Souad Chaabouni
email: souad.chaabouni@labri.fr

Jenny Benois-Pineau

Ofer Hadar
email: hadar@bgu.ac.il

Ben Amar

Deep Learning for Saliency Prediction in Natural Video

Keywords: Deep learning, saliency map, optical flow, convolution network, contrast features

The purpose of this paper is the detection of salient areas in natural video by using the new deep learning techniques. Salient patches in video frames are predicted first. Then the predicted visual fixation maps are built upon them. We design the deep architecture on the basis of CaffeNet implemented with Caffe toolkit. We show that changing the way of data selection for optimisation of network parameters, we can save computation cost up to 12 times. We extend deep learning approaches for saliency prediction in still images with RGB values to specificity of video using the sensitivity of the human visual system to residual motion. Furthermore, we complete primary colour pixel values by contrast features proposed in classical visual attention prediction models. The experiments are conducted on two publicly available datasets. The first is IRCCYN video database containing 31 videos with an overall amount of 7300 frames and eye fixations of 37 subjects. The second one is HOLLYWOOD2 provided 2517 movie clips with the eye fixations of 19 subjects. On IRCYYN dataset, the accuracy obtained is of 89.51%. On HOLLYWOOD2 dataset, results in prediction of saliency of patches show the improvement up to 2% with regard to RGB use only. The resulting accuracy of 76, 6% is obtained. The AUC metric in comparison of predicted saliency maps with visual fixation maps shows the

Introduction

Deep learning has emerged as a new field of research in machine learning, providing learning at multiple levels of abstraction for mining the data such as images, sound and text [START_REF] Deng | DEEP LEARNING: Methods and Applications[END_REF]. Although, it is hierarchically created usually on the basis of neural networks, deep learning presents a philosophy to model the complex relationships between data [START_REF] Lecun | Gradient-Based Learning Applied to Document Recognition[END_REF], [START_REF] Bruna | Invariant Scattering Convolution Networks[END_REF]. Since recently, deep learning has become the most exciting field which attracts many researchers. First, to understand the new deep networks in itself ( [START_REF] Jia | Caffe: Convolutional Architecture for Fast Feature Embedding[END_REF], [START_REF] Krizhevsky | ImageNet Classification with Deep Convolutional Neural Networks[END_REF], [START_REF] Bengio | Representation Learning: A Review and New Perspectives[END_REF], [START_REF] Glorot | Understanding the difficulty of training deep feedforward neural networks[END_REF], [START_REF] Schulz | Exploiting Local Structure in Boltzmann Machines[END_REF], [START_REF] Scherer | Accelerating Large-scale Convolutional Neural Networks with Parallel Graphics Multiprocessors, large scale machine learning :Parralelism and massive datasets[END_REF], [START_REF] Scherer | Evaluation of Pooling Operations in Convolutional Architectures for Object Recognition[END_REF],), such as the important question in building a deep convolutional network, is the optimization of pooling layer [START_REF] Avila | Pooling in image representation: The visual codeword point of view[END_REF]. Second, to use that deep network in their original domain such as object recognition [START_REF] Uetz | Large-scale Object Recognition with CUDAaccelerated Hierarchical Neural Networks[END_REF], [START_REF] Pinto | Beyond Simple Features: A Large-Scale Feature Search Approach to Unconstrained Face Recognition[END_REF], [START_REF] Simonyan | Two-Stream Convolutional Networks for Action Recognition in Videos[END_REF], [START_REF] Cheng | Silhouette Analysis for Human Action Recognition Based on Supervised Temporal t-SNE and Incremental Learning, Image Processing[END_REF], multi-task learning [START_REF] Uetz | Large-scale Object Recognition with CUDAaccelerated Hierarchical Neural Networks[END_REF]. As a definition, neural networks are generally multilayer generative networks formed to maximize the probability of input data with regard to target classes.

The predictive power of Deep Convolutional Neural Networks (CNN) is interesting for the use in the problem of prediction of visual attention in visual content, i.e. saliency of the latter. Indeed, several saliency models have been proposed in various fields of research such as psychology and neurobiology, which are based on the feature integration theory ( [START_REF] Treisman | A Feature-Integration Theory of Attention[END_REF], [START_REF] Bruce | Saliency based on information maximization[END_REF], [START_REF] Gao | On the plausibility of the discriminant center-surround hypothesis for visual saliency[END_REF], [START_REF] Gao | Integrated learning of saliency, complex features, and object detectors from cluttered scenes[END_REF], [START_REF] Hou | Dynamic visual attention: Searching for coding length increments[END_REF], [START_REF] Kanan | SUN: Top-down saliency using natural statistics[END_REF], [START_REF] Marat | Modelling spatiotemporal saliency to predict gaze direction for short videos[END_REF], [START_REF] Meur | Predicting visual fixations on video based on low-level visual features[END_REF], [START_REF] Torralba | 80 million tiny images: A large dataset for non-parametric object and scene recognition[END_REF], [START_REF] Seo | Static and space-time visual saliency detection by self-resemblance[END_REF], [START_REF] Boujut | Fusion of Multiple Visual Cues for Visual Saliency Extraction from Wearable Camera Settings with Strong Motion[END_REF], [START_REF] Itti | A Model of Saliency-Based Visual Attention for Rapid Scene Analysis[END_REF],..). These research models the so-called "bottom-up" saliency with the theory that suggests the visual characteristics of low-level as luminance, color, orientation and movement to provoke human gaze attraction [START_REF] Borji | State-of-the-art in Visual Attention Modeling[END_REF]. The "bottom-up" models have been extensively studied in the literature [START_REF] Borji | State-of-the-art in Visual Attention Modeling[END_REF]. They suffer from insufficiency of low-level features in the feature integration theory framework, especially when the scene contains significant content and semantic objects. In this case, the so-called "top-down" attention [START_REF] Pinto | Bottom-up and top-down attention are independent[END_REF] be-2 comes prevalent, the human subject observes visual content progressively with increasing the time of looking of the visual sequence. Supervised machine learning techniques help in detection of salient regions in images predicting attractors on the basis of seen data [START_REF] Vig | Large-Scale Optimization of Hierarchical Features for Saliency Prediction in Natural Images[END_REF]. Various recent research is directed towards the creation of a basic deep learning model ensuring the detection of salient areas.

We can cite here [START_REF] Shen | Learning to Predict Eye Fixations for Semantic Contents Using Multi-layer Sparse Network[END_REF], [START_REF] Simonyan | Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps[END_REF] and [START_REF] Vig | Large-Scale Optimization of Hierarchical Features for Saliency Prediction in Natural Images[END_REF]. While a significant effort has been already done for building such models from still images, very few models have been built for video content for saliency prediction with supervised learning [START_REF] Han | Spatial and temporal visual attention prediction in videos using eye movement data[END_REF]. It has a supplementary dimension: the temporality expressed by apparent motion in the image plane.

In this paper, we present a new approach with Deep CNN that ensures the learning of salient areas in order to predict the saliency maps in videos. The paper is organized as follows. Section 2 describes the related work of different deep learning models used to detect salient areas in images or to classify images by content. Section 3 presents our proposed method for detection of salient regions with a deep learning approach. Pixel-wise computation of predicted visual attention/saliency maps is then introduced. In section 4 we present results and comparison with reference methods of the state-of-the-art. Section 5 concludes the paper and outlines the perspectives of this research.

Related work

Deep learning architectures which have been recently proposed for the prediction of salient areas in images differ essentially by the quantity of convolution and pooling layers, by the input data, by pooling strategies, by the nature of the final classifiers and the loss functions to optimize, but also by the formulation of the problem. The attempt to predict visual attention reveals the binary classification problem of areas in images as "salient" and "non-salient". It corresponds to the visual experiment with free instructions, when the subjects are simply asked to look at the content. Shen [START_REF] Shen | Learning to Predict Eye Fixations for Semantic Contents Using Multi-layer Sparse Network[END_REF] proposed a deep learning model to extract salient areas in images. It allows firstly to learn the relevant char-acteristics of the saliency of natural images, and secondly to predict the eye fixations on objects with semantic content. The proposed model is formed by three layer sequences of "filtering" and "pooling", followed by a layer of linear SVM classifier providing ranked "salient" or "non-salient" regions of the input image. With the filtering by sparse coding and the max pooling, this model approximates human gaze fixations. In Simonyan's work [START_REF] Simonyan | Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps[END_REF] the saliency of image pixels is defined with regard to a given class in image taxonomy as a relevance of the image for the class.

Therefore the classification problem is multi-class, and can be expressed as a "task-dependent" visual experiment, where the subjects are asked to look for an object of a given taxonomy in the images. The creation of the saliency map for each class using deep CNN with optimisation of parameters by stochastic gradient descent, presents the challenge of this research [START_REF] Simonyan | Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps[END_REF]. After a step of generating the map that maximizes the score of the specific class, the saliency map of each class is defined by the amplitude of the weight calculated from the convolution network with a single layer.

The learning model of salient areas proposed by Vig [START_REF] Vig | Large-Scale Optimization of Hierarchical Features for Saliency Prediction in Natural Images[END_REF] In our work we also seek for predicting saliency of image regions for HVS.

While in [START_REF] Simonyan | Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps[END_REF] only primary RGB pixel values are taken for class-based saliency prediction, we use several combinations of primary (input) features such as residual motion and primary spatial features, inspired by feature integration theory as in [START_REF] Harel | Graph-Based Visual Saliency[END_REF], [START_REF] Itti | A Model of Saliency-Based Visual Attention for Rapid Scene Analysis[END_REF], [START_REF] Treisman | A feature-integration theory of attention[END_REF], [START_REF] Boujut | Fusion of Multiple Visual Cues for Visual Saliency Extraction from Wearable Camera Settings with Strong Motion[END_REF]. The sensitivity of HVS to residual motion in dynamic visual scenes is used for saliency prediction in video [START_REF] Boujut | Fusion of Multiple Visual Cues for Visual Saliency Extraction from Wearable Camera Settings with Strong Motion[END_REF]. For training of deep CNN in our two class classification problem we use human fixations maps as in [START_REF] Vig | Large-Scale Optimization of Hierarchical Features for Saliency Prediction in Natural Images[END_REF] to select positive and negative samples.

