11institutetext: CRH-EHESS, Paris, France
11email<EMAIL_ADDRESS>EPITA Research &
Development Laboratory (LRDE), Le Kremlin-Bicêtre, France
22email<EMAIL_ADDRESS>LASTIG,
Univ. Gustave Eiffel, IGN-ENSG, F-94160 Saint-Mandé, France
33email<EMAIL_ADDRESS>
# Entry Separation using a Mixed Visual and Textual Language Model:
Application to 19th century French Trade Directories

B. Duménieu 11 0000-0002-2517-2058 E. Carlinet 22 0000-0001-5737-5266 N.
Abadie 33 0000-0001-8741-2398 J. Chazalon 22 0000-0002-3757-074X

###### Abstract

When extracting structured data from repetitively organized documents, such as
dictionaries, directories, or even newspapers, a key challenge is to correctly
segment what constitutes the basic text regions for the target database.
Traditionally, such a problem was tackled as part of the layout analysis and
was mostly based on visual clues for dividing (top-down) approaches. Some
agglomerating (bottom-up) approaches started to consider textual information
to link similar contents, but they required a proper over-segmentation of
fine-grained units. In this work, we propose a new pragmatic approach whose
efficiency is demonstrated on 19th century French Trade Directories. We
propose to consider two sub-problems: coarse layout detection (text columns
and reading order), which is assumed to be effective and not detailed here,
and a fine-grained entry separation stage for which we propose to adapt a
state-of-the-art Named Entity Recognition (NER) approach. By injecting special
visual tokens, coding, for instance, indentation or breaks, into the token
stream of the language model used for NER purpose, we can leverage both
textual and visual knowledge simultaneously. Code, data, results and models
are available at https://github.com/soduco/paper-entryseg-icdar23-code,
https://huggingface.co/HueyNemud/ (icdar23-entrydetector* variants).

###### Keywords:

Entry separation layout analysis text token classification multi-modal
features

##  1 Introduction

Discriminating semantically consistent content blocks, such as entries in
dictionaries or directories, or articles in newspapers, is a challenging task
for historical document analysis. Most of the approaches proposed by the
Document Analysis and Recognition (DAR) community tackled the problem by
extracting and analyzing visual clues, mainly considering the problem as a
refined version or, rather, the extension of the layout analysis problem. This
makes much sense, as proper block segmentation is a prerequisite, but misses
important textual clues.

In this work, we investigate the integration of visual features into an
encoder-only transformer architecture (BERT here), originally targeted to
text, to form a multi-modal _entry separation_ system. Our core idea and
contribution consists therefore in injecting special tokens like, for
instance, “<small_left_spacing>”, “<large_right_spacing>”, “<line_break>”, or
“<column_break>”, in the stream of regular tokens computed from the textual
content and process the resulting enriched stream with classical Natural
Language Processing (NLP) techniques to predict split positions.

Such textual content is assumed to be extracted from documents like the ones
illustrated in fig. 1, i.e. a pretty clean document with a regular structure
that a dedicated (visual only) preliminary layout analysis may segment
correctly in columns. This is an important, pragmatic assumption, as this
intermediate representation level is, in practice, very stable in our
experiments, and, furthermore, enables us to create a multicolumn, multipage
stream of text to later _separate_ into entries. Please note that here we use
the term _separate_ instead of _segment_ to avoid confusion between the
“cutting” of the text stream and the geometric decomposition of the page
image.

This paper features the following contributions:

  1. 1.

We introduce a pragmatic approach to _separate_ semantically consistent parts
of text (validated on directory entries) by combining visual and textual clues
in a stream of text tokens, leveraging existing NLP techniques.

  2. 2.

We release a new public dataset under an open license to reproduce our results
and enabling the comparison of _content separation_ systems.

After a survey of the techniques that can be used for a _separation_ task in
Section 2, we detail our approach in Section 3. We then introduce the new
public data set and the metrics that we use to benchmark our _separation_
system in Section 4. Lastly, we describe our experimental protocol and the
results obtained in Section 5.

##  2 Related Works

Document Layout Analysis (DLA) approaches aim at identifying and labeling
homogeneous regions of documents. DLA can be viewed as the first step in a
document understanding pipeline. Subsequent stages, such as OCR or Named
Entity Recognition (NER), process the document regions produced by this stage,
and the quality of their results depends directly on the quality of the
document segmentation done by the DLA step.

###  2.1 Top-down DLA strategies

The survey of DLA approaches proposed by [2] identifies three main strategies.
Among them, top-down strategies consider the whole document level and split it
into smaller regions based on some rules. They stop when there is no smaller
region left or when some predefined stopping criterion has been reached.
Classical approaches are generally based on a variant of XY cuts [34]. This
recursive cutting approach was improved with better preprocessing [15] and
with a recovery of the reading order [30, 41]. These approaches are very fast
and can perform reasonably well on clean, well-separated document layouts.
However, they consider only superficial visual clues and cannot perform a
fine-grained decomposition of a document or classify regions precisely, as the
semantics involved are unavailable to them. Indeed, once some over-
segmentation is reached, it is often necessary to reassemble document
fragments using a dedicated process. This is the case, for example, of the
article separation technique for historical newspapers used in the NewsEye
project [31] which links blocks according to their content similarity to
recover articles and reading order.

