stability and stabilization of
infinite dimensional fractional dynamical linear systems in the Hilbert space
$L^{2}(\Omega)$ with Caputo derivatives of fractional order $0<\alpha<1$. In
particular, we characterize exponential and strong stability for fractional
Caputo systems on infinite-dimensional spaces.

The remainder of this paper is organized as follows. In Section 2, some basic
knowledge of fractional calculus and some preliminary results, which will be
used throughout the paper, are given. In Section 3, we prove results on the
global asymptotic and exponential stability of Caputo-time fractional
differential linear systems. In contrast with available results in the
literature, which are restricted to systems of integer order or to fractional
systems in the finite dimensional state space $\mathbb{R}^{n}$, here we study
a completely different class of systems: we investigate fractional linear
systems where the state space is the Hillbert space $L^{2}(\Omega)$. We also
characterize the stabilization of a controlled Caputo diffusion linear system
via a decomposition method. Section 4 presents the main conclusions of the
work and some interesting open questions that deserve further investigations.

##  2 Preliminaries and Notation

In this section, we introduce several definitions and results of fractional
calculus that are used in the sequel.

[Kilbas et al. (2006)] Let $0<\alpha<1$ and $T>0$. The Caputo derivative of
fractional order $\alpha$ for an absolutely continuous function $y(\cdot)$ on
$[0,T]$ can be defined as follows:

${}^{C}D_{t}^{\alpha}y(t)=\dfrac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-s)^{-\alpha}\dfrac{d}{ds}y(s)\,\mathrm{d}s,$

where $\Gamma(1-\alpha)$ is the Euler Gamma function.

[Zhou and Jiao (2010)] For any given function $g\in L^{2}(0,T,L^{2}(\Omega))$,
we say that function $y\in C(0,T,L^{2}(\Omega))$ is a mild solution of the
system

$\left\\{\begin{array}[]{ll}{}^{C}D_{t}^{\alpha}y(t)=Ay(t)+g(t)&t\in]0,+\infty[\\\
y(0)=y_{0}&y_{0}\in L^{2}(\Omega)\end{array}\right.$ (1)

if it satisfies

$y(t)=S_{\alpha}(t)y_{0}+\int_{0}^{t}(t-s)^{\alpha-1}K_{\alpha}(t-s)g(s)\,\mathrm{d}s,$
(2)

where

$S_{\alpha}(t)=\int_{0}^{+\infty}\Psi_{\alpha}(\theta)S(t^{\alpha}\theta)\,\mathrm{d}\theta$
(3)

and

$K_{\alpha}(t)=\alpha\int_{0}^{+\infty}\theta\Psi_{\alpha}(\theta)S(t^{\alpha}\theta)\,\mathrm{d}\theta$
(4)

with

$\Psi_{\alpha}(\theta)=\frac{1}{\alpha}\theta^{-1-\frac{1}{\alpha}}T_{\alpha}(\theta^{-\frac{1}{\alpha}}),$
(5)

$(S(t))_{t\geq 0}$ the strongly continuous semigroup generated by operator
$A$, and $T_{\alpha}$ the probability density function defined on $(0,\infty)$
by

$T_{\alpha}=\frac{1}{\pi}\sum_{n=1}^{+\infty}(-1)^{n}\theta^{\alpha
n-1}\frac{\Gamma(n\alpha+1)}{n!}\sin(n\pi\alpha).$

[Mainardi et al. (2007)] The probability density function $T_{\alpha}$ defined
on $(0,\infty)$ satisfies

$T_{\alpha}(\theta)\geq 0,\quad\theta\in(0,\infty),\quad\text{ and
}\int_{0}^{+\infty}T_{\alpha}(\theta)\,\mathrm{d}\theta=1.$

[Erdélyi et al. (1981)] The Mittag–Leffler function of one parameter is
defined as

$E_{\eta}(z)=\sum_{n=0}^{+\infty}\frac{z^{n}}{\Gamma(\eta
n+1)}\quad\text{with}\quad Re(\eta)>0,\quad z~{}\in\mathbb{C}.$

[Erdélyi et al. (1981)] The Mittag–Leffler function of two parameters is
defined as

$E_{\eta,\beta}(z)=\sum_{n=0}^{+\infty}\frac{z^{n}}{\Gamma(\eta
n+\beta)}\quad\text{with}\quad Re(\eta)>0,\quad\beta>0,\quad z\in\mathbb{C}.$

The Mittag–Leffler function appears naturally in the solution of fractional
differential equations and in various applications: see Erdélyi et al. (1981)
and references therein. The exponential function is a special case of the
Mittag–Leffler function Joshi et al. (2020): for $\beta=1$ one has
$E_{\eta,1}(z)=E_{\eta}(z)$ and $E_{1,1}(z)=e^{z}$.