Prediction of visual saliency with deep CNN

Hence we design a deep CNN to classify regions in video frames into two classes salient and non-salient. Then on the basis of these classifications, a visual fixation map will be predicted. Before describing the architecture of our proposed deep CNN, we introduce the definition of saliency of regions and explain how we extract positive and negative examples for training the CNN.

Extraction of salient and non-salient patches.

We define a salient patch in a video frame on the basis of interest expressed by the subjects. The latter is measured by the magnitude of a visual attention map built upon gaze fixations which are recorded during a psycho-visual experiment in free-viewing conditions. The maps are built by the method of Wooding [START_REF] Wooding | Eye movements of large populations: II. Deriving regions of interest, coverage, and similarity using fixation maps[END_REF]. Such a map represents a multi-Gaussian surface normalized by its global maximum. To train the network it is necessary to extract salient and non-salient patches from training video frames with available Wooding maps. A squared patch P of parametrized size t × t is considered "salient" if the visual attention map W value in its center is above a threshold. A patch P is a vector in R t×t×n , where n stands for the quantity of primary feature maps serving as an input to the CNN. In case when RGB planes of a colour video sequences are used, n = 3.

The choice of the parameter t obviously depends on the resolution of video, but also is constrained by the computational capacity to process a huge amount of data. In this work we considered t = 100 for SD video. More formally, a binary label is associated with pixels X of each patch P i using equation (1):

l(X) =    1 if W (x 0,i , y 0,i ) ≥ τ J 0 otherwise (1)
with (x 0,i , y 0,i ) the coordinates of the center of the patch. We select a set of thresholds, starting by the global maximum value of the normalized attention map and then relax threshold as in equation( 2):

   τ 0 = max(W (x, y), 0) τ (j+1) = τ j -τ j (2)
Here 0 < < 1 is a relaxation parameter, j = 0, • • • , J, and J limits the relaxation of saliency. It was chosen experimentally as J = 5, while = 0.04.

In such a manner, salient patches are progressively selected up to non-salient areas, where non-salient patches are extracted randomly. The process of extraction of salient patches in the frames of training videos is illustrated in figure 1.

The tables 1, 2 present the group of salient patches on the left and nonsalient patches on the right, each row presents some examples of patches taken from each frame of video sequence denoted by "SRC" in IRCCYN1 dataset, and "actioncliptrain" in the HOLLYWOOD 2 data set.

Primary feature maps for saliency prediction in video

On the contrary to still natural images where saliency is "spatial", based on color contrasts, saturation contrasts, intensity contrasts • • • , the saliency of the video is also based on the motion information of the objects with regard to the background. Therefore, in the following we present primary motion features we consider and then briefly describe spatial primary features (colours, contrasts) we use.

Motion feature maps

Visual attention is not attracted by the motion in general, but by the difference between the global motion in the scene, expressing the camera work, and the "local" motion, that one of a moving object. This difference is called the "residual motion" [START_REF] Marat | Modelling spatiotemporal saliency to predict gaze direction for short videos[END_REF]. To create the feature map of residual motion in videos, we used the model developed in [START_REF] Brouard | Cartes de saillance spatio-temporelle basées contrastes de couleur et mouvement relatif[END_REF], [START_REF] Marat | Modelling spatiotemporal saliency to predict gaze direction for short videos[END_REF], [START_REF] González-Díaz | Fusion of Multiple Visual Cues for Object Recognition in Videos[END_REF]. This model allows the calculation of the residual motion in three steps: the optical flow estimation -→ M c (x, y), the estimation of the global motion -→ M θ (x, y), from optical flow accordingly to the first order complete affine model θ and finally, the computation of residual motion according to equation(3):

-→ M r (x, y) = -→ M θ (x, y) - -→ M c (x, y) (3)
The sensitivity of HVS to motion is selective. Daly [START_REF] Daly | Engineering observations from spatiovelocity and spatiotemporal visual models[END_REF] proposes a non-linear model of sensitivity accordingly to the speed of motion. In our work, we use a simplified version: the primary motion feature is the magnitude of residual motion (3) in a given pixel, and leave the decision on the saliency of the patch to the CNN classifier. For spatial primary features we resort to the work in [START_REF] Brouard | Cartes de saillance spatio-temporelle basées contrastes de couleur et mouvement relatif[END_REF] which yeilds coherent results accordingly to our studies in [START_REF] Boujut | Fusion of Multiple Visual Cues for Visual Saliency Extraction from Wearable Camera Settings with Strong Motion[END_REF].

Primary spatial features

The choice of features from [START_REF] Brouard | Cartes de saillance spatio-temporelle basées contrastes de couleur et mouvement relatif[END_REF] is conditioned by relatively low computational cost and their good performance we have stated in [START_REF] Boujut | Fusion of Multiple Visual Cues for Visual Saliency Extraction from Wearable Camera Settings with Strong Motion[END_REF]. The authors propose seven color contrast descriptors. As the color space 'Hue Saturation Intensity' (HSI) is more appropriate to describe the perception and color interpretation by humans, the descriptors of the spatial saliency are built in this color space. Five of these seven local descriptors depend on the value of the hue, saturation and/or intensity of the pixel. These values are determined for each frame I of a video sequence, from a saturation factor f sat and an intensity factor f int , calculated using the equations ( 4),( 5):

f sat (I, i, j) = Sat(I, i) + Sat(I, j) 2 × (k min + (1 -k min ) • Sat(I, i)) (4)
f int (I, i, j) = Int(I, i) + Int(I, j) 2 × (k min + (1 -k min ) • Int(I, i)) (5)
Here Sat(I, i) is the saturation of the pixel i at coordinates (x i , y i ) and the value at Sat(I, j) is the saturation of the pixel at coordinates (x j , y j ) adjacent to the pixel i. The constant k min = 0, 21 sets the minimum value for the protection of the interaction of pixel i when the saturation approaches zero [START_REF] Brouard | Cartes de saillance spatio-temporelle basées contrastes de couleur et mouvement relatif[END_REF]. Contrast descriptors are calculated by equations (6. . . 13):

1. color contrast: the first input of the saliency of a pixel is obtained from the two factors of saturation and intensity. This descriptor X 1 (I, i) is calculated for each pixel i and its eight connected neighbors j of the frame I, as in equation( 6):

X 1 (I, i) = j∈ηi f sat (I, i, j) • f int (I, i, j) (6)
2. hue contrast: a hue angle difference on the color wheel can produce a contrast. In other words, this descriptor is related to the pixels having a hue value far from their neighbors (the largest angle difference value is equal to 9

X 2 (I, i) = j∈ηi f sat (I, i, j) • f int (I, i, j) • ∆ hue (I, i, j) (7)
The difference in color ∆ hue between the pixel i and its neighbor j = 8 is calculated accordingly to equations ( 8) and ( 9) :

∆ hue =      ∆ µ (I, i, j) if ∆ µ (I, i, j) ≤ 0.5 1 -∆ µ (I, i, j) else (8) ∆ µ (I, i, j) = |Hue(I, i) -Hue(I, j)| (9)
3. contrast of opponents: the colors located on the opposite sides of the hue wheel creating a very high contrast. An important difference in tone level will make the contrast between active color (hue < 0, 5 180 • ) and passive, more salient. This contribution to the salience of the pixel i is defined by equation [START_REF] Scherer | Evaluation of Pooling Operations in Convolutional Architectures for Object Recognition[END_REF]:

     X 3 (I, i) = j∈ηi f sat (I, i, j) • f int (I, i, j) • ∆ hue (I, i, j)
if Hue(I, i) < 0.5 and Hue(I, j) ≥ 0.5 [START_REF] Scherer | Evaluation of Pooling Operations in Convolutional Architectures for Object Recognition[END_REF] 4. contrast of saturation: occurs when low and high color saturation regions are close. Highly saturated colors tend to attract visual attention, unless a low saturation region is surrounded by a very saturated area. It is defined by equation ( 11):

X 4 (I, i) = j∈ηi f sat (I, i, j) • f int (I, i, j) • ∆ sat (I, i, j) (11)
with ∆ sat denoting the saturation difference between the pixel i and its 8 neighbor j, see equation ( 12):

∆ sat (I, i, j) = |Sat(I, i) -Sat(I, j)| (12)
5.contrast of intensity: a contrast is visible when dark colors and shiny ones coexist. The bright colors attract visual attention unless a dark region is completely surrounded by highly bright regions. The contrast of intensity is defined by equation ( 13):

X 5 (I, i) = j∈ηi f sat (I, i, j) • f int (I, i, j) • ∆ int (I, i, j) (13)
With ∆ int denotes the difference of intensity between the pixel i and its 8 neighbor j

∆ int (I, i, j) = |Int(I, i) -Int(I, j)| (14)
6. dominance of warm colors: the warm colors -red, orange and yelloware visually attractive. These colors (hue < 0.125 45 • ) are still visually appealing, although the lack of contrast (hot and cold colors in the area) is observed in the surroundings. This feature is defined by equation ( 15):

V 6 (I, i) =      Sat(I, i) • Int(I, i) if 0 ≤ Hue(I, i) < 0.125 0 otherwise (15)
7. dominance of brightness and saturation: highly bright, saturated colors are considered attractive regardless of their hue value. The feature is defined by equation ( 16):

V 7 (I, i) = Sat(I, i) • Int(I, i) (16)
The normalization

(V 1•••5 (I, i) = X1•••5 |ηi| ) of the first five descriptors (X 1•••5
) by the number of neighboring pixels (|η i | = 8) is performed. In [START_REF] Boujut | Fusion of Multiple Visual Cues for Visual Saliency Extraction from Wearable Camera Settings with Strong Motion[END_REF], [START_REF] Daly | Engineering observations from spatiovelocity and spatiotemporal visual models[END_REF] it is reported that mixing a large quantity of different features increases the performance of prediction. This is why it is attractive to mix primary features (1-7) with those which have been used in previous works of saliency prediction [START_REF] Simonyan | Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps[END_REF], that is simple RGB planes of a video frame.