###  2.2 Bottom-up DLA strategies

Bottom-up strategies are the second type of DLA approach listed in [2]. They
start from small document features, such as pixels, words, or connected
components, and group them based on some rules to create homogeneous and
consistent regions until some predefined stopping criterion is reached. Hybrid
strategies, which take advantage of top-down and bottom-up approaches, are the
last type of strategies mentioned by [2]. The docstrum method [37] is one of
the first bottom-up approaches that has been proposed. It relies on a nearest-
neighbor clustering of document components, based on measurements of skew,
within-line, and between-line spaces, to identify the distinct regions. More
generally, machine learning-based approaches that classify document elements
according to regions can be viewed as bottom-up approaches. Many traditional
machine learning approaches have been leveraged, either directly on pixel
intensities [4, 9], on hand-crafted features based on Gradient Shape Feature
(GSF) [8] or Scale Invariant Feature Transform (SIFT) [11, 10]. These visual
feature-based solutions have yielded better results than the former. More
recently, deep learning-based approaches have become popular [3, 12, 38],
establishing new baselines for many DLA applications. Contrary to classical
machine learning methods, they generate their own visual features based on
pixel data to classify pixel into document regions. Hence, such methods can
form homogeneous regions with unconstrained shapes, and sometimes classify
them at the same time. However, the classes remain rather limited, and these
methods are preferably used as a preprocessing to form stable regions which
will be later processed according to specific rules.

###  2.3 Multimodal DLA approaches

Early deep learning-based DLA approaches have focused on visual information
only to solve the most common DLA tasks. [48] innovated by proposing a fully
multi-modal convolutional network, involving also a text embedding map to
perform a page semantic segmentation task. [26] handles texts segments as
graph nodes and uses a GCN to compute text segment embeddings capturing their
visual and textual context. These embeddings are then used jointly with text
embeddings in a key information extraction task. The recent advent of
Transformer-based models has led to the development of multimodal approaches,
which jointly leverage visual, textual, and layout information in the self-
supervised pretraining step. The LayoutML [46] approach also includes, in
addition to a textual embedding, two additional types of embeddings for the
input tokens in the original BERT [7] architecture: 1) a 2D position embedding
representing each token bounding box coordinates in the document local
coordinates system, provided by the OCR preprocessing step, and 2) a visual
embedding represented by an image patch of each token. The former is thus
intended to capture the tokens relative positions in the document, and the
latter its visual properties (color, font, etc.). The model is then pretrained
on two self-supervised tasks. For the Masked Visual-Language Model task, the
model has to predict masked tokens based on their context tokens and 2D
position embeddings. In the Multilabel Document Classification task, the model
has to predict each document category using all the available embeddings. The
model is finally fine-tuned for specific DLA tasks (form understanding,
receipt understanding, and document classification) using well-known annotated
benchmark datasets. The LayoutMLv2 [45], LayoutMLv3 [19], StructuralML [24],
XYLayoutLM [14], BROS [18], and LayoutXLM [47], LiLT [44] approaches build on
this first proposal and propose new embeddings, attention mechanisms or
pretraining tasks to improve the model’s ability to take into account text,
layout and visual content at the same time. In practice, these approaches
produce state-of-the-art results for region tagging, but at the cost of a
training with many annotations, including object positions. Furthermore, their
architecture is not well suited to capture complex textual structures, and it
is necessary to use a dedicated network as a downstream step to extract
structured information. The Document Attention Network [6] is an end-to-end
DLA, OCR, and structured information extraction system that produces a
structured output directly from a document image. Such an approach is very
promising but still suffers from some limitations. It is complex to train and
can not yet deal with a series of pages at the same time. Furthermore, it is
not possible yet to provide corrections at an intermediate step, like fixing a
column position, and may not offer the modularity we expect from a system
dedicated to the processing of historical documents, where both the capability
to reuse existing, off-the-shelf component and the ability to be corrected (to
propose an assisted digitization process) are of high importance. As a result,
dividing the content separation problem into two sub-problems seems a
pragmatic, but unexplored approach for mostly textual documents with a rather
clean and simple structure: 1\. use an existing DLA stage for a coarse
segmentation and extract a textual stream; and 2. use a dedicated text
processing stage to isolate and tag relevant content.

###  2.4 Named Entity Recognition in Text Sequences

Named Entity Recognition (NER) [33] is a common natural language processing
task that aims to locate and classify portions of text that represent entities
of predefined categories such as people, places, organizations, dates, etc.
The early approaches were mainly based on hand-crafted rules designed to
retrieve lexico-syntactic patterns [36] or dictionary entries [29, 27]
representing named entities. With the use of a comprehensive dictionary and
significant efforts in rule engineering, these approaches give good results
for clean, domain-specific texts [33]. Supervised approaches include
traditional machine learning approaches based on carefully chosen text
features and deep learning approaches that automatically build their own
features for classification from raw text. As highlighted in the survey
proposed by [25], in recent years, deep learning-based NER approaches have
outperformed both supervised and unsupervised feature-based approaches,
achieving state-of-the-art results. BERT-based approaches (for Bidirectional
Encoder Representations from Transformers) [7], use the Transformer attention
model [43] for language modeling. They take advantage of the transfer learning
principle: training a neural network on some known generic tasks — here
Masked-Language Modeling and Next Sentence Prediction — and then fine-tuning
it for a more specific purpose. To fine-tune a BERT model for the NER task,
one only needs to feed the output vector of each token into a classification
layer that associates a label to it. Currently, transformer-based models give
the best results for the NER task [25]. Recently, Hao et al. [16] showed that
language models were general interfaces that can merge multiple modalities.
Modern transformer libraries make it easy to add and use custom tokens. While
using images embedding directly might be a possibility, directly generating
tokens based on visual properties of the text sequence has not yet been
proposed, to our knowledge.

###  2.5 DLA tasks and benchmark datasets

Many datasets have been proposed by the community to evaluate DLA approaches.
They are designed to train and evaluate models on one, two, or three well-
targeted task types. For example, the FUNSD [21] and XFUN [47] datasets are
designed for form understanding, which includes text field detection, OCR,
semantic annotation, and linking of document regions. RVL-CDIP [17] is made up
of images of documents of various types, annotated to classify them into
categories. SROIE [20] and CORD [39] provide scanned images of annotated
receipts for text localization, OCR, and key extraction tasks. DocVQA [28]
combines document images, questions, and their expected answers for a VQA use
case. SciTSR [5] contains several thousand tables extracted from scientific
articles and annotated for a structure extraction task. Some of these
datasets, such as XFUN, have documents written in various languages.