[Mainardi (2014)] The Mittag–Leffler function $E_{\alpha}(-t^{\alpha})$ is
completely monotonic: for all $0<\alpha<1$, for all $n\in\mathbb{N}$ and
$t>0$, one has

$(-1)^{n}\frac{d^{n}}{dt^{n}}E_{\alpha}(-t^{\alpha})\geq 0.$

[Schneider (1996)] The generalized Mittag–Leffler function
$E_{\eta,\beta}(-x)$, $x\geq 0$, is completely monotonic for $\eta,\beta>0$ if
and only if $\eta\in(0,1]$ and $\beta\geq\eta$.

[Podlubny (1999)] Let $\beta>0$, $0<\eta<2$, and $\mu$ be an arbitrary real
number such that $\frac{\pi\eta}{2}<\mu<\min\\{\pi,\pi\eta\\}$. Then, the
following asymptotic expressions hold:

  * •

if $|arg(z)|\leq\mu$ and $|z|>0$, then

$|E_{\eta,\beta}(z)|\leq
M_{1}(1+|z|)^{(1-\beta)/\eta}e^{Re(z^{\frac{1}{{\eta}}})}+\frac{M_{2}}{1+|z|};$
(6)

  * •

if $\mu<|arg(z)|\leq\pi$ and $|z|\geq 0$, then

$|E_{\eta,\beta}(z)|\leq\frac{M_{2}}{1+|z|},$ (7)

where $M_{1}$ and $M_{2}$ are positive constants.

##  3 Main Results

Our main goal is to study the stability and provide stabilization for a class
of abstract Caputo-time fractional differential linear systems.

###  3.1 Stability of Time Fractional Differential Systems

Let $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$, $n=1,2,3,\ldots$,
and let us consider the following abstract time fractional order differential
system:

$\left\\{\begin{array}[]{ll}{}^{C}D_{t}^{\alpha}z(t)=Az(t),&t\in\,]0,+\infty[,\\\
z(0)=z_{0},&z_{0}\in L^{2}(\Omega),\end{array}\right.$ (8)

where ${}^{C}D_{t}^{\alpha}$ is the left-sided Caputo fractional derivative of
order $0<\alpha<1$, the second order operator $A:D(A)\subset
L^{2}(\Omega)\longrightarrow L^{2}(\Omega)$ is linear, with dense domain and
such that the coefficients do not depend on time $t$, and such that it is also
the infinitesimal generator of the $C_{0}$-semi-group $(S(t))_{t\geq 0}$ on
the Hilbert state space $L^{2}(\Omega)$ endowed with its usual inner product
$<\cdot,\cdot>$ and the corresponding norm $\Arrowvert\cdot\Arrowvert$. The
unique mild solution of system (8) can be written, from Lemma 2, as

$z(t)=S_{\alpha}(t)z_{0},$

where $S_{\alpha}(t)$ is defined by (3).

We begin by proving the following lemma, which will be used thereafter.

Let $A$ be the infinitesimal generator of a $C_{0}$-semi-group $(S(t))_{t\geq
0}$ on the Hilbert space $L^{2}(\Omega)$. Assume that there exists a function
$h(\cdot)\in L^{2}(0,+\infty;\mathbb{R}^{+})$ satisfying

$\Arrowvert S_{\alpha}(t+s)z\Arrowvert\leq h(t)\Arrowvert
S_{\alpha}(s)z\Arrowvert,\quad\forall\,t,s\geq 0,\quad\forall\,z\in
L^{2}(\Omega).$ (9)

Then the operators $(S_{\alpha}(t))_{t\geq 0}$ are uniformly bounded.

###### Proof.

To prove that $(S_{\alpha}(t))_{t\geq 0}$ are bounded, we have to show that

$\forall z\in L^{2}(\Omega)~{}\underset{t\geq 0}{\sup}\Arrowvert
S_{\alpha}(t)z\Arrowvert<\infty.$ (10)

By _reductio ad absurdum_ , let us suppose that (10) does not hold, which
means that there exists a sequence $(t_{s}+\tau_{n})$, $t_{s}>0$ and
$\tau_{n}\longrightarrow+\infty$, satisfying

$\Arrowvert
S_{\alpha}(t_{s}+\tau_{n})z\Arrowvert\longrightarrow+\infty~{}as~{}n\longrightarrow+\infty.$
(11)

From relation

$\displaystyle\int_{0}^{+\infty}\Arrowvert
S_{\alpha}(s+\tau_{n})z\Arrowvert^{2}\,\mathrm{d}s=\displaystyle\int_{\tau_{n}}^{+\infty}\Arrowvert
S_{\alpha}(s)z\Arrowvert^{2}\,\mathrm{d}s,~{}~{}0\leq s<+\infty,$

it follows that the right-hand side goes to $0$ as $n\longrightarrow+\infty$.
Using Fatou’s Lemma yields