The network design

In this section we present the architecture of a deep CNN we designed for our two class classification problem: prediction of a saliency of a patch in a given video frame. It includes five layers of convolution, three layers of pooling, five layers of Rectified Linear Units (RELU), two normalisation layers, and one layer of Inner product followed by a loss layer as illustrated in Figure 2. The final classification is ensured by a soft-max classifier in equation ( 17). This function is a generalization of the logistic function that compresses a vector of arbitrary real values of K dimension to a vector of the same size but with actual values in the range (0, 1).

f (x i ) = e xi j e xj (17)
Figure 3 shows the order of layers in our proposed network. The CNN architecture was implemented using the Caffe software [START_REF] Jia | Caffe: Convolutional Architecture for Fast Feature Embedding[END_REF]. We created our network architecture made on three patterns (see figure 3 with a step of normalisation between each one. Each pattern contains a linear/nonlinear cascading operation (convolution, pooling, RELU). For the first pattern we chose a cascading operation different than the two following patterns.

The first operation cascade is represented as the succession of convolution layer, pooling layer followed by a RELU layer. In fact, the applying of the pooling operation before the RELU layer does not change the final results because the two layers compute the function of maximum, however, it ensures the decrease of the execution time of the prediction as the step of pooling reduces the number of nodes. The two convolution layers stacked before the pooling layer for the followed pattern ensures the development of more complex features that will be more "expressive" before the destructive Pool operation. In the following, we will describe the most crucial layers which are convolution, pooling and local response normalisation.

Convolution layers

In order to extract the most important information for further analysis or exploitation of image patches, the convolution with a fixed number of filters that is based on the natural functioning of the HVS is needed. It is necessary to determine the size of the convolution kernel to be applied to each pixel of the input image to highlight areas of the image. Gaussian filters were used to create all of the feature maps of the convolution layer. The number of filters, in other words the number of kernels, convolved with the input image is the number of the obtained feature maps. Three stages are conceptually necessary to create the convolution layer. The first refers to the convolution of the input image by linear filters. The second is to add a bias term. And finally, the application of a nonlinear function (here we have used the rectified linear function f (x) = max(x, 0)). Generally, the equation of convolution can be written as [START_REF] Gao | On the plausibility of the discriminant center-surround hypothesis for visual saliency[END_REF]:

X l j = f ( i∈Mj X l-1 i * ω l ij + B l j ) (18)
with X l j : the activity of the unit j according to the layer l, X i represents a selection of the input feature maps, B l j is the additive bias of the unit j in the features maps of the layer l, ω l ij : presents the synaptic weights between unit j of the layer l and l -1.

Pooling layers

To reduce the computational complexity for the upper layers, and provide a form of translation invariance, pooling summarizes the outputs of neighboring groups of neurons on the same kernel map. The size of the region of 'pooling' reduces the size of each feature map as input by the acquisition of a value for each region. We use max-pooling, see equation ( 19):

h n j (x, y) = max x,ȳ∈N h n-1 j (x, ȳ) (19)
Here N denotes the neighbourhood of (x,y).

Local response normalization layers

LRN layer normalizes values of feature maps which are calculated through the neurons having unbounded activations to detect the high-frequency characteristics with a high response of the neuron, and to amortize answers that are uniformly greater in a local area. The output computation is presented is presented in equation 20:

f (U x,y f ) = U x,y f (1 + α N 2 min(S,x-[N/2]+N ) x =max(0,x-[N/2]) min(S,y-[N/2]+N ) y =max(0,y-[N/2]) (U x ,y f ) 2 ) β (20)
Here U x,y f represents the value of the feature map at (x, y) coordinates and the sums are taken in the neighbourhood of (x, y) of size N ×N , α and β regulate normalisation strength.

Training and validation of the model

To solve the learning problem and to validate the network with the purpose to generate a robust model of salient area recognition, the solver of Caffe [START_REF] Jia | Caffe: Convolutional Architecture for Fast Feature Embedding[END_REF] is iteratively optimizing the network parameters in forward-backward loop. The optimisation method used is that one of stochastic gradient. The parameterization of the solver requires setting the learning rate and the number of iterations at training and testing steps.

The numbers of training and testing iterations are defined according to the "batch size" parameter of Caffe [START_REF] Jia | Caffe: Convolutional Architecture for Fast Feature Embedding[END_REF]. The batch size presents the number of images that is salient and non-salient patches in our case, processed at an iteration.

This number depends on two parameters:

• The power of the GPU/RAM of the used machine,

• The number of patches available for each database.

The number of iterations is computed according to equation ( 21

Generation of a pixel-wise saliency map

The designed and trained Deep CNN predicts for a patch in a video frame if it is salient for a human observer. Despite the interest of this problem for selection of important areas in images for further pattern recognition tasks, for finer, pixel-wise saliency prediction in video, the transformation of sparse classifier responses into a dense predicted saliency map is needed. The response for each patch is given by the soft-max classifier, see figure 2 and equation ( 17) in section 3.3. The value of classifier which is interpreted as a probability to belong to the saliency class, can be considered as a predicted saliency of a patch. Then a Gaussian is centred on the patch center with a pick value of 10f (i) 2πσ 2 with the spread parameter σ chosen of a half-size of the patch. Hence a sparse saliency map is predicted. In order to densify the map we classify densely sampled patches with a half-patch overlap and then interpolate obtained values.

Examples of predicted saliency maps using RGB only features (3K model), RGB features and Residual motion features(4Kmodel), Wooding gaze-fixation maps and popular saliency prediction models of Itti [START_REF] Itti | A Model of Saliency-Based Visual Attention for Rapid Scene Analysis[END_REF](named "GBVS") and

Harell [START_REF] Harel | Graph-Based Visual Saliency[END_REF](named "SignatureSal") are depicted in table 3. Visual evaluation of the maps shows that the proposed method yields maps more similar to Wooding maps built on gaze fixations. Indeed GBVS and SignatureSal are pixel-wise maps, while our maps are built upon salient patches. Further evaluation will be presented in the next section 4.

#f rame78 #f rame122 #f rame166 #f rame217 #f rame253 #f rame298 #f rame342 #f rame386 #f rame430 #f rame474 #f rame518 #f rame571 #f rame603 #f rame650
4. Experiments and results

Datasets

To learn the model, we have used two different datasets, the IRCCYN [START_REF] Boulos | Region-of-Interest Intra Prediction for H.264/AVC Error Resilience[END_REF] and the HOLLYWOOD [41] [42].

IRCCYN database contains 31 SD videos and gaze fixations of 37 subjects.

Evaluation of patches prediction with deep CNN

The network was implemented using a powerful graphic card Tesla K40m and processor (2 × 14 cores). Therefore a sufficiently large amount of patches, 256, was used per iteration, see the batch size parameter in equation [START_REF] Kanan | SUN: Top-down saliency using natural statistics[END_REF]. After a fixed number of training iterations, a model validation step is implemented.

335

At this stage the accuracy of the model at the current iteration is computed.

F irst experiment. To evaluate our deep network and to prove the importance of the addition of the residual motion map, we have created two models with the same parameter settings and architecture of the network: the first one contained R, G and B, primary pixel values in patches. We denote it as 340 DeepSaliency3k. The DeepSaliency4k presents the model using RGB and the normalized magnitude of residual motion as input data. The following figures 9 and 10 illustrate the variations of the accuracy along iterations of the both models 3k and 4k for each used database "IRCCYN" and "HOLLYWOOD". 71.6% ± 0.018 73.2% ± 0.020

In the IRCCYN database, we found a higher accuracy with both models used. The maximum value of accuracy obtained on the IRCCYN dataset is 89.5% at the iteration 15000 with the 3k model and 89.2% at the iteration 13000 on the 4k model, see table 4. We can explain the not improvement of the accuracy by the low number of videos in the IRCCYN dataset [START_REF] Srivastava | Dropout: A Simple Way to Prevent Neural Networks from Overfitting[END_REF].

For the HOLLYWOOD database, adding residual motion map improves the accuracy with almost 2% on the 4k model compared to the 3k model. The resulting accuracy of our proposed network along a fixed number of iterations shows the interest of adding the residual motion as a new feature together with spatial feature maps R, G and B. Nevertheless, the essential of accuracy is obtained with purely spatial features(RGB). This is why we add spatial contrast features which have been proposed in classical visual saliency prediction framework [START_REF] Brouard | Cartes de saillance spatio-temporelle basées contrastes de couleur et mouvement relatif[END_REF] in the second experiment below.

Second experiment. The second experiment for saliency prediction is conducted when limiting the maximal number of iterations to prevent us from falling into overfitting problem. Instead of increasing the number of training iterations with a limited number of data samples before each validation iteration, as this is the case in the work of [START_REF] Krizhevsky | ImageNet Classification with Deep Convolutional Neural Networks[END_REF], we pass all the training set before the validation of the parameters and limit the maximal number of iterations in the whole training process. This drastically decreases (12 times approximately)

the training complexity, without the loss of accuracy (see tables 4 and 5 for 3k and 4k models). In order to evaluate the performance of contrast features in a deep learning spatio-temporal model, we test the "8K" model first. Its input layers are composed of 7 contrast features, as described in section 3.2.2 and of the residual motion map. The results are presented in table 5 and illustrated in figure 11. It can be seen, that contrasts features only combined with motion 370 yield poorer performance with regard to 3K and 4K models. Therefore, we keep primary colour information in the further HSV8K and RGB8K models.