The set of scanned and annotated newspaper articles proposed by the NewsEye
project for the ICPR 2020 competition on text block segmentation [31] shows to
correspond the most to our goal of separating directory entries. More
specifically, the dataset provided for the “simple” task, which only deals
with regular multicolumn newspaper pages, without images or tables, has a
layout that looks like directory or dictionary entries. Moreover, the task to
be performed, which consists in grouping the detected lines of text into
semantically consistent paragraphs, has a purpose similar to that of
separating directory entries. But directory or dictionary entries are short
texts written according to a well-defined pattern, where case, punctuation,
line breaks, and indents play a strong structuring role. Newspaper pages, even
without images, include more visual information to organize the articles, such
as blank spaces or horizontal and vertical lines. Therefore, it seems useful
to produce and publish an open benchmark dataset for entry separation in this
type of document.

##  3 A Mixed Visual and Textual Language Model

We detail in this section our proposal for a mixed _visual and textual_
language model. We consider it as a sort of response to the CharGrid [22]
approach from the point of view of natural language processing (NLP). CharGrid
introduced an elegant way to process administrative documents, such as
invoices, using both textual and visual information. It did so by projecting
textual information (simplified character codes) into the image domain: pixels
were “colored” according to the bounding box of the character they fall into.
This enabled the authors to use classical Computer Vision (CV) object
detection techniques based on the Faster-RCNN architecture [40].

This was an interesting case of the Document Analysis and Recognition (DAR)
community benefiting from CV advances. In the case of single-page documents
whose information is mostly carried by the spatial organization of contents,
this approach produced great results. However, for documents whose information
is carried mainly by textual content, such as the trade directories we
introduced earlier (fig. 1), such an approach is limited, and it is necessary
to consider approaches that emerged from the field of NLP such as BERT [7] to
capture rich syntactic structures. Like Katti et al. showed how to leverage
Faster-RCNN for DAR, we propose a way to leverage BERT (an encoder-only
transformer which implements a non-causal language model) for DAR.

We propose a method where visual tokens are injected in the regular token
stream (usually built from text content) and capture extra visual clues which
can assist the network in discriminating and tagging input’s fragments as
expected.

Our proposed approach is composed of the following steps:

  1. 1.

_Run a _coarse_ document layout segmentation._ In this paper, we assume that
the performance of this preliminary step is strong enough to further process
its results. In practice, we found that for mostly textual documents whose
structure is relatively simple and clean, some ad hoc, top-down image
splitting followed by OCR’s line detection performs well. We use the line
extraction algorithm from the same OCR system that we use in step 2 because
OCR models are trained on data produced by some specific line extraction
algorithm, and we witnessed performance degradation when using other detection
techniques. A single example line may look like this, with its enclosing
polygon drawn in dashed red:

  2. 2.

_Run character recognition on the set of pages we intend to process._ We store
the transcription for each detected text line, along with its exact position
on the page and the blocks it belongs to. In the experiments we report here,
we only kept column information and ran the OCR system column-by-column. In
our previous example, the following (noisy) text may be obtained:
"Bailly, coutelier. S.Honuré, 416."

  3. 3.

_Compute extra visual clues which will be merged into the text stream as extra
tokens._ Visual clues can be of several kinds and will be detailed hereafter.
In the experiments we report here, we add new tokens to represent breaks for
line or columns, left-hand line indentation, and right-hand horizontal space
left at the end of a line. In our running example, the following items would
be generated, omitting their position attributes for clarity:
[<LSPACE size=0.0056/>, <RSPACE size=0.3490/>, <LINE_BREAK/>]

  4. 4.

_Prepare a categorical encoding of parametrized tokens._ Some tokens may carry
numerical values, such as the length of horizontal spaces, and we need to
perform a categorical encoding before using them as inputs. We know that some
techniques directly use numerical tokens, but we did not experiment them. In
our running example, the categorical tokens may look like:
[<LSPACE_SMALL>, <RSPACE_LARGE>, <LINE_BREAK/>]

  5. 5.

_Merge visual and textual content in textual representation._ In practice,
merging visual and textual clues boils down to adding a few special text
strings at the right position in the input text stream, and configuring the
tokenizer with the new tokens and their associated special strings. In our
running example, the resulting text content would look like this:
"<LSPACE_SMALL>Bailly, coutelier … <LINE_BREAK/>"

  6. 6.

_Tokenize the textual representation._ Text tokenization is a necessary step
to capture frequent text patterns and build better NLP models. Input text
undergoes some chunking process, where each chunk is assigned the identifier
of the pattern it matches with. Modern NLP libraries make it easy to add new
tokens from special substrings. First, we can define rules to substitute some
arbitrary text from the original text stream with a special token, and second
we can easily change the weights of the input layer of the BERT transformer to
deal with new tokens, without changing already-trained weights assigned to
other tokens. In our running example, tokenization can look like this, with
the numbers between brackets being token unique identifiers:
[<LSPACE_SMALL>(1001), "_Bail"(118), "ly"(386), ","(10) …]

  7. 7.

_Process the token sequence using some BERT-like, encoder-only transformer._
We implement a token classification task which assigns a label to each token.
Possible labels and how they can be assigned to tokens is detailed hereafter.
This last step assigns categories to text spans or isolated tokens. This
enables us to split and tag the input stream simultaneously. Another advantage
is that it is possible to reuse a pre-trained network and fine-tune it for a
particular document type with a limited annotation cost: annotating text
splits is very fast and easy, as it only consists in inserting tags in text.
Also, the amount of annotation required to train a network is within reach of
a small team (as the experiments show), and the computation cost of a text
transformer, both in terms of training and inference, is well suited for
digital humanities applications. In our running example, some possible
labeling for simultaneous entry separation and NER can be:
[EBEGIN, I-PER, I-PER, O …]

##### Inputs tokens.

The approach we propose enables the input stream to contain a sequence of
“physical” tokens of several kinds. (We indicate in bold the tokens we use in
the experiments we report in section 5.)