$\underset{n\longrightarrow+\infty}{\lim~{}\inf}~{}\Arrowvert
S_{\alpha}(s+\tau_{n})z\Arrowvert=0~{}~{}\forall~{}s>0.$

Therefore, for some $s_{0}<t_{s}$, we may find a subsequence $\tau_{n_{k}}$
such that

$\underset{k\longrightarrow+\infty}{\lim}\Arrowvert
S_{\alpha}(s_{0}+\tau_{n_{k}})z\Arrowvert=0.$

By virtue of condition (9), one obtains

$\Arrowvert S_{\alpha}(t_{s}+\tau_{n_{k}})z\Arrowvert\leq
h(t_{s}-s_{0})\Arrowvert
S_{\alpha}(s_{0}+\tau_{n_{k}})z\Arrowvert\underset{k\longrightarrow+\infty}{\longrightarrow
0},$

which contradicts (11). The intended conclusion follows from the uniform
boundedness principle. ∎

Let $z_{0}\in L^{2}(\Omega)$. System (8) is said to be exponentially stable if
there exist two strictly positive constants, $M>0$ and $\omega>0$, such that

$\Arrowvert z(t)\Arrowvert\leq Me^{-\omega t}\Arrowvert
z_{0}\Arrowvert,\quad\forall t\geq 0.$

The next theorem provides necessary and sufficient conditions for exponential
stability of the abstract fractional order differential system (8).

Suppose that the operators $(S_{\alpha}(t))_{t\geq 0}$ fulfill assumption (9)
and

$\forall z\in L^{2}(\Omega)~{}~{}~{}~{}~{}~{}\Arrowvert
S_{\alpha}(t+s)z\Arrowvert\leq\Arrowvert
S_{\alpha}(t)z\Arrowvert\cdot\Arrowvert S_{\alpha}(s)z\Arrowvert,\quad\forall
t,s\geq 0.$ (12)

Then, system (8) is exponentially stable if, and only if, for every $z\in
L^{2}(\Omega)$ there exists a positive constant $\delta<\infty$ such that

$\displaystyle\int_{0}^{+\infty}\Arrowvert
S_{\alpha}(t)z\Arrowvert^{2}\,\mathrm{d}t<\delta.$ (13)

###### Proof.

One has

$\begin{split}t\Arrowvert
S_{\alpha}(t)z\Arrowvert^{2}&=\displaystyle\int_{0}^{t}\Arrowvert
S_{\alpha}(t)z\Arrowvert^{2}\,\mathrm{d}s\\\
&=\displaystyle\int_{0}^{t}\Arrowvert
S_{\alpha}(t-s+s)z\Arrowvert^{2}\,\mathrm{d}s.\end{split}$

Combining assumption (9), Lemma 3.1, and condition (13), one gets

$\begin{split}t\Arrowvert
S_{\alpha}(t)z\Arrowvert^{2}&\leq\displaystyle\int_{0}^{t}\Arrowvert
S_{\alpha}(s)z\Arrowvert^{2}\Arrowvert
S_{\alpha}(t-s)z\Arrowvert^{2}\,\mathrm{d}s\\\ &\leq N\delta\Arrowvert
z\Arrowvert^{2}\end{split}$

for some $N>0$. Therefore, for $t$ sufficiently large, it follows that

$\Arrowvert S_{\alpha}(t)\Arrowvert<1.$

Then, there exists $t_{1}>0$ such that

$\ln\Arrowvert S_{\alpha}(t)\Arrowvert<0,\quad\forall t\geq t_{1}.$

Thus,

$\omega_{0}=\underset{t\geq 0}{\inf}~{}\dfrac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}<0.$

Now, let us show that

$\omega_{0}=\underset{t\longrightarrow+\infty}{\lim}\frac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}.$ (14)

Let $t_{s}>0$ be a fixed number and
$N^{{}^{\prime}}=\underset{t\in[0,t_{s}]}{\sup}{\Arrowvert
S_{\alpha}(t)\Arrowvert}$. Thus, for each $t>t_{s}$, there exists
$m\in\mathbb{N}$ such that $mt_{s}\leq t\leq(m+1)t_{s}$. From (12), it follows
that