First Experiment Second Experiment

VideoName Gaze-fix vs Deep3k Gaze-fix vs Deep4k Gaze-fix vs Deep3k Gaze-fix vs Deep4k clipTest1 0, 58612 ± 0, 19784 0, 61449 ± 0, 17079 0, 55641 ± 0, 20651 0, 77445 ± 0, 14233 clipTest56 0, 74165 ± 0, 17394 0, 75911 ± 0, 12509 0, 65480 ± 0, 19994 0, 82034 ± 0, clipTest105 0, 35626 ± 0, 33049 0, 74312 ± 0, 19479 0, 66285 ± 0, 20553 0, 74740 ± 0, ClipTest200 0, 50643 ± 0, 241466 0, 59407 ± 0, 20188 0, 53926 ± 0, 21976 0, 69309 ± 0, ClipTest250 0, 548647 ± 0, 240311 0, 754679 ± 0, 15476 0, 41965 ± 0, 28409 0, 72621 ± 0, ClipTest300 0, 58236 ± 0, 22632 0, 66156 ± 0, 16352 0, 33808 ± 0, 19672 0, 79186 ± 0, 09732 ClipTest350 0, 67679 ± 0, 29777 0, 739803 ± 0, 16859 0, 47971 ± 0, 40607 0, 80467 ± 0, 15750 ClipTest500 0, 58351 ± 0, 20639 0, 75242 ± 0, 15365 0, 36761 ± 0, 36777 0, 82230 ± 0, 15196 ClipTest704 0, 59292 ± 0, 18421 0, 68858 ± 0, 16278 0, 46192 ± 0, 21286 0, 76831 ± 0, ClipTest752 0, 41710 ± 0, 11422 0, 63240 ± 0, 16870 0, 25651 ± 0, 25830 0, 58621 ± 0, ClipTest803 0, 67961 ± 0, 24997 0, 82489 ± 0, 14023 0, 55019 ± 0, 18646 0, 87474 ± 0, ClipTest849 0, 39952 ± 0, 31980 0, 67103 ± 0, 20623 0, 30190 ± 0, 27491 0, 81148 ± 0, Gaze-fix vs DeepHSV8k clipTest1 0, 55641 ± 0, 20651 0, 77445 ± 0, 14233 0, 58518 ± 0, 17991 0, 725073 ± 0, 168168 0, 76923 ± 0, 09848 clipTest56 0, 65480 ± 0, 19994 0, 82034 ± 0, 12727 0, 78106 ± 0, 090992 0, 82244 ± 0, 07295 0, 81651 ± 0, 06100 ClipTest105 0, 66285 ± 0, 20553 0, 74740 ± 0, 14689 0, 71597 ± 0, 11538 0, 63652 ± 0, 22207 0, 81365 ± 0, 08808 ClipTest200 0, 53926 ± 0, 21976 0, 69309 ± 0, 16428 0, 74225 ± 0, 19740 0, 77948 ± 0, 17523 0, 68396 ± 0, 17425 ClipTest250 0, 41965 ± 0, 28409 0, 72621 ± 0, 15028 0, 51697 ± 0, 21393 0, 84299 ± 0, 10787 0, 69886 ± 0, 13633 ClipTest300 0, 33808 ± 0, 19672 0, 79186 ± 0, 09732 0, 79265 ± 0, 10030 0, 74878 ± 0, 12161 0, 83009 ± 0, 08418 ClipTest350 0, 47971 ± 0, 40607 0, 80467 ± 0, 15750 0, 78924 ± 0, 16506 0, 72284 ± 0, 16996 0, 80009 ± 0, 232312 ClipTest500 0, 36761 ± 0, 36777 0, 82230 ± 0, 15196 0, 68157 ± 0, 15676 0, 85621 ± 0, 16137 0, 88067 ± 0, 09641 ClipTest704 0, 46192 ± 0, 21286 0, 76831 ± 0, 11186 0, 80725 ± 0, 11455 0, 78256 ± 0, 09523 0, 79551 ± 0, 071867 ClipTest752 0, 25651 ± 0, 25830 0, 58621 ± 0, 21568 0, 78029 ± 0, 08851 0, 59356 ± 0, 17804 0, 76665 ± 0, 07837 ClipTest803 0, 55019 ± 0, 18646 0, 87474 ± 0, 06946 0, 84338 ± 0, 06868 0, 88170 ± 0, 10827 0, 85641 ± 0, 06181 ClipTest849 0, 30190 ± 0, 27491 0, 81148 ± 0, 10363 0, 70777 ± 0, 08441 0, 91089 ± 0, 05217 0, 71224 ± 0, 07434

Table 8: The comparison of AUC metric gaze fixations 'Gaze-fix' vs predicted saliency 'GBVS', 'SignatureSal' and 'Seo') and our DeepRGB8k for the videos from HOLLYWOOD dataset VideoName Gaze-fix vs GBVS Gaze-fix vs SignatureSal Gaze-fix vs Seo Gaze-fix vs DeepRGB8k clipTest1 0, 81627 ± 0, 10087 0, 69327 ± 0, 13647 0, 50090 ± 0, 06489 0, 725073 ± 0, 168168 clipTest56 0, 76594 ± 0, 11569 0, 75797 ± 0, 08650 0, 64172 ± 0, 11630 0, 82244 ± 0, 07295 clipTest105 0, 63138 ± 0, 16925 0, 57462 ± 0, 13967 0, 54629 ± 0, 12330 0, 63652 ± 0, 22207 clipTest200 0, 75904 ± 0, 17022 0, 87614 ± 0, 10807 0, 65675 ± 0, 13202 0, 77948 ± 0, 17523 clipTest250 0, 74555 ± 0, 09992 0, 69339 ± 0, 11066 0, 47032 ± 0, 10193 0, 84299 ± 0, 10787 clipTest300 0, 82822 ± 0, 11143 0, 81271 ± 0, 12922 0, 75965 ± 0, 13658 0, 74878 ± 0, 12161 clipTest350 0, 65136 ± 0, 16637 0, 68849 ± 0, 249027 0, 57134 ± 0, 12408 0, 72284 ± 0, 16996 clipTest500 0, 82347 ± 0, 13901 0, 84531 ± 0, 15070 0, 75748 ± 0, 15382 0, 85621 ± 0, 16137 ClipTest704 0, 80168 ± 0, 08349 0, 85520 ± 0, 06826 0, 57703 ± 0, 07959 0, 78256 ± 0, 09523 ClipTest752 0, 73288 ± 0, 17742 0, 54861 ± 0, 15555 0, 71413 ± 0, 13138 0, 59356 ± 0, 17804 ClipTest803 0, 86825 ± 0, 106833 0, 87556 ± 0, 06896 0, 73847 ± 0, 14879 0, 88170 ± 0, 10827 ClipTest849 0, 75279 ± 0, 15518 0, 91888 ± 0, 07070 0, 55145 ± 0, 12245 0, 91089 ± 0, 05217

Proposed DeepRGB8K saliency model turns to be winner more systmatically (6/12 clips) than each reference model.

Discussion

Visual saliency prediction with deep CNN is still a recent while intensive research. The major bottle-neck in it is the computation power and memory requirements. We have shown, that a very large amount of iterations -hundreds of thousands are not needed for prediction of interesting patches in video frames. Indeed, to get better maximal accuracy smaller amount of iterations is needed, and the maximal number of iterations can be limited (17400 in our When applying a supervised learning approach to visual saliency prediction in video, one has to keep in mind that gaze-fixation maps, which serve for selection of training "salient" regions in video frames, not only express the "bottom-up" attention. Humans are attracted by stimuli, but in case of video when understanding a visual scene with time, they focus on the objects of interest, thus reinforcing the "top-down" mechanisms of visual attention [START_REF] Han | Spatial and temporal visual attention prediction in videos using eye movement data[END_REF].

Hence, the prediction of patches of interest by a supervised learning, we mix all mechanisms: bottom-up and top-down.

In order to re-inforce the bottom-up sensitivity of HVS to contrasts, we completed the input data layers by specific contrast features well studied in classical saliency prediction models. As we could not state the improvement of performance in prediction of saliency of patches in video frames in average (see table 5) a more detailed experience clip -by-clip was performed on a sample of clips from HOLLYWOOD dataset when comparing resulting predicted saliency maps. This series of experiments resumed in table 9, shows that indeed adding features, expressing local color contrast slightly improves performances with regard to the reference bottom-up spatial (GBVS, SignatureSal) and spatiotemporal models (Seo)). Hence, the mean improvement of the complete model with motion, contrast features and primary HSV colour pixel values with regard to Itti, Harell and Seo models are 0.00677, 0.01560, 0.15862 respectively. 1614 0, 00677 ± 0, 16922 0, 01560 ± 0, 19025 0, 15862 ± 0, 21036

Conclusion

Hence, in this paper, we proposed a deep convolutional network to predict salient areas (patches) in video content and built dense predicted visual saliency maps upon them. We built an adequate architecture on the basis of Caffe CNN.

While the aspiration of the community consisted in the use of primary features such as RGB planes only for visual attention prediction in images, we have shown that for video, adding of features expressing sensitivity of the human visual system to residual motion, is important. Furthermore, we also completed the RGB pixel values by low-level features of contrast and colour which are easy to compute and have proven efficient in former spatio-temporal predictors of visual attention. The results are better, nevertheless, the gain is not strong.

Therefore, it is clear that for further research it is important to better explore the link between known physiological mechanisms of human vision and the design of a CNN. The central bias hypothesis namely needs to be explored.

  tackles prediction of saliency of pixels for a human visual system (HVS) and corresponds to a free-viewing visual experience. It comprises two phases. First, a random bank of uniform filters is used to generate multiple representations of localized input images. The second phase provides the combination of different localized representations. The training step is summarized by the random token, from the combined representation of each image, of regions composed of ten pixels, and granted to each region a saliency class by reference to the density fixations map. The integration of this set in a SVM classifier allows the creation of the learning model. The learning model of salient areas is composed by the SVM trained on the combination of feature maps that are obtained using of different architectures of deep network.