  * •

Regular _text tokens_ produced by the tokenizer of the NLP model: they belong
to a previously trained (context-free) lexicon, according to the SentencePiece
[23] tokenization approach. The CamemBERT model that we used was trained with
these particular tokens.

  * •

_Structural tokens_ : left/right space, line, column and page breaks, vertical
spaces, etc.

  * •

_Graphical tokens_ : horizontal rules, decorations, bullets, etc.

  * •

_Presentational tokens_ in the form of pairs of start and end tags surrounding
the content described: text size, text style, text family, etc.

Such an approach is very flexible. The only delicate part is to identify the
correct position where each token should be inserted in the text stream. For
documents whose content is mostly textual, this information is easily derived
from the coarse structure, the reading order, and the polygons of the text
lines. It may also be suitable to process digital native documents like PDFs,
where reading order is recovered using coarse layout analysis, and style
attributes are easily obtained.

##### Output labels and annotation policies.

The output stream contains the sequence of predictions, assigning a label to
each element of the input stream. We use matching pairs of labels to indicate
the beginning and the end of the semantic constructs we intend to detect. Even
if assigning a “cut” label to a particular token is possible, for instance, to
mark separation points in a stream of directory entries, we believe that a
pair-of-tags approach is safer in production. Indeed, real input streams may
contain unexpected structures, and we want to be able to discard them from our
automated extraction, rather than just partitioning the input. It is also
possible to perform content separation (of headings, entries, page numbers…)
and structured content extraction from entries at the same time, using the
same network. This is a very attractive feature both in terms of computation
efficiency (the same network is called once to produce several outputs) and
ability to leverage more training annotation: extents of headings and entries
at one level, and of key entities in entries at another level. However,
deciding which token will carry each label is not straightforward, and several
strategies are possible. Using joint labeling or any hierarchical
classification technique, more than one label may be assigned to a given input
token, but this technique requires some special label preparation or a custom
architecture. Another option is to take advantage of the extra “visual” tokens
and label them with some structural tags. For instance, the tokens for left-
and right-line spaces may be used to annotate beginning and end of entries.
Note that when we use such tokens, they are always present even if the space
is null: in the particular case of the left space indent, we would use several
tokens in the categorical encoding of the possible space sizes, reserving one
token for zero-width space.

##  4 Evaluation Framework

We present here the dataset and metrics used to assess the performance on
real, noisy OCR text of the entry separation task and the NER task. The NER
task may be run in conjunction with the entry separation task.

###  4.1 Dataset

Figure 1:  Examples of directory layouts and contents: 1) Duverneuil et La
Tynna 1806 - index by name; 2) Deflandre 1828 - index by activity ; 3) Bottin
1851 - index by street name

In this work, we assume that a general-purpose segmentation technique can be
used to detect text columns reliably; hence, we will not measure the
performance on this stage on the pipeline and instead consider that text
columns are already properly segmented. Of course, such assumptions can cause
a cascade of errors when the first stages of a pipeline fail, but in our
experience, such segmentation is quite reliable on printed contents like the
ones used hereafter. The dataset used in the experiments is augmented from the
19th Paris trade directories dataset from [1]. The directories are available
from different libraries in Paris and have been independently digitized at
various levels of quality. Directories contain social information and
localization of people that are mainly organized into lists. For instance, the
entry “Batton (D.-A.) , professeur au Conservatoire de musique et de
déclamation, Saint-Georges, 47.” from the _Didot (1854)_ contains the person’s
name, his surname, his title (_Légion d’Honneur_), his activity (professor at
the Conservatory of music and declamation), the street number (47) and the
street name (Saint-Georges) of his living. This information is part of the
annotations provided by the dataset. The dataset covers a long period of time,
and the directories originate from several publishers. Therefore, their
contents (layout, methods of printing) vary greatly and the formatting of
entries as well (see fig. 1). Even within the same directory, the indentation
rules may vary from page to page. Therefore, our experiments focus on the task
of properly placing directory entries _separators_ in the token stream we
generate. As previously explained, this stream can be composed of tokens of
different nature, depending on whether we add special visual tokens and on how
we decide to model such visual information.

The dataset consists of 80 annotated pages, containing a total of 8 900
directory entries. Entries are made of 1.4 text lines on average, and most
entries are single lines. This means that a dummy classifier which predicts
entry separators for each line would reach an F-score of 83.3% (metrics are
detailed hereafter). 80% of the pages are assigned to the train set, 15% for
the test set, and the remaining 5% are used for validation.

Results we report are computed from noisy OCR inputs, even if the reading
order is assumed to be correct. This means that we projected the target entry
separation and entity tags over a steam which contains noisy OCR content,
while such annotation was made on clean text. Several tools are available to
project NER targets, or generally tag positions, annotated on manually
corrected OCR text onto noisy, raw OCR results [1, 42]. When OCR results are
very poor, annotation alignment tends to fail and related items are discarded.
Hence, this evaluation protocol reports results for acceptable OCR noise, and
only measures errors introduced by the stream annotation stage we propose.
Errors due to upstream stages should also be measured to estimate the global
performance of such a system.