$\begin{split}\Arrowvert S_{\alpha}(t)\Arrowvert&=\Arrowvert
S_{\alpha}(mt_{s}+(t-mt_{s}))\Arrowvert\\\ &\leq\Arrowvert
S_{\alpha}(mt_{s})\Arrowvert\Arrowvert
S_{\alpha}(t-mt_{s})\Arrowvert,\end{split}$

which yields

$\frac{\ln\Arrowvert S_{\alpha}(t)\Arrowvert}{t}\leq\frac{\ln\Arrowvert
S_{\alpha}(mt_{s})\Arrowvert}{t}+\frac{\ln\Arrowvert
S_{\alpha}(t-mt_{s})\Arrowvert}{t}.$

Using again (12), it results that

$\frac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}\leq\frac{mt_{s}}{t}\frac{\ln\Arrowvert
S_{\alpha}(t_{s})\Arrowvert}{t_{s}}+\frac{\ln\Arrowvert
N^{{}^{\prime}}\Arrowvert}{t}.$

Since $\dfrac{mt_{s}}{t}\leq 1$ and $t_{s}$ is arbitrary, one obtains

$\underset{t\longrightarrow+\infty}{\lim~{}\sup}~{}\frac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}\leq\underset{t>0}{\inf}~{}\frac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}\leq\underset{t\longrightarrow+\infty}{\lim~{}\inf}~{}\frac{\ln\Arrowvert
S_{\alpha}(t)\Arrowvert}{t}.$

Consequently, (14) holds. Hence, we conclude that for all
$\omega\in\,]0,-\omega_{0}[$, there exists $M>0$ such that

$\forall z\in L^{2}(\Omega)\quad\Arrowvert S_{\alpha}(t)z\Arrowvert\leq
Me^{-\omega t}\Arrowvert z\Arrowvert,~{}~{}\forall t\geq 0,$

which means that system (8) is exponentially stable. The converse is obvious.
∎

When $\alpha=1$, the conditions (9) and (12) are verified, and we retrieve
from our Theorem 3.1 the results established in Curtain and Zwart (1995);
Pritchard and Zabczyk (1981) about the exponential stability of system (8) on
$\Omega$, which is equivalent to

$\int_{0}^{+\infty}\Arrowvert
S(t)z\Arrowvert^{2}\,\mathrm{d}t<\infty,\quad\forall z\in L^{2}(\Omega).$

Let $z_{0}\in L^{2}(\Omega)$. System (8) is said to be strongly stable if its
corresponding solution $z(t)$ satisfies

$\Arrowvert z(t)\Arrowvert\longrightarrow
0~{}~{}as~{}~{}t\longrightarrow+\infty.$

In our next theorem, we provide sufficient conditions that guaranty the strong
stability of the fractional order differential system (8). The result
generalizes the asymptotic result established by Matignon for finite
dimensional state spaces, where the dynamics of the system $A$ is considered
to be a matrix with constant coefficients in $\mathbb{R}^{n}$ Matignon (1996).
In contrast, here we tackle the stability for a different class of systems.
Precisely, we consider fractional systems where the system dynamics $A$ is a
linear operator generating a strongly continuous semigroup in the infinite
dimensional state space $L^{2}(\Omega)$.

Let $(\lambda_{p})_{p\geq 1}$ and $(\phi_{p})_{p\geq 1}$ be the eigenvalues
and the corresponding eigenfunctions of operator $A$ on $L^{2}(\Omega)$. If
$A$ is a symmetric uniformly elliptic operator, then system (8) is strongly
stable on $\Omega$.

###### Proof.

Since $A$ is a symmetric uniformly elliptic operator, it follows that system
(8) admits a weak solution defined by

$z(t)=\sum_{p=1}^{+\infty}E_{\alpha}(\lambda_{p}t^{\alpha})\langle
z_{0},\phi_{p}\rangle\phi_{p}\quad\forall\,z_{0}\in L^{2}(\Omega),$

where $(\lambda_{p})_{p\geq 1}$ satisfy

$0>\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{j}\geq\cdots,\lim\limits_{j\longrightarrow\infty}=-\infty,$

and $(\phi_{p})_{p\geq 1}$ forms an orthonormal basis in $L^{2}(\Omega)$
Sakamoto and Yamamoto (2011); Courant and Hilbert (1953). Using the fact that
function $E_{\alpha}(-t^{\alpha})$ is completely monotonic, for all
$\alpha\in(0,1)$ and $t>0$ (Lemma 2), yields

$\begin{split}\Arrowvert
z(t)\Arrowvert&=\left\Arrowvert\displaystyle\sum\limits_{{p=1}}^{+\infty}E_{\alpha}(\lambda_{p}t^{\alpha})\langle
z_{0},\phi_{p}\rangle\phi_{p}\right\Arrowvert\\\
&\leq|E_{\alpha}(\lambda_{1}t^{\alpha})|\Arrowvert
z_{0}\Arrowvert.\end{split}$