Figure 1 :

 1 Figure 1: Extraction of salient patches for training

Figure 2 :

 2 Figure 2: Architecture and design of the deep saliency framework.

Figure 3 :

 3 Figure 3: Architecture of video saliency convolution network

  ): iterations numbers = epochs × T otal images number batch size[START_REF] Kanan | SUN: Top-down saliency using natural statistics[END_REF] here batch size represents the number of images for each network switching, epochs presents how many times the totality of the dataset is switched by the network.It is interesting to visualize the purely spatial features computed by the designed CNN in case when the network is configured to predict saliency only with primary RGB values as this it the goal instead of aspiration of the overall deep learning approach to saliency prediction. As the feature integration theory states, the HVS is sensitive to orientations and contrasts. This is what we observe in features going through layers of the network. The output of convolution layers (see figures 4, 5 and 6) yields more and more contrasted and structured patterns. In this figure convi and convii stands for consecutive convolution layers without pooling layers in between.

Figure 4 :

 4 Figure 4: (a) Input patch, (b) the output of first convolution layer and (c) the output of the first pooling layer.

Figure 5 :

 5 Figure 5: The output of the 2nd convolution layer data of ' Conv2' and 'Conv22' .

Figure 6 :

 6 Figure 6: The output of third convolution layer ' Conv3' and 'Conv33'.

From the overall set

  of 7300 frames, we have extracted 6837 salient patches and 6741 non-salient patches. We have used 10850 patches (5454 were salient and 5396 were non-salient) at the training step. For the testing step we have used 2728 patches (1383 salient patches and 1345 non-salient ones) respectively. The HOLLYWOOD database contains 823 training videos and 884 video for the validation step. The number of subjects with recorded gaze fixations varies according to each video up to 19 subjects. The spatial resolution of videos varies as well. The distribution of resolutions is presented in figures 7 and 8). In another terms the HOLLYWOOD dataset contains 229825 frames for training and 257733 frames for testing. From the frames of training step we have extracted 222863 salient patches and 221868 non-salient patches. During the testing phase, we have used 251294 salient patches and 250169 non-salient patches respectively.

Figure 7 :

 7 Figure 7: Histogram of video resolutions (W × H) of "HOLLYWOOD" database in training step.

Figure 8 :

 8 Figure 8: Histogram of video resolutions (W × H) of "HOLLYWOOD" database in testing step.

Figure 9 :

 9 Figure 9: Accuracy vs iterations of the both models 3k and 4k for "IRCCYN" database.

Figure 10 :

 10 Figure 10:Accuracy vs iterations of the both models 3k and 4k for "HOLLYWOOD" database.

Figure 11 :

 11 Figure 11: Second experiment: Learning of contrast feature -Accuracy vs iterations of 3k, 4k, 8k, RGB8k and HSV8k for "HOLLYWOOD" database.

4. 3 .

 3 Evaluation of predicted visual saliency mapsIn the literature, various evaluation criteria were used to determine the level of similarity between visual attention maps and gaze fixations of subjects like the normalized scanpath saliency 'NSS', Pearson Correlation Coefficient 'PCC', and the area under the ROC curve 'AUC' [44][START_REF] Engelke | Comparative study of fixation density maps[END_REF]. The Area under the ROC Curve measures the precision and accuracy of a system with the goal of categorizing entities into two distinct groups based on their features. The image pixels may belong either to the category of pixels fixated by subjects, either to the category of pixels that are not fixated by any subject. More the area is large, more the curve deviates from the line of the random classifier (area 0.5) and approaches to the ideal bend of the classifier (area 1.00). A value of AUC close to 1 indicates a correspondence between the predicted saliency and the eye positions. While a value close to 0.5 presents a random generation of the salient areas by the model computing the saliency maps. Therefore the objective and subjective saliency differs strongly. In our work, visual saliency being predicted by a deep CNN classifier, we have computed the hybrid AUC metric between predicted saliency maps and gaze-fixations as in[START_REF] Le Meur | Methods for comparing scanpaths and saliency maps: strengths and weaknesses[END_REF]. The results of the experiments are presented in the tables 6 and 7 below on an arbitrary chosen subset of 12 videos from HOLLYWOOD dataset. The figures depicted in the tables correspond to the maximum value obtained during the training and validation (as presented in tables 4 and 5

  case) accompanied by another data selection strategy: all data from training set are passed before each validation iteration of the learning, see tables 4, 5. Next, we have shown that in case of a sufficient training set, adding primary motion features improves prediction accuracy up to 2% in average on a very large data set (HOLLYWOOD test) containing 257733 video frames. Hence the deep CNN captures the sensitivity of Human Visual System to motion.

Table 1 :

 1 Training data from IRCCYN data base

	Salient patch #SRC {1..6}	Non-salient patch #SRC {1..6}

Table 2 :

 2 Training data from HOLLYWOOD data base

Salient patch #actioncliptrain {1,2,11,105,171,596} Non-salient patch #actioncliptrain {1,2,11,105,171,596}

Table 3 :

 3 Different saliency map of testing frame from 576i50 videos of IRCCYN database.

	nbr f rame	Frame	Wooding	Deep3k	Deep4k	GBVS	SignatureSal
	#f rame34

Table 4 :

 4 The accuracy results on IRCCYYN and HOLLYWOOD dataset in the first experi-

	ment
			3k model	4k model
		min (#iter)	50.9% (#0)	50.7% (#0)
	IRCCYN	max (#iter)	89.5% (#15000) 89.2% (#13000)
		avg ± std	86.3% ± 0.075 85.9% ± 0.075
		min (#iter)	50.1% (#0)	49.8% (#0)
	HOLLYWOOD	max (#iter)	74.8% (#3000)	76.6% (#3000)

avg ± std

Table 5 :

 5 The accuracy results on HOLLYWOOD dataset during the second experiment

		3k model	4k model	8k model	RGB8k model	HSV8k model
	min (#iter)	49.8% (#0)	55.6% (#0)	49.8% (#0)	50.1% (#0)	50.1% (#0)
	max (#iter)	75.1% (#5214)	76.6% (#5214)	72.9% (#12166)	76.9% (#5214)	73.5% (#3476)
	avg ± std	71.6% ± 0.072	73.6% ± 0.060	70.1% ± 0.067	73.5% ± 0.078	70.5% ± 0.068

  ). For the first experiment the maximal number of iterations was

		in clipTest250. The quality of the prediction of patches (see table 4, 5 and
	405	figure 11) DeepRGB8K outperforms DeepHSV8k. Therefore, for comparison
		with reference models from the state of the art, GBV S, SignatureSal and
		spatio-temporal model by Seo [25], named "Seo" we use DeepRGB8K model,
		see table 9 below.
		set to 174000 and for the second experiment, this number was fixed 10 times
		lower. From table 6 it can be stated that i)adding primary motion features,
		such as residual motion improves the quality of predicted visual attention maps
		whatever is the training of the network. The improvement is systematic and
		goes up to 38% in case of clipTest105 (in the first experiment);ii) the way to
		train the network, we propose with lower number of iterations and all training
		data used does not strongly affect the performances. Indeed, with 4k model the
		results are better for almost all clips, see highlighted figures in table 6. In table
		7 we compare all our predicted saliency models with gaze fixations. It comes
		out that more complex models yield better results: up to 42% of improvement

Table 6 :

 6 The comparison, with AUC metric, of the two experiments for 3K and 4K saliency models vs gaze fixations 'Gaze-fix' on a subset of HOLLYWOOD dataset

Table 7 :

 7 The comparison metric of gaze fixations 'Gaze-fix' vs Deep saliency '3k', '4k', '8k' ,

	'RGB8k' and 'HSV8k') for the video from HOLLYWOOD
	VideoName	Gaze-fix vs Deep3k	Gaze-fix vs Deep4k	Gaze-fix vs Deep8k	Gaze-fix vs DeepRGB8k

Table 9 :

 9 The mean improvement of the complete model.

available in ftp://ftp.ivc.polytech.univ-nantes.fr/IRCCyN IVC Eyetracker SD

12/ 2 available in http://www.di.ens.fr/∼laptev/actions/hollywood2/


Delphine Bresch-Pietri

Nicolas Petit

Miroslav Krstic

Prediction-based control for nonlinear state-and input-delay systems with the aim of delay-robustness analysis

This paper investigates prediction-based control for nonlinear systems subject to both pointwise input-and (potentially) distributed state-delays. We address infinity-norm stability analysis of the corresponding closed-loop system reformulating both delays as transport Partial Differential Equations (PDEs) and transforming the resulting distributed state. We show how the performed analysis can be extended to establish robustness to delay uncertainties. We illustrate the merit of this design with numerical simulation of a prey-predator population dynamics.

I. INTRODUCTION

Time-delays systems are often involved in the modeling of systems exhibiting periodical behaviors, as it is the case in biology [START_REF] Murray | Mathematical biology i: An introduction[END_REF] or population dynamics [START_REF] Berezansky | Delay differential logistic equation with harvesting[END_REF], [START_REF] Erneux | Applied Delay Differential Equations[END_REF], [START_REF] Kar | Selective harvesting in a prey-predator fishery with time delay[END_REF] for example. This lag, which could be either pointwise or distributed, accounts for physical distance and related transport, blood circulation, cell generation, etc. Most of the time, those delays affect both the input and the state. The same conclusion applies, e.g., for systems involving a recycle loop as it is commonly the case in the process industry (see [START_REF] Meyer | An experimental application of time delay compensation techniques to distillation column control[END_REF], [START_REF] Chauvin | Transient control of a diesel engine airpath[END_REF], [START_REF] Bresch-Pietri | Practical delay modeling of externally recirculated burned gas fraction for sparkignited engines[END_REF]). While most of the recycle processes can reasonably be modeled as linear dynamics, this is not the case of most biological systems.