###  4.2 Metrics

We consider the entry separation problem, as well as the NER problem, as token
classification problems. This makes the evaluation very simple as we consider
a prediction to be correct only if it exactly matches the position of some
expected one. For the entry separation task, only one token is labeled for
each “entry begin” and “entry end” markers. For the NER task, we predict start
and end positions similarly for each entity, and positions must match exactly
to be considered as correct. This evaluation protocol has the advantage of not
making any assumption on the importance of small errors: all errors are
treated equal, and in the particular case of directory entries, this makes
much sense, as a lost separator is a lost entry. Therefore, we report standard
classification metrics such as precision, recall, and F-score. For entry
detection, we consider two classes: entry-start cuts and entry-end cuts, and
metrics are computed in a straightforward way, as only one token can carry the
correct information. For NER, each entity is considered as a sample and
metrics are reported accordingly, disregarding the number of tokens it
contains. We use the SeqEval [35] library to compute our metrics.
We are aware of the evaluation process used in the ICPR 2020 Competition on
Text Block Segmentation on a NewsEye Dataset [32], which internally uses the
coverage metric of the READ-BAD evaluation framework [13]. We believe that
this metric, which was designed to assess the quality of text baseline
detection, is too permissive, as it tolerates small boundary variations and
fragmentation. It also requires a matching step, which seems superfluous in
our case.

##  5 Experimental Validation

Figure 2: Illustration of possible input data. In all cases, input data is
stream of tokens from which we have to detect _entry_ delimiters. In the left
case (A), only the textual information (no line breaks, column breaks…) is
used, as a pure text token stream. On the other side (case B), only the visual
layout information is used with _breaks_ and left/right blank spaces of the
lines encoded as categorical tokens. The middle case (C) shows the combination
of both kind of information, plus the annotations of the text entities: names,
cardinal, location…Below each case, we show the stream of tokens that is
passed to the system for detecting entries.

In this section, we describe the experimental setup we use to demonstrate the
feasibility and performance of our proposed approach. As already mentioned, we
focus on processing noisy text extracted by OCR, once coarse layout is
detected. In the case of trade directories, this text content represents
directory entries in the order they appear, across rows, columns, and pages.
Among this continuous stream of text, we first want to identify tokens that
delineate directory entries. Then, in a second step, we seek to jointly
identify these separation tokens, as well as those that compose named
entities. We thus have two tasks to perform: one to recognize tokens that
delineate text structure (entry separation task), and the other to recognize
semantic information (NER task). In the rest of this section, we make explicit
which extra tokens are used, detail the progress of the experiments we
conducted, and discuss their outcomes.

###  5.1 New tokens definitions

To help the language model recognize the entry beginning and end, we inject
new tokens into the tokenizer’s vocabulary. <break> which represents all
breaks in the text stream like new lines, column breaks, and page breaks, and
<lhspace-x> and <rhspace-y> which represent left- and right-line indents
detected by the coarse layout detection algorithm. $x$ and $y$ are integers
that quantize the value of the indent. We measure indent in two ways: 1. its
width expressed as a percentage of the total line width (referred to as
_‘absolute’_ values), and 2. the width difference with the previous indent on
the same side of the text (_‘relative’_ values).

We performed a categorical encoding of left and right spaces, both for
absolute and relative variants, before feeding them to the tokenizer. The
threshold values used were determined manually in the training set
exclusively. Distributions for absolute and relative spaces are displayed in
fig. 3.

Figure 3: Distributions for left and right spaces on the training set. _Left_
: values for absolute measures. _Right_ : values for measures relative to the
previous line. Right relative spaces were not considered in our experiments.
Red dashed vertical bars indicate thresholds used for categorical encoding.

The values of the relative left indents are binned into $]-\infty;-0.01[$,
$[-0.01;+0.01[$, and $[+0.01;+\infty[$ to produce three categorical tokens.
Similarly, the values of the absolute left indents are binned into
$[0.00;0.02[$, $[0.02;0.08[$, and $[0.08;1.00]$, and the values of the
absolute right indents into $[0.00;0.05[$, $[0.05;0.08[$, and $[0.08;1.00]$.
We aimed at producing a small amount of tokens to ease model’s training. For
absolute spaces, our logic was to produce three tokens: one for clear _“no
indentation”_ another for clear _“indentation”_ cases, and the last for
ambiguous cases. For relative spaces, we also tried to capture 3 simple
configurations: _“no significant change”_ , _“significant negative
difference”_ and _“significant positive difference”_.

Finally, we also used a <textline> token which represents all the textual
content found between two consecutive indents. This particular case is
intended to replace the actual textual content in the input to assess the
performance of visual-only separation, with minimal alteration of the network.

###  5.2 Experiments

To assess the interest of each injected visual feature, we add them one by one
and compare the results obtained by each strategy. Then, we perform a second
set of experiments to jointly recognize entry boundaries and named entities in
the text. An entry boundary is modeled with two dedicated NER labels: EBEGIN
and EEND. In all the following experiments, the entry boundary labels are
marked on text tokens only, except for experiment 1.7 and 2.1.

The first set of experiments (xp-1.X) explores the ability of a token
classification model to recognize entry boundaries with a varying number of
visual information tokens injected in the text and different configuration of
indents. It starts with the raw text stream (xp-1.1) and adds more and more
visual clues. Experiment 1.2 adds visual break separators, at the line, column
or page level, and experiments 1.3 to 1.6 add space tokens to express the
indentation level. We propose to assign two kinds of values to those spaces:
1. the (Absolute) distance between the line borders and the column borders (as
a percentage of the column width), 2. the (Relative) offset of the current
line borders with the border of the previous line (also as a percentage of the
current line.). Although the first valuation represents the level of
indentation or if a line reaches the right border of the column, the
_relative_ approach could eventually detect a break in the alignment. In
experiment 1.7, we study the opposite approach of xp-1.1, where we evaluate
the system only on the visual content (all spaces and breaks, but no text).
The following table summarizes the information embedded in the input for each
experiment.