Moreover, from Lemma 2, it follows that

$\Arrowvert z(t)\Arrowvert\leq\frac{M_{2}}{1-\lambda_{1}t^{\alpha}}\Arrowvert
z_{0}\Arrowvert\longrightarrow 0\quad\text{ as }\quad t\longrightarrow+\infty$

for some $M_{2}>0$. Hence, system (8) is strongly stable on $\Omega$. ∎

Let us consider, on $\Omega=]0,1[$, the following one-dimensional fractional
diffusion system defined by

$\left\\{\begin{array}[]{lll}{}^{C}D_{t}^{0.5}z(x,t)=\dfrac{\partial^{2}z}{\partial
x^{2}}(x,t),&x\in\Omega,\quad t\in\,]0,+\infty[,\\\ z(0,t)=z(1,t)=0,&\forall
t>0,\\\ z(x,0)=z_{0},&x\in\Omega,\end{array}\right.$ (15)

where the second order operator $A=\dfrac{\partial^{2}}{\partial x^{2}}$ has
its spectrum given by the eigenvalues $\lambda_{p}=-(p\pi)^{2}$, $p\geq 1$,
and the corresponding eigenfunctions are
$\phi_{p}(x)=\sqrt{\frac{2}{1+(p\pi)^{2}}}\sin(p\pi x)$, $p\geq 1$. Operator
$A$ generates a $C_{0}$-semi-group $(S(t))_{t\geq 0}$ defined by

$S(t)z_{0}=\sum_{p=1}^{+\infty}e^{\lambda_{p}t}\langle
z_{0},\phi_{p}\rangle\phi_{p}.$

Moreover, the solution of system (15) is given by

$S_{0.5}(t)z_{0}=\sum_{p=1}^{+\infty}E_{0.5}(\lambda_{p}t^{0.5})\langle
z_{0},\phi_{p}\rangle\phi_{p}.$

One has that operator $A$ is symmetric and uniformly elliptic. Consequently,
from our Theorem 3.1, we deduce that system (15) is strongly stable on
$\Omega$. This is illustrated numerically in Figure 1 for $z(x,0)=\sin(\pi
x)$, $t=0.1$, $t=0.15$, $t=0.2$, and $t=1.0$.

Figure 1: The state of system (15) for $z(x,0)=\sin(\pi x)$, $t=0.1$,
$t=0.15$, $t=0.2$, and $t=1.0$, illustrating the fact that (15) is strongly
stable on $\Omega=]0,1[$.

###  3.2 Stabilization of Time Fractional Differential Systems

Let $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$, $n=1,2,3,\ldots$.
We consider the following Caputo-time fractional differential linear system:

$\left\\{\begin{array}[]{ll}{}^{C}D_{t}^{\alpha}z(t)=Az(t)+Bu(t),&t\in]0,+\infty[,\quad
0<\alpha<1,\\\ z(0)=z_{0},&z_{0}\in L^{2}(\Omega),\end{array}\right.$ (16)

with the same assumptions on $A$ as in Section 3.1 and where $B$ is a bounded
linear operator from $U$ into $L^{2}(\Omega)$, where $U$ is the space of
controls, assumed to be a Hilbert space. By Lemma 2, the unique mild solution
$z(\cdot)$ of system (16) is defined by

$z(t)=S_{\alpha}(t)z_{0}+\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}K_{\alpha}(t-s)Bu(s)\,\mathrm{d}s,$
(17)

where $S_{\alpha}(t)$ and $K_{\alpha}(t)$ are given, respectively, by (3) and
(4) .

System (16) is said to be exponentially (respectively strongly) stabilizable
if there exists a bounded operator $K\in\mathcal{L}(L^{2}(\Omega),U)$ such
that the system

$\left\\{\begin{array}[]{ll}{}^{C}D_{t}^{\alpha}z(t)=(A+BK)z(t),&t\in\,]0,+\infty[,\\\
z(0)=z_{0},&z_{0}\in L^{2}(\Omega),\end{array}\right.$ (18)

is exponentially (respectively strongly) stable on $\Omega$.

It is clear that the exponential stabilization of system (16) implies the
strong stabilization of (16). Note that the concept is general: when
$\alpha=1$, we obtain the classical definitions of stability and
stabilization.

Let $(S^{k}(t))_{t\geq 0}$ be the strongly continuous semi-group generated by
$A+BK$, where $K\in\mathcal{L}(L^{2}(\Omega),U)$ is the feedback operator. The
unique mild solution of system (16) can be written as

$z(t)=S^{k}_{\alpha}(t)z_{0}$

with

$S^{k}_{\alpha}(t)=\displaystyle\int_{0}^{+\infty}\Psi_{\alpha}(\theta)S^{k}(t^{\alpha}\theta)\,\mathrm{d}\theta,$

where $\Psi_{\alpha}(\theta)$ is defined by (5).