In this paper, we design a prediction-based controller for nonlinear systems subject to both input and state delays. Predictor-based control strategies, more commonly known as Smith Predictor (see [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF], [START_REF] Smith | A controller to overcome dead time[END_REF], [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF]) are state-ofthe-art for systems with constant input time-delays (see for instance [START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF], [START_REF] Bresch-Pietri | Adaptive control scheme for uncertain time-delay systems[END_REF], [START_REF] Gu | Survey on recent results in the stability and control of time-delay systems[END_REF], [START_REF] Jankovic | Recursive predictor design for linear systems with time delay[END_REF], [START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF], or [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] and the reference therein). Grounding on the use of a prediction of the system state on a time horizon equal to the input delay, this technique aims at compensating this delay, which notably improves the transient performances. However, while the literature on control of either input delay or state delay systems is extremely wealth, systems with both input and state delays have seldom been studied, except from very recent contributions [START_REF] Bekiaris-Liberis | Simultaneous compensation of input and state delays for nonlinear systems[END_REF], [START_REF] Bresch-Pietri | Prediction-based control for linear systems with input-and state-delay-robustness to delay mismatch[END_REF], [START_REF] Kharitonov | An extension of the prediction scheme to the case of systems with both input and state delay[END_REF].

In this work, we consider nonlinear dynamics with (potentially) distributed state delay subject to pointwise inputdelay. We assume that we know a nominal feedback law which stabilizes the input-delay-free nominal system (and potentially compensates the state delay by relying on past values of the system state). The strategy that we advocate is similar to the one proposed in [START_REF] Bekiaris-Liberis | Simultaneous compensation of input and state delays for nonlinear systems[END_REF]. It consists in the use of such a nominal feedback law, which, instead of being applied to the current state, takes as input a suitable prediction of the system state.

While the control design and the stability result we propose are very close to the ones in [START_REF] Bekiaris-Liberis | Simultaneous compensation of input and state delays for nonlinear systems[END_REF], the stability analysis we present relies on a different methodology which enables one to pursue a robustness analysis in the future. In details, our approach is grounded on PDEs tools that were proposed lately to address input delay uncertainties (see [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF], [START_REF] Krstic | Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch[END_REF]) and were extended in [START_REF] Bresch-Pietri | Prediction-based control for linear systems with input-and state-delay-robustness to delay mismatch[END_REF] to handle both state-and inputdelay uncertainties. In this paper, we build on those previous contributions to propose a PDE framework accounting for nonlinearities and distributed delays. Modeling both actuator and the state delays as transport PDEs coupled with the original Ordinary Differential Equation (ODE), we rely on a backstepping transformation of the distributed input to analyze the closed-loop stability. To formulate the corresponding target system, one needs to study an implicit functional PDE. While nonlinearity is responsible for the implicit nature, the functional one originates from the fact that the state-delay is distributed. We then carry out an L ∞ analysis for the closedloop system. This is the main contribution of the paper.

While we do not formally investigate the delay-robustness problem, we show how the proposed methodology can be applied to analyze it. The objective of this paper is to present this new PDE framework by providing an alternative proof to the one in [START_REF] Bekiaris-Liberis | Simultaneous compensation of input and state delays for nonlinear systems[END_REF] as a preliminary step.

The paper is organized as follows. In Section II, we introduce the problem under consideration before providing the prediction-based control we propose in Section III. Then, we present the stability analysis of the closed-loop dynamics in Section IV before sketching in Section V how it can be extended to analyze robustness to delay uncertainties. Finally, we illustrate the merits of our designs in Section VI with simulations of a prey-predator model.

Notations.

Let n be a positive integer and µ a given positive constant. In the sequel, for u

∈ C ([0, 1] × R, R n ) and ϕ ∈ C (I, R n ) for any interval I ⊂ R, we denote u(t) p,µ = 1 0 e pµx |u(x,t)| p dx 1 p (1) u(t) ∞,µ = max x∈[0,1] |e µx u(x,t)| = lim p→∞ u(t) p,µ (2)
ϕ ∞ = max s∈I |ϕ(s)| (3)
and we use the standard notations

• p = • p,0
and

• ∞ = • ∞,0 . For (D 1 , D 2 ) ∈ R 2 and a function X, we write X t,1 : s ∈ [-D 1 , 0] → X(t + s) and X t,2 : s ∈ [-D 2 , 0] → X(t + s). Finally, for u ∈ C ([0, 1] × R, R n ), we define u(•,t) : x ∈ [0, 1] → u(x,t) for t ≥ 0 and, for v ∈ C ([0, 1] 2 × R, R n ), we define v(x, •,t) : y ∈ [0, 1] → v(x, y,t) for (x,t) ∈ [0, 1] × R and v(•, ,t) : (x, y) ∈ [0, 1] 2 → v(x, y,t) for (x,t) ∈ [0, 1] × R.

II. PROBLEM STATEMENT

Consider the (potentially) unstable system

Ẋ(t) = f (X t,1 ,U(t -D 2 )) (4) in which X ∈ R n , U is scalar and D 2 > D 1 > 0.
In the following, we assume that f is locally Lipschitz and that its partial derivative with respect to its first argument (Gâteaux derivative) is locally Lipschitz.

The dynamics (4) can encompass various complex delay classes. Those can be either distributed or pointwise (however, we do not allow the delay to be time-varying 1 ).

The control objective is to stabilize (4) following a prediction-based approach to compensate the input delay. With this aim in view, we first characterize further the system under consideration.

Assumption 1: The system (4) is forward complete.

Assumption 2: There exists a feedback law κ : C 0 ([-D 1 , 0], R n ) → R which is a class C 1 function such that the dynamics Ẋ(t) = f (X t,1 , κ(X t,1 )) (5)
is globally exponentially stable, i.e.(see [START_REF] Kolmanovskii | Introduction to the theory and applications of functional differential equations[END_REF], [START_REF] Pepe | Converse lyapunov-krasovskii theorems for systems described by neutral functional differential equations in hale's form[END_REF]), there exist a continuous functional

W : C ([-D 1 , 0], R n ) and constants C 1 ,C 2 ,C 3 > 0 such that C 1 ϕ ∞ ≤ W (ϕ) ≤ C 2 ϕ ∞ (6) |∂ ϕ W (ϕ)| ≤ C 3 (7)
and, moreover, the functional W is differentiable along the trajectories of the closed-loop system (5) and

Ẇ (t) ≤ -W (t) (8)
Assumption 1 guarantees that (4) does not escape in finite time and, in particular, before the closed-loop input reaches the system at time t = D 2 . This is a necessary assumption to guarantee that the control designed in the sequel will stabilize the plant for any delay value.

Assumption 2 guarantees the existence of a nominal feedback law for the input-delay free system. We assume that this feedback law is exponentially stabilizable. Alternatively, one could consider an asymptotically stabilizable feedback law as it is done in [START_REF] Bekiaris-Liberis | Simultaneous compensation of input and state delays for nonlinear systems[END_REF]. However, the nominal closed-loop dynamics should then also satisfy an Input-to-State Stability property, which is not needed here. Note that we consider that this feedback map is a function of the distributed state X t,1 to allow state-delay compensation.

In the sequel, we build on this nominal feedback law and design a prediction strategy to compensate the input-delay.

III. NOMINAL CONTROL DESIGN

Consider the distributed state prediction

P t (τ) = (9)    X(τ + D 2 ) if t -D 1 -D 2 ≤ τ ≤ t -D 2 X(t) + τ t-D 2 f (P t s,1 ,U(s))ds if t -D 2 ≤ τ ≤ t for t ≥ 0 and τ ∈ [t -D 1 -D 2 ,t]
. We now use this prediction as argument for the nominal input-delay free control law in lieu of the original distributed state

U(t) =κ(P t t,1 ) (10)
Theorem 1: Consider the closed-loop system consisting of the plant (4) satisfying Assumptions 1-2 and the control law [START_REF] Chauvin | Transient control of a diesel engine airpath[END_REF] involving the prediction [START_REF] Bresch-Pietri | Prediction-based control for linear systems with input-and state-delay-robustness to delay mismatch[END_REF]. Define the functional

Γ(t) = X t,1 ∞ + U t,2 ∞ (11)
There exists a class K L function β such that, for

(X 0,1 ,U 0,2 ) ∈ C 0 ([-D 2 , 0]) × C 0 ([-D 1 , 0]), Γ(t) ≤ β (Γ(0),t) , t ≥ 0 (12)
In order to properly understand the choice of the control law, we provide several comments next.

First, it is worth noticing that P t t,1 defined through ( 9) is a D 2 units of time ahead prediction of the distributed system state X t,1 . Indeed, integrating (4) between t and τ + D 2 , one obtains

X(τ + D 2 ) =X(t) + τ+D 2 t f (X s,1 ,U(s -D 2 ))ds =X(t) + τ t-D 2 f (X s+D 2 -D 1 ,1 ,U(s))ds (13)
in which the second equation has been obtained performing a change of variable. One can observe that this equation is equivalent to [START_REF] Bresch-Pietri | Prediction-based control for linear systems with input-and state-delay-robustness to delay mismatch[END_REF]. Hence, formally, one obtains

P t (τ) = X(τ +D 2 ) for τ ∈ [t -D 1 ,t] as D 1 < D 2 .
Plugging this control law into the original dynamics, one naturally infers that the closed-loop dynamics should be (5) which is asymptotically stable by assumption.