Experiment | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | 1.7
---|---|---|---|---|---|---|---
Text |  |  |  |  |  |  |
Visual breaks |  |  |  |  |  |  |
Left spaces |  |  | A | R | A | R | R
Right spaces |  |  |  |  | A | A | A

In experiments 2.X, we propose to train the model for both named entity
recognition and entry separation at the same time. The two extra labels,
EBEGIN (EntryBEGIN) and EEND (EntryEND), are added to the set of labels to
predict by the NER. Experiment 2.1 is thus very similar to xp-1.6, but those
labels are set on the left and right spaces. In contrast, in experiment 2.2,
we propose to evaluate the system on a joint labeling of the first and last
words of the entry (which can, for example, be (ACT, EBEGIN)). Such approach
relies on new labels composed of the product of entry separation marks and NER
labels.

###  5.3 Training protocol and parameters

All experiments were run 3 times with different random seeds. We report the
averaged results over these runs. The base network is the public CamemBERT
variant from [1] pretrained on noisy OCR text from the same trade directories
using an unsupervised masked language modeling task. This network was not
trained for NER on trade directories, but was able to capture some lexical and
syntactic structures of the dataset during pretraining.

For each experiment, the network was trained with batches of randomly cropped
sequences of tokens of variable sizes. We generated these tokens according to
each of the variants presented earlier in this section. Each network was
trained with up to 7500 training steps, and the F score was evaluated every
300 steps. Training was stopped when this metric did not improve for 5
consecutive evaluations. We use the AdamW optimizer as recommended, with a
learning rate of $10^{-4}$ and a weight decay of $10^{-5}$.

###  5.4 Results and discussion

Table 1: Performances, over 3 runs, of the models trained to recognize entry boundaries, with different experimental setups. The models are sorted in ascending order of the macro F-score (in %) of the entities EBEGIN and EEND, computed as the harmonic mean of the average precision and recall for both entities over all runs. Finally, the global NER F-score for all named entities is given for experiments 2.1 and 2.2. Abs. stands for _absolute indents values_ , rel. for _absolute indents values_ | Experimental setup | EBEGIN & EEND (%) |
---|---|---|---
# experiment |  text |  ner on text content |  visual breaks |  Left spaces |  right spaces |  Precision |  Recall |  F-score |  Overall NER F-score
2.2 | yes | yes | yes | rel. | abs. | 94.7 | 56.3 | 70.6 | 93.0
1.1 | yes | no | no | no | no | 97.3 | 95.5 | 96.4 | -
1.7 | no | no | yes | rel. | abs. | 98.4 | 97.9 | 98.2 | -
1.2 | yes | no | yes | no | no | 99.5 | 98.0 | 98.7 | -
1.4 | yes | no | yes | rel. | no | 99.7 | 98.2 | 98.9 | -
1.3 | yes | no | yes | abs. | no | 99.7 | 98.3 | 99.0 | -
1.6 | yes | no | yes | rel. | abs. | 99.4 | 98.6 | 99.0 | -
1.5 | yes | no | yes | abs. | abs. | 99.6 | 98.5 | 99.0 | -
2.1 | yes | yes | yes | rel. | abs. | 99.3 | 99.0 | 99.2 | 94.0

From the group of experiments 1.X conducted on the entry separation task
without NER, several conclusions can be drawn. Experiment 1.1 proves that
text-only entry separation is a solid baseline, reaching an F-score of 96.4%
on the entry separation task from noisy OCR text. Experiments 1.2 to 1.6 show
that progressive integration of visual (structural) information steadily
improves the quality of entry separation, reaching an F-score of 99.0% for the
best configuration, composed of line and column breaks and absolute left and
right spaces. This configuration is tied with the variant with relative left
spaces, letting us believe that the network captures similar information from
absolute and relative variants in these documents. Experiment 1.7, on the
other hand, shows that using structural tokens only forms a solid baseline
that reaches an F-score of 98.2%. This is due to the short nature of entries
in the dataset and the capacity of the network to model sequence state very
efficiently, but may not be so significant on datasets with larger units. This
first group proves the feasibility of our proposed approach and the positive
interaction visual and textual information can have when projected into the
same stream of tokens, processed by a mixed language model.

The group of experiment 2.X conducted on a joint task of entry separation and
NER, proves that such unified process can be accomplished by a single network,
leveraging the training effort and the computation required to extract named
entities. Experiment 2.1 shows that extra tokens injected into the input
stream can also be useful to simplify the output objective of the network: by
trying to label the spacing tokens as entry beginning and end, we avoid the
need for the complex join labels used in experiment 2.2 and reach the best
absolute performance in terms of entry segmentation with an F-score of 99.2%,
while reaching a good NER performance. Experiment 2.2, on the other hand,
shows that it is possible to annotate only textual tokens, but at the cost of
an increase of target labels. Such a system is harder to converge and reaches
a lower performance. We do not report the results for an experiment which
would perform NER and entry separation at the same time from text only without
joint labeling, as it would require adding another network head and would not
make a fair comparison with experiment 2.1 because of the extra weights.

##  6 Conclusion and perspectives

We have assessed the importance of combining visual and textual information
for the particular case of entry separation in 19th century trade directories
with raw OCR text. As expected, by mixing indentation, break, and layout
elements with the regular text stream, we outperform the systems that would
use only a single of those modalities. Nevertheless, the originality of this
paper lies in the way we have mixed this information. While the current trend
is to convert text content into visual “pixel” content, we choose the opposite
way: turning visual clues into text tokens. This approach has many advantages.
Compared to visual transformers, it is simpler, lighter to fine-tune using
some pre-trained network, and requires less annotated data, which is easier to
produce. Also, as a NER system is necessary to extract structured information,
we maximize computation efficiency by performing two tasks at the same time
with this same network, benefiting from richer annotations during training.
Eventually, such system can process an infinite stream of tokens, and reduces
the burden of handling content units spanning across lines, columns, or pages.

Code, data, results and models are available at
https://github.com/soduco/paper-entryseg-icdar23-code,
https://huggingface.co/HueyNemud/
(icdar23-entrydetector* variants).