Let $A+BK$ generate a strongly continuous semi-group $(S^{k}(t))_{t\geq 0}$ on
$L^{2}(\Omega)$. If the operator $(S^{k}_{\alpha}(t))_{t\geq 0}$ satisfies
conditions (9) and (12) and if

$\forall z\in
L^{2}(\Omega)~{}~{}~{}~{}~{}~{}\displaystyle\int_{0}^{+\infty}\Arrowvert
S^{k}_{\alpha}(t)z\Arrowvert^{2}\,\mathrm{d}t<~{}\infty$

holds, then system (16) is exponentially stabilizable on $\Omega$.

###### Proof.

The proof is similar to the proof of Theorem 3.1. ∎

Let $(\lambda^{k}_{p})_{p\geq 1}$ and $(\phi^{k}_{p})_{p\geq 1}$ be the
eigenvalues and the corresponding eigenfunctions of operator $A+BK$ on
$L^{2}(\Omega)$. If $A+BK$ is a symmetric uniformly elliptic operator, then
system (16) is strongly stabilizable on $\Omega$.

###### Proof.

The proof is similar to the proof of Theorem 3.1. ∎

Let us consider, on $\Omega=]0,1[$, the following fractional differential
system of order $\alpha=0.2$:

$\begin{cases}{}^{C}D_{t}^{0.2}z(x,t)=\displaystyle\frac{1}{100}\dfrac{\partial^{2}z}{\partial
x^{2}}(x,t)+\displaystyle\frac{1}{2}z(x,t)+BKz(x,t),&(x,t)\in\Omega\times]0,+\infty[,\\\
z(0,t)=z(1,t)=0,&\forall t>0,\\\ z(x,0)=z_{0},&x\in\Omega,\end{cases}$ (19)

with the linear bounded operator $B=I$ and where we take $K=-B^{*}=-I$. The
operator

$A+BK=\frac{1}{100}\dfrac{\partial^{2}}{\partial x^{2}}-\frac{1}{2},$

with spectrum given by the eigenvalues
$\lambda^{k}_{p}=-\frac{1}{2}-\frac{1}{100}(p\pi)^{2}$, $p\geq 1$, and the
corresponding eigenfunctions
$\phi^{k}_{p}(x)=\sqrt{\frac{2}{1+(p\pi)^{2}}}\cos(p\pi x)$, $p\geq 1$,
generates a $C_{0}$-semi-group $(S^{k}(t))_{t\geq 0}$ defined by

$S^{k}(t)z_{0}=\displaystyle\sum_{p=1}^{+\infty}e^{\lambda^{k}_{p}t}\langle
z_{0},\phi^{k}_{p}\rangle\phi^{k}_{p}.$

Furthermore, the solution of system (19) can be written as

$z(t)=S^{k}_{0.2}(t)z_{0}=\displaystyle\sum_{p=1}^{+\infty}E_{0.2}(\lambda^{k}_{p}t^{0.2})\langle
z_{0},\phi^{k}_{p}\rangle\phi^{k}_{p}.$

It is clear that $A+BK$ is a symmetric and uniformly elliptic operator. Hence,
from Theorem 3.2, we deduce that system (19) is strongly stabilizable on
$\Omega$, i.e., the system

$\begin{cases}{}^{C}D_{t}^{0.2}z(x,t)=\displaystyle\frac{1}{100}\dfrac{\partial^{2}z}{\partial
x^{2}}(x,t)+\displaystyle\frac{1}{2}z(x,t)+Bu(t),&(x,t)\in\Omega\times]0,+\infty[,\\\
z(0,t)=z(1,t)=0,&\forall t>0,\\\ z(x,0)=z_{0},&x\in\Omega,\end{cases}$

is strongly stabilizable by the feedback control $u(t)=-B^{*}z(t)$. Figure 2
shows, for $z(x,0)=x(x-1)$, that the state $z(x,t)$ of system (19) is unstable
at $t=0$. Moreover, we see that the state evolves close to 0 at $t=10$.
Numerically, the state is stabilized by $u(t)=-B^{*}z(t)$ with an error equal
to $1.75\times 10^{-04}$.

Figure 2: The state of system (19) for $z(x,0)=x(x-1)$, $t=0$, and $t=10$,
illustrating the fact that (19) is unstable at $t=0$ but it is stabilized at
$t=10$ on $\Omega=]0,1[$.

###  3.3 Decomposition Method

Now, we study the stabilization of system (16) using the decomposition method,
which consists in decomposing the state space and the system using the
spectral properties of operator $A$.