Second, note that we define the prediction as a function of two arguments: P t 1 (τ) is the prediction X(τ + D 2 ) computed at time t 1 and using X(t 1 ) as a starting point. A priori, this prediction could be different from P t 2 (τ) for a different t 2 . However, this is not the case here, as we consider no model uncertainty, but it is likely to be the case in practical applications. In particular, for the nominal case under consideration here, the definition of the distributed prediction ( 9) is formally equivalent to the following one given in [START_REF] Bekiaris-Liberis | Simultaneous compensation of input and state delays for nonlinear systems[END_REF]

P(τ) = X(t) + τ t-D 2 f (P s,1 ,U(s))ds , ( 14
) for t -D 2 ≤ τ ≤ t with t > D 2 P(τ) = (15)    X(τ + D 2 ) if -D 1 -D 2 ≤ τ ≤ -D 2 X(0) + τ -D 2 f (P s,1 ,U(s))ds if -D 2 ≤ τ ≤ 0
in which P is then expressed as a function of one argument to emphasize the fact that it actually does not depend on the time at which the computation is done. However, (9) constantly incorporates measured delayed state while ( 14)-( 15) is initialized with them and then only integrates the dynamics. Thus, in all likelihood, the formulation we propose should improve the robustness of the prediction to model mismatch or delay uncertainty (even if the induced number of computations would then increase). Further, it facilitates the stability analysis, as will appear in the following. Finally, even if the equation ( 9) may seem implicit at first glance, this prediction is actually well-defined and the solution always exists and is unique, as the solution of the differential equation ( 4). Further, more interestingly, it is also practically computable, relying on suitable discretization scheme of the integral (see [START_REF] Van Assche | Some problems arising in the implementation of distributed-delay control laws[END_REF] for a study on the effect of this discretization scheme on the closed-loop stability of linear systems and [START_REF] Karafyllis | Numerical schemes for nonlinear predictor feedback[END_REF] where nonlinear dynamics are addressed and a time-varying discretization methodology is proposed. Alternatively, one can rely on a a low-pass filter addition [START_REF] Mondié | Finite spectrum assignment of unstable time-delay systems with a safe implementation[END_REF] for linear systems or on an approximate predictor as done in [START_REF] Karafyllis | Stabilization by means of approximate predictors for systems with delayed input[END_REF].).

We now provide an alternative proof to this theorem, relying on PDEs reformulation.

IV. PROOF OF THEOREM 1 WITH PDES TECHNIQUE

A. PDEs reformulation and backstepping transformation

As a first step in our analysis, we define the distributed variables ζ (x,t) = X(t + D 1 (x -1)) and u(x,t) = U(t + D 2 (x -1)) for x ∈ [0, 1]. The plant (4) can then be reformulated as the following ODE-PDE cascade

               Ẋ(t) = f 0 (ζ (•,t), u(0,t)) D 1 ∂ t ζ (x,t) =∂ x ζ (x,t) ζ (1,t) =X(t) D 2 ∂ t u(x,t) =∂ x u(x,t) u(1,t) =U(t) (16)
in which we used

f 0 (ζ (•,t), u(0,t)) = f (X t,1 , u(0,t)) (17)
which is a simple change of arguments in the function definition. Similarly, in the sequel, we denote

κ 0 (ζ (•,t)) =κ(X t,1 ) (18)
Now define a distributed prediction, for x ∈ [0, 1] and t ≥ 0,

p(x,t) =X(t) + D 2 x 0 f 0 (χ(y, •,t), u(y,t))dy (19) χ(x, y,t) =    ζ y + x D 2 D 1 ,t if xD 2 + (y -1)D 1 ≤ 0 p x + D 1 D 2 (y -1),t if xD 2 + (y -1)D 1 ≥ 0 for 0 ≤ y ≤ 1 ( 20
)
along with the following backstepping transformation of the distributed input u(x,t)

w(x,t) =u(x,t) -κ 0 (χ(x, •,t)) , 0 ≤ x ≤ 1 (21)
Lemma 1: The infinite-dimensional backstepping transformation [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF] together with the control law [START_REF] Chauvin | Transient control of a diesel engine airpath[END_REF] transform ( 16) into the target system

Ẋ(t) = f 0 (ζ (•,t), κ 0 (ζ (•,t)) + w(0,t)) (22) D 1 ∂ t ζ (x,t) =∂ x ζ (x,t) (23) ζ (1,t) =X(t) (24) D 2 ∂ t w(x,t) =∂ x w(x,t) (25) w(1,t) =0 (26) Proof:
We start this proof by noticing that p(x,t) = P t (t + D 2 (x -1)) for x ∈ [0, 1] and that χ(x, y,t) = P t (t + D 2 (x -1) + D 1 (y -1)) for (x, y) ∈ [0, 1] 2 (this could be obtained performing a change of variable in [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF] and using the fact that the solution to the differential equation ( 4) or, equivalently, to the integral equation ( 9) is unique). From there, it follows that U(t) = κ 0 (χ(1, •,t)). Consequently, from the definition [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF], one obtains [START_REF] Pepe | Converse lyapunov-krasovskii theorems for systems described by neutral functional differential equations in hale's form[END_REF]. Further, from [START_REF] Krstic | Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch[END_REF], one gets χ(0, y,t) =ζ (y,t) [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] and [START_REF] Meyer | An experimental application of time delay compensation techniques to distillation column control[END_REF] follows with [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF]. Now, taking space-and time-derivatives of [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF], one obtains

∂ t p(x,t) = f (X t,1 , u(0,t)) + D 2 x 0 ∂ f 0 ∂ χ (χ(y, •,t), u(y,t)) • ∂ t χ(y, •,t) + ∂ f 0 ∂ u (χ(y, •,t), u(y,t)) • ∂ t u(y,t) dy (28)
∂ x p(x,t) =D 2 f 0 (χ(x, •,t), u(x,t)) =D 2 f 0 (χ(0, •,t), u(0,t)) + x 0 ∂ f ∂ χ (χ(y, •,t), u(y,t)) • ∂ x χ(y, •,t) + ∂ f ∂ u (χ(y, •,t), u(y,t)) • ∂ x u(y,t) dy (29)
Observing that χ(0, [START_REF] Krstic | Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch[END_REF] and that

•,t) = ζ (•,t) from
f 0 (ζ (•,t), u(0,t)) = f (X t,1 , u(0,t)), it follows that D 2 ∂ t p(x,t) -∂ x p(x,t) = (30) D 2 x 0 ∂ f 0 ∂ χ (χ(y, •,t), u(y,t)) • (D 2 ∂ t χ(y, •,t) -∂ x χ(y, •,t)) dy
Now, taking spatial-and time-derivatives of (20), one gets

∂ t χ(x, y,t) =    ∂ t ζ y + x D 2 D 1 ,t if xD 2 + (y -1)D 1 < 0 ∂ t p x + D 1 D 2 (y -1),t if xD 2 + (y -1)D 1 > 0 (31) ∂ x χ(x, y,t) =    D 2 D 1 ∂ x ζ y + x D 2 D 1 ,t if xD 2 + (y -1)D 1 < 0 ∂ x p x + D 1 D 2 (y -1),t if xD 2 + (y -1)D 1 > 0 (32)
and, thus, using ( 23) and (30),

D 2 ∂ t χ(x, y,t) -∂ x χ(x, y,t) =      0 if xD 2 + (y -1)D 1 < 0 D 2 x+ D 1 D 2 (y-1) 0 ∂ f 0 ∂ χ (χ(ξ , •,t), u(ξ ,t)) • (D 2 ∂ t χ(ξ , •,t) -∂ x χ(ξ , •,t)) dξ if xD 2 + (y -1)D 1 > 0 (33) Thus, defining r(x, •,t) = D∂ t χ(x, •,t) -∂ x χ(x, •,t) and, for (s, x,t) ∈ [-D 1 /D 2 , 0] × [0, 1] × R, r x (s, •,t) = r(s + x, •,t) if s + x ≥ 0 0 otherwise (34)
there exists a continuous function g :

[0, 1] × R → C (C ([-D 1 /D 2 , 0] × [0, 1] × R, R), C ([0, 1] × R, R)) such that ∂ x r(x, •,t) =g(x,t) • r x (•, ,t) (35) r 0 (•, ,t) =0 (36)
As the plant (4) is forward complete, p and thus χ exist for all time. Hence, the functional g is defined for all time. From there, one obtains that the solution of this system is global and unique. Solving this equation, it follows that r(x,t) = 0 or, equivalently,

D 2 ∂ t χ(x, •,t) =∂ x χ(x, •,t) (37)
Finally, taking space-and time-derivatives of ( 21), one obtains

∂ t w(x,t) =∂ t u(x,t) - dκ dχ (χ(x, •,t)) • ∂ t χ(x, •,t) (38)
∂ x w(x,t) =∂ x u(x,t) - dκ dχ (χ(x, •,t)) • ∂ x χ(x, •,t) (39)
Therefore, with ( 16) and (37), the transport PDE (25) follows straightforwardly.

We now exploit the backstepping transformation [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF] to provide a Lyapunov analysis of the stability of the closedloop system. The reason for this choice is linked to the boundary condition [START_REF] Pepe | Converse lyapunov-krasovskii theorems for systems described by neutral functional differential equations in hale's form[END_REF] as will appear in the sequel.