####  6.0.1 Acknowledgements

This work is supported by the French National Research Agency (ANR), as part
of the SODUCO project, under Grant ANR-18-CE38-0013.

## References

  * [1] Abadie, N., Carlinet, E., Chazalon, J., Duménieu, B.: A benchmark of named entity recognition approaches in historical documents - application to 19th century french directories. In: Uchida, S., Barney, E., Eglin, V. (eds.) Proc. of Document Analysis Systems. DAS 2022. Lecture Notes in Computer Science, vol. 13237. Springer, Cham (May 2022). https://doi.org/10.1007/978-3-031-06555-2_30
  * [2] Binmakhashen, G.M., Mahmoud, S.A.: Document layout analysis: a comprehensive survey. ACM Computing Surveys (CSUR) 52(6), 1–36 (2019)
  * [3] Chen, K., Seuret, M., Hennebert, J., Ingold, R.: Convolutional neural networks for page segmentation of historical document images. In: 2017 14th IAPR International Conference on Document Analysis and Recognition (ICDAR). vol. 1, pp. 965–970. IEEE (2017)
  * [4] Chen, K., Wei, H., Hennebert, J., Ingold, R., Liwicki, M.: Page segmentation for historical handwritten document images using color and texture features. In: 2014 14th International Conference on Frontiers in Handwriting Recognition. pp. 488–493. IEEE (2014)
  * [5] Chi, Z., Huang, H., Xu, H.D., Yu, H., Yin, W., Mao, X.L.: Complicated table structure recognition. arXiv preprint arXiv:1908.04729 (2019)
  * [6] Coquenet, D., Chatelain, C., Paquet, T.: DAN: a segmentation-free document attention network for handwritten document recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence pp. 1–17 (2023). https://doi.org/10.1109/tpami.2023.3235826
  * [7] Devlin, J., Chang, M.W., Lee, K., Toutanova, K.: BERT: Pre-training of deep bidirectional transformers for language understanding. In: Proc. of NAACL-HLT. pp. 4171–4186 (2019)
  * [8] Diem, M., Kleber, F., Sablatnig, R.: Text classification and document layout analysis of paper fragments. In: 2011 International Conference on Document Analysis and Recognition. pp. 854–858. IEEE (2011)
  * [9] Fischer, A., Baechler, M., Garz, A., Liwicki, M., Ingold, R.: A combined system for text line extraction and handwriting recognition in historical documents. In: 2014 11th IAPR International Workshop on Document Analysis Systems. pp. 71–75. IEEE (2014)
  * [10] Garz, A., Diem, M., Sablatnig, R.: Detecting text areas and decorative elements in ancient manuscripts. In: 2010 12th International Conference on Frontiers in Handwriting Recognition. pp. 176–181. IEEE (2010)
  * [11] Garz, A., Sablatnig, R., Diem, M.: Layout analysis for historical manuscripts using sift features. In: 2011 International Conference on Document Analysis and Recognition. pp. 508–512. IEEE (2011)
  * [12] Grüning, T., Leifert, G., Strauß, T., Michael, J., Labahn, R.: A two-stage method for text line detection in historical documents. International Journal on Document Analysis and Recognition (IJDAR) 22(3), 285–302 (2019)
  * [13] Grüning, T., Labahn, R., Diem, M., Kleber, F., Fiel, S.: READ-BAD: A New Dataset and Evaluation Scheme for Baseline Detection in Archival Documents. In: 2018 13th IAPR International Workshop on Document Analysis Systems (DAS). pp. 351–356 (Apr 2018). https://doi.org/10.1109/DAS.2018.38
  * [14] Gu, Z., Meng, C., Wang, K., Lan, J., Wang, W., Gu, M., Zhang, L.: Xylayoutlm: Towards layout-aware multimodal networks for visually-rich document understanding. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 4583–4592 (2022)
  * [15] Ha, J., Haralick, R.M., Phillips, I.T.: Recursive xy cut using bounding boxes of connected components. In: Proceedings of 3rd International Conference on Document Analysis and Recognition. vol. 2, pp. 952–955. IEEE (1995)
  * [16] Hao, Y., Song, H., Dong, L., Huang, S., Chi, Z., Wang, W., Ma, S., Wei, F.: Language models are general-purpose interfaces. https://doi.org/10.48550/arXiv.2206.06336, http://arxiv.org/abs/2206.06336
  * [17] Harley, A.W., Ufkes, A., Derpanis, K.G.: Evaluation of deep convolutional nets for document image classification and retrieval. In: 2015 13th International Conference on Document Analysis and Recognition (ICDAR). pp. 991–995. IEEE (2015)
  * [18] Hong, T., Kim, D., Ji, M., Hwang, W., Nam, D., Park, S.: Bros: A pre-trained language model focusing on text and layout for better key information extraction from documents. In: Proceedings of the AAAI Conference on Artificial Intelligence. vol. 36, pp. 10767–10775 (2022)
  * [19] Huang, Y., Lv, T., Cui, L., Lu, Y., Wei, F.: Layoutlmv3: Pre-training for document ai with unified text and image masking. In: Proceedings of the 30th ACM International Conference on Multimedia. pp. 4083–4091 (2022)
  * [20] Huang, Z., Chen, K., He, J., Bai, X., Karatzas, D., Lu, S., Jawahar, C.: Icdar2019 competition on scanned receipt ocr and information extraction. In: 2019 International Conference on Document Analysis and Recognition (ICDAR). pp. 1516–1520. IEEE (2019)
  * [21] Jaume, G., Ekenel, H.K., Thiran, J.P.: Funsd: A dataset for form understanding in noisy scanned documents. In: 2019 International Conference on Document Analysis and Recognition Workshops (ICDARW). vol. 2, pp. 1–6. IEEE (2019)
  * [22] Kat ti, A.R., Reisswig, C., Guder, C., Brarda, S., Bickel, S., Höhne, J., Faddoul, J.B.: Chargrid: Towards Understanding 2D Documents (Sep 2018), https://arxiv.org/abs/1809.08799v1