Let $\xi>0$ be fixed and assume that there are at most finitely many
nonnegative eigenvalues of $A$ and each with finite dimensional eigenspace. In
other words, assume there exists $l\in\mathbb{N}$ such that

$\sigma(A)=\sigma_{u}(A)\cup\sigma_{s}(A),$ (20)

where $\sigma_{u}(A)=\sigma(A)\cap\\{\lambda_{p},~{}~{}~{}p=1,2,\ldots,l\\}$,
$\sigma_{s}(A)=\sigma(A)\cap\\{\lambda_{p},~{}~{}~{}p=l+1,l+2\ldots\\}$ with
$\lambda_{l}\geq 0$ and $\lambda_{l+1}\leq-\xi$. Because the sequence
$(\phi_{p})_{p\geq 1}$ forms a complete and orthonormal basis in
$H=L^{2}(\Omega)$, it follows that the state space $H$ can be decomposed as

$H=H_{u}\oplus H_{s},$ (21)

where $H_{u}=PH=\mathrm{span}\\{\phi_{1},\phi_{2},\ldots,\phi_{l}\\}$ and
$H_{s}=(I-P)H=\mathrm{span}\\{\phi_{l+1},\phi_{l+2},\ldots\\}$ with
$P\in\mathcal{L}(H)$ the projection operator Kato (1966). Hence, system (16)
can be decomposed into the following two sub-systems:

$\begin{cases}{}^{C}D_{t}^{\alpha}z_{u}(t)=A_{u}z_{u}(t)+PBu(t),\\\
z_{0u}=Pz_{0},\end{cases}$ (22)

and

$\begin{cases}{}^{C}D_{t}^{\alpha}z_{s}(t)=A_{s}z_{s}(t)+(I-P)Bu(t),\\\
z_{0s}=(I-P)z_{0},\end{cases}$ (23)

where $A_{s}$ and $A_{u}$ are the restrictions of $A$ on $H_{s}$ and $H_{u}$,
respectively, and are such that $\sigma(A_{s})=\sigma_{s}(A)$,
$\sigma(A_{u})=\sigma_{u}(A)$, and $A_{u}$ is a bounded operator on $H_{u}$.

Our next result asserts that stabilization of system (16) is equivalent to the
one of system (22).

Let the spectrum $\sigma(A)$ of $A$ satisfy the above spectrum decomposition
assumptions (20) for some $\xi>0$ and $A_{s}$ be a symmetric uniformly
elliptic operator. If system (22) is strongly stabilizable by the control

$u(t)=D_{u}z_{u}(t)$ (24)

with $D_{u}\in\mathcal{L}(H,U)$ such that

$\Arrowvert z_{u}(t)\Arrowvert\leq C~{}t^{-\mu},\quad\mu,C>0,$ (25)

then system (16) is strongly stabilizable using the feedback control
$v(t)=D_{u}z_{u}(t)$.

###### Proof.

Using the fact that system (22) is strongly stabilizable by control (24), and
inequality (25) yields

$\Arrowvert z_{u}(t)\Arrowvert\longrightarrow 0\text{ as
}t\longrightarrow+\infty$ (26)

and

$\Arrowvert u(t)\Arrowvert\leq C\Arrowvert D_{u}\Arrowvert t^{-\mu},$ (27)

the unique weak solution of system (23) can be written in the space $H_{s}$ as

$z_{s}(t)=\sum_{p=l+1}^{+\infty}E_{\alpha}(\lambda_{p}t^{\alpha})\langle
z_{0s},\phi_{p}\rangle\phi_{p}+\sum_{p=l+1}^{+\infty}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}E_{\alpha,\alpha}(\lambda_{p}(t-s)^{\alpha})\langle(I-P)Bu(s),\phi_{p}\rangle\phi_{p}\,\mathrm{d}s$

since $A_{s}$ is a symmetric uniformly elliptic operator Sakamoto and Yamamoto
(2011). Using the spectrum decomposition relation (20), Lemma 2, and Lemma 2,
one has that

$E_{\alpha}(\lambda_{p}t^{\alpha})\leq E_{\alpha}(-\xi t^{\alpha})\quad\text{
for all }p\geq l+1$ (28)

and

$E_{\alpha,\alpha}(\lambda_{p}(t-s)^{\alpha})\leq
E_{\alpha,\alpha}(-\xi(t-s)^{\alpha})\quad\text{ for all }p\geq l+1.$ (29)

Then, feeding system (23) by the same control $u(t)=D_{u}z_{u}(t)$ and using
(27)–(29), it follows that