B. Lyapunov analysis

Define the following Lyapunov-Krasovskii functional candidate

V p (t) = µ -1 + 1 2p W (t) 2p (40)
+ Db 2p p ( X 0,1 µ,2p + w(0) µ,2p )

1 0 e 2pµx w(x,t) 2p dx in which the functional W has been introduced in Assumption 2, p ∈ N, µ > 1 and

b p ( X 0,1 µ,2p + w(0) µ,2p ) = (41) max µ -1 + 1 2p C 3 ( X 0,1 µ,2p + w(0) µ,2p
), 1 and : R + → R + is defined as the Lipschitz constant of f (•, κ(•)) with respect to its second argument over the ball B(0, R). In the following, for the sake of clarity, we omit the arguments of b p and . Taking a time-derivative of this functional and using integrations by parts, one obtains

Vp (t) = (2p(µ -1) + 1)W 2p-1 (42) × ∂ X t,1 W (X t,1 ) • f (X t,1 , κ(X t,1 ) + w(0,t)) -2pµb 2p p 1 0 e 2pµx w(x,t) 2p dx -b 2p p w(0,t) 2p ≤ -(2p(µ -1) + 1)W (t) 2p -2pµb 2p p w(t) 2p 2p,µ + (2p(µ -1) + 1)W 2p-1 C 3 |w(0,t)| -b 2p
p w(0,t) 2p in which we have also used [START_REF] Bresch-Pietri | Delay-adaptive control for nonlinear systems[END_REF] and [START_REF] Bresch-Pietri | Practical delay modeling of externally recirculated burned gas fraction for sparkignited engines[END_REF]. Further, using Young inequality, one obtains

(2p(µ -1) + 1)W 2p-1 C 3 |w(0,t)| (43) ≤(2p -1)W (t) 2p + µ -1 + 1 2p 2p C 2p 3 2p w(0,t) 2p
Therefore, plugging (43) into (42) and using (41), it follows Vp (t) ≤ -2pµV p (t) , t ≥ 0 (44) and thus, for t ≥ 0,

V p (t) 1 2p ≤ e -µt (µ -1 + 1 2p 1 2p W (0) (45) +b p D 1 0 e 2pµx w(x, 0) 2p dx 1 2p
This gives, for t ≥ 0,

(µ -1) 1 2p W (t) + b p D 1 0 e 2pµx w(x,t) 2p dx 1 2p (46) ≤ 2e -µt µ - 1 2 1 2p W (0) + b p D 1 0 e 2pµx w(x, 0) 2p dx 1 2p
Now, define

Γ 0 (t) =W (t) + b w(t) µ,∞ (47)
Γ(t) =W (t) + w(t) ∞ (48) in which b ∆ = max {(µ -1)C 3 ( X 0,1 ∞ + w(0) ∞ ), 1}
. Taking the limit of (46) as p tends to infinity, one finally gets Γ 0 (t) ≤ 2e -µt Γ 0 (t) , t ≥ 0 (49) and thus, using [START_REF] Bresch-Pietri | Delay-adaptive predictor feedback for systems with unknown long actuator delay[END_REF] and the fact that b ≥ 1 by definition, the existence of a K ∞ function α such that

Γ(t) ≤ α( Γ(0))e -µt (50)
which is exponentially stable.

For simulation, we consider D 1 = 1s, D 2 = 2s and d = 1 and choose λ 1 = λ 2 = 1. We choose a discrete sampling time equal to 10ms and approximate integrals involved in predictions with a Backward Euler scheme.

Fig. 1 depicts the corresponding results. After being turned on at t = 3s, the control kicks in at time t = 5s which is consistent with the input-delay value. This can be observed through the fact that X 2 remains to its zero equilibrium before that time. Then, the first effect on the first coordinate can be noticed at time t = 6s which is also consistent with the state delay value. Finally, one can check that asymptotic stabilization is achieved with a smooth (tunable) transient when delays are known. This transient worsens with delay uncertainties but, as Claim 1 state it, stabilization is still achieved for sufficiently small delay estimation errors.

To improve corresponding transient performances and potentially increase delay robustness, existing delay-adaptive techniques involving time-varying estimates (see [START_REF] Bresch-Pietri | Adaptive control scheme for uncertain time-delay systems[END_REF], [START_REF] Bresch-Pietri | Delay-adaptive predictor feedback for systems with unknown long actuator delay[END_REF]) for both delays in the same time should be investigated within the new framework proposed in this paper in future works.

  Evolution of the prey population.

  Evolution of the predator population.

Fig. 1 .

 1 Fig. 1. Simulation results for the closed-loop system consisting of the plant (64) and the control law (58),(65) with g(x) = 1 -x 50 , p(x) = 2 10+x , d = 1, D 1 = 1s, D 2 = 2s, λ 1 = λ 2 = 1 and the initial condition X(0) = [0.1 0]. Control is turned on at t = 3s. We consider three cases: (i) the state-and input-delays are known; (ii) both delays are underestimated with D = ( D1 , D2 ) = (0.8, 1.8); and (iii) both delays are overestimated with D = ( D1 , D2 ) = (1.2, 2.2).

Even if the methodology we propose could be extended to statedependent delays or to time-varying delays a model of which is known. This is a direction of future work.

ACKNOWLEDGEMENT

The authors are indebted to Iasson Karafyllis for crucial advice to carry out the infinity-norm Lyapunov analysis.

C. Equivalence between functionals Γ and Γ As a final step to prove Theorem 1, the fact that Γ in [START_REF] Erneux | Applied Delay Differential Equations[END_REF] and Γ in (48) are equivalent remains to be shown. With this aim in view, we first prove the following lemma.

Lemma 2: There exist class K ∞ functions α 1 and α 2 such that

Proof: From ( 19) and [START_REF] Krstic | Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch[END_REF], there exists a class K ∞ function α 3 such that

Second, from [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF], one obtains the existence of a class K ∞ function α 4 such that

Gathering ( 53)-( 54), one obtains the existence of α 2 such that (52) holds. The one of α 1 such that (51) holds follows similar lines. Now, observing that,

one gets, using (6) and Lemma 2, the existence of class K ∞ functions α 5 and α 6 such that

From there, using (50), one obtains the existence of a class K L function β such that Theorem 1 holds. This concludes the proof.

V. ON DELAY-ROBUSTNESS OF THE PREDICTION-BASED CONTROL LAW ( 9)- [START_REF] Chauvin | Transient control of a diesel engine airpath[END_REF] In this section, we investigate how the previously proposed design can be extended in the case of (state-and input-) delays uncertainty. We start by formulating a new assumption. We now consider that the two delays are uncertain and use D1 and D2 as state-delay and input-delay estimates, respectively (with ( D1 , D2 ) ∈ [D, D] 2 ). The previously proposed design is then modified as follows.

A. Control design

The estimated distributed state prediction is now

and the control law is

Claim 1: Consider the closed-loop system consisting of the plant (4) satisfying Assumptions 1-2 and the control law [START_REF] Chauvin | Transient control of a diesel engine airpath[END_REF] involving the prediction [START_REF] Bresch-Pietri | Prediction-based control for linear systems with input-and state-delay-robustness to delay mismatch[END_REF]. Define the functional

Further, the functions ∆ * 1 and ∆ * 2 are decreasing functions.

B. Elements of proof of Claim 1 and discussion

For the sake of clarity, we do not provide here the proof of Claim 1 which is lengthy and technical. However, we aim at underlying its milestones which should lead to the above statement.

Grounding on [START_REF] Bresch-Pietri | Prediction-based control for linear systems with input-and state-delay-robustness to delay mismatch[END_REF], we propose to rely on the estimated distributed input û(x,t) = U(t + D2 (x -1)) along with the estimated distributed state ζ = X(t + D1 (x -1)) to carry out the analysis. Then, one can introduce as in the previous section the backstepping transformation ŵ(x,t) = û(x,t)κ 0 ( χ(x, •,t))

(61) in which the distributed prediction is now p(x,t) =X(t) + D2

x 0 f 0 ( χ(y, •,t), û(y,t))dy (62)

This would lead to a target system similar to ( 22)-( 26) but including transport equations corresponding to distributed errors (distributed state and input errors) and with additive source terms. To handle those source terms and the nonlinearity of the systems, it is necessary to carry out the analysis with a modified H p 2 -norm as done in [START_REF] Bresch-Pietri | Delay-adaptive control for nonlinear systems[END_REF] (for p = 1) instead of a modified L p norm as it is the case in the previous section. Hence, the conclusion obtained in [START_REF] Bresch-Pietri | Delay-adaptive control for nonlinear systems[END_REF] for inputdelay systems would extend to the state-and input-delays case under the form of Claim 1.

In a nutshell, elements and techniques from [START_REF] Bresch-Pietri | Delay-adaptive control for nonlinear systems[END_REF], [START_REF] Bresch-Pietri | Prediction-based control for linear systems with input-and state-delay-robustness to delay mismatch[END_REF] and the infinity norm analysis carried out in the previous section yield the proof of Claim 1.

VI. APPLICATION TO POPULATION DYNAMICS -NUMERICAL SIMULATIONS

Time-delays are of particular interest for logistic and populations modeling as they are capable of generating periodic solutions of considerable complexity, dynamics which are often observed in those fields.

To illustrate the interest of the proposed approach, we consider the following (simplified) prey-predator model with predator harvesting

in which X 1 is the population of preys, X 2 the predators one, g the growth rate of the preys in the absence of predators, xp(x) is the response function and d is the death rate of predators in the absence of prey. In the above, the state delay D 1 accounts for the fact that the predation requires maturity of individuals (young predators do not kill preys); the delay D 2 accounts for the fact that the predators are neither introduced at the same physical location nor harvested before a certain size/age. Following [START_REF] Kar | Selective harvesting in a prey-predator fishery with time delay[END_REF], we consider g(x) = 1 -x 50 and p(x) = 2 10+x . The dynamics (64) satisfy Assumption 1 and exhibit a nonzero equilibrium point X r with X r 1 solution of the equation X r 1 p(X r 1 ) = d and r 2 = g(X r 1 )/p(X r 1 ). Furthermore, using the backstepping technique for ODEs, one obtains that a feedback law satisfying Assumption 2 is

)) (66) in which λ 1 and λ 2 are given positive constants and

Π(s)(g(Π(s)) -X 2 (s)p(Π(s)))ds (67) for τ ∈ [t -D 1 ,t] which only depends on X t,1 . The timederivative v in (65) can be formally calculated using (67) and the first component of (64). In details, this feedback law aims at compensating the state-delay through the prediction Π = X 1 (t + D 1 ) and the corresponding nominal input-delay-free closed-loop system is then