$\begin{split}\Arrowvert z_{s}(t)\Arrowvert&\leq E_{\alpha}(-\xi
t^{\alpha})\Arrowvert z_{0s}\Arrowvert+C\Arrowvert D_{u}\Arrowvert\Arrowvert
I-P\Arrowvert\Arrowvert
B\Arrowvert\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}s^{-\mu}E_{\alpha,\alpha}(-\xi(t-s)^{\alpha})\,\mathrm{d}s\\\
&\leq E_{\alpha}(-\xi t^{\alpha})\Arrowvert z_{0s}\Arrowvert+C\Arrowvert
D_{u}\Arrowvert\Arrowvert I-P\Arrowvert\Arrowvert
B\Arrowvert\displaystyle\sum\limits_{{n=1}}^{+\infty}\displaystyle\int_{0}^{t}\dfrac{(-\xi)^{n}(t-s)^{\alpha
n+\alpha-1}s^{-\mu}\,\mathrm{d}s}{\Gamma(\alpha n+\alpha)}\\\ &\leq
E_{\alpha}(-\xi t^{\alpha})\Arrowvert z_{0s}\Arrowvert+C\Arrowvert
D_{u}\Arrowvert\Arrowvert I-P\Arrowvert\Arrowvert
B\Arrowvert\displaystyle\sum\limits_{{n=1}}^{+\infty}\dfrac{(-\xi)^{n}t^{\alpha
n+\alpha-\mu}}{\Gamma(\alpha n+\alpha-\mu-1)\Gamma(1-\mu)^{-1}}\\\ &\leq
E_{\alpha}(-\xi t^{\alpha})\Arrowvert
z_{0s}\Arrowvert+C\Gamma(1-\mu)\Arrowvert D_{u}\Arrowvert\Arrowvert
I-P\Arrowvert\Arrowvert B\Arrowvert t^{\alpha-\mu}E_{\alpha,\alpha-\mu+1}(-\xi
t^{\alpha}).\end{split}$

Lemma 2 implies that

$\Arrowvert z_{s}(t)\Arrowvert\leq\frac{M_{1}}{1+\xi t^{\alpha}}\Arrowvert
z_{s0}\Arrowvert+C\Gamma(1-\mu)\Arrowvert D_{u}\Arrowvert\Arrowvert
I-P\Arrowvert\Arrowvert B\Arrowvert\frac{M_{2}t^{\alpha-\mu}}{1+\xi
t^{\alpha}}$

for some $M_{1},M_{2}>0$. Therefore,

$\Arrowvert z_{s}(t)\Arrowvert\longrightarrow 0\text{ as
}t\longrightarrow+\infty.$ (30)

On the other hand, we have that

$\Arrowvert z(t)\Arrowvert=\Arrowvert
z_{s}(t)+z_{u}(t)\Arrowvert\leq\Arrowvert z_{s}(t)\Arrowvert+\Arrowvert
z_{u}(t)\Arrowvert.$ (31)

Combining (26), (30), and (31), we deduce the strong stabilization of system
(16). ∎

##  4 Conclusions and Future Work

We investigated the stability problem of infinite dimensional time fractional
differential linear systems under Caputo derivatives of order
$\alpha\in(0,1)$, where the state space is the Hillbert space $L^{2}(\Omega)$.
We proved necessary and sufficient conditions for exponential stability and
obtained a characterization for the asymptotic stability, which is guaranteed
if the system dynamics is symmetric and uniformly elliptic. Moreover, some
stabilization criteria were also proved. Finally, we investigated the strong
stabilization of the system via a decomposition method, where an explicit
feedback control is obtained. Illustrative examples were given, showing the
effectiveness of the theoretical results. As future work, we intend to extend
our work to the class of infinite dimensional time fractional differential
nonlinear systems. Various other questions are still open and deserve further
investigations, such as, studying boundary stability and gradient stability
for time fractional differential linear systems or considering the more recent
notion of $\Lambda$-fractional derivative Lazopoulos and Lazopoulos (2019),
and thus obtaining a geometrical interpretation.

Each author equally contributed to this paper, read and approved the final
manuscript.

This research was funded by Moulay Ismail University (H.Z.); by Hassan II
Academy of Science and Technology, project N 630/2016 (A.B.); and by The
Portuguese Foundation for Science and Technology, R&D unit CIDMA, within
project UIDB/04106/2020 (D.F.M.T.).

###### Acknowledgements.

This research is part of first author’s Ph.D. project, which is carried out at
Moulay Ismail University, Meknes, and has began during a one-month visit of
Zitane to the R&D Unit CIDMA, Department of Mathematics, University of Aveiro,
Portugal, June 2019. The hospitality of the host institution is here
gratefully acknowledged. The authors are strongly grateful to three anonymous
referees for their suggestions and invaluable comments. The authors declare no
conflict of interest. References

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