  * [23] Kudo, T., Richardson, J.: SentencePiece: A simple and language independent subword tokenizer and detokenizer for Neural Text Processing. In: Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing: System Demonstrations. pp. 66–71. Association for Computational Linguistics, Brussels, Belgium (Nov 2018). https://doi.org/10.18653/v1/D18-2012, https://aclanthology.org/D18-2012
  * [24] Li, C., Bi, B., Yan, M., Wang, W., Huang, S., Huang, F., Si, L.: StructuralLM: Structural pre-training for form understanding. In: Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers). pp. 6309–6318. Association for Computational Linguistics, Online (Aug 2021). https://doi.org/10.18653/v1/2021.acl-long.493, https://aclanthology.org/2021.acl-long.493
  * [25] Li, J., Sun, A., Han, J., Li, C.: A survey on deep learning for named entity recognition. IEEE Transactions on Knowledge and Data Engineering 34(1), 50–70 (2020)
  * [26] Liu, X., Gao, F., Zhang, Q., Zhao, H.: Graph convolution for multimodal information extraction from visually rich documents. In: Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Industry Papers). pp. 32–39. Association for Computational Linguistics, Minneapolis, Minnesota (Jun 2019). https://doi.org/10.18653/v1/N19-2005, https://aclanthology.org/N19-2005
  * [27] Mansouri, A., Affendey, L.S., Mamat, A.: Named entity recognition approaches. TAL 52(1), 339–344 (2008)
  * [28] Mathew, M., Karatzas, D., Jawahar, C.: Docvqa: A dataset for vqa on document images. In: Proceedings of the IEEE/CVF winter conference on applications of computer vision. pp. 2200–2209 (2021)
  * [29] Maurel, D., Friburger, N., Antoine, J.Y., Eshkol-Taravella, I., Nouvel, D.: Casen : a transducer cascade to recognize french named entities. TAL 52(1), 69–96 (2011)
  * [30] Meunier, J.L.: Optimized xy-cut for determining a page reading order. In: Eighth International Conference on Document Analysis and Recognition (ICDAR’05). pp. 347–351. IEEE (2005)
  * [31] Michael, J., Weidemann, M., Laasch, B., Labahn, R.: Icpr 2020 competition on text block segmentation on a newseye dataset. In: Pattern Recognition. ICPR International Workshops and Challenges: Virtual Event, January 10-15, 2021, Proceedings, Part VIII. p. 405–418. Springer-Verlag, Berlin, Heidelberg (2021). https://doi.org/10.1007/978-3-030-68793-9_30
  * [32] Michael, J., Weidemann, M., Laasch, B., Labahn, R.: ICPR 2020 Competition on Text Block Segmentation on a NewsEye Dataset. In: Del Bimbo, A., Cucchiara, R., Sclaroff, S., Farinella, G.M., Mei, T., Bertini, M., Escalante, H.J., Vezzani, R. (eds.) Pattern Recognition. ICPR International Workshops and Challenges. pp. 405–418. Lecture Notes in Computer Science, Springer International Publishing, Cham (2021). https://doi.org/10.1007/978-3-030-68793-9_30
  * [33] Nadeau, D., Sekine, S.: A survey of named entity recognition and classification. Lingvisticae Investigationes 30(1), 3––26 (2007)
  * [34] Nagy, G., Seth, S.C.: Hierarchical representation of optically scanned documents. In: " International conference on Pattern Recognition (1984)
  * [35] Nakayama, H.: seqeval: A python framework for sequence labeling evaluation (2018), https://github.com/chakki-works/seqeval
  * [36] Nouvel, D., Antoine, J.Y., Friburger, N., Soulet, A.: Recognizing named entities using automatically extracted transduction rules. In: 5th Language and Technology Conference. pp. 136–140. Poznan, Poland (2011)
  * [37] O’Gorman, L.: The document spectrum for page layout analysis. IEEE Transactions on pattern analysis and machine intelligence 15(11), 1162–1173 (1993)
  * [38] Oliveira, S.A., Seguin, B., Kaplan, F.: dhsegment: A generic deep-learning approach for document segmentation. In: 2018 16th International Conference on Frontiers in Handwriting Recognition (ICFHR). pp. 7–12. IEEE (2018)
  * [39] Park, S., Shin, S., Lee, B., Lee, J., Surh, J., Seo, M., Lee, H.: Cord: a consolidated receipt dataset for post-ocr parsing. In: Workshop on Document Intelligence at NeurIPS 2019 (2019)
  * [40] Ren, S., He, K., Girshick, R., Sun, J.: Faster R-CNN: towards real-time object detection with region proposal networks. In: Proceedings of the 28th International Conference on Neural Information Processing Systems \- Volume 1. NIPS’15, vol. 28, pp. 91–99. MIT Press, Cambridge, MA, USA (2015)
  * [41] Sutheebanjard, P., Premchaiswadi, W.: A modified recursive xy cut algorithm for solving block ordering problems. In: 2010 2nd International Conference on Computer Engineering and Technology. vol. 3, pp. V3–307. IEEE (2010)
  * [42] Teklia: Nerval, a python package for NER evaluation on noisy text. online (2022), https://gitlab.com/teklia/ner/nerval
  * [43] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, Ł., Polosukhin, I.: Attention is all you need. In: Adv. Neural Inf. Process. Syst. pp. 5998–6008 (2017)
  * [44] Wang, J., Jin, L., Ding, K.: LiLT: A simple yet effective language-independent layout transformer for structured document understanding. In: Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). pp. 7747–7757. Association for Computational Linguistics, Dublin, Ireland (May 2022). https://doi.org/10.18653/v1/2022.acl-long.534, https://aclanthology.org/2022.acl-long.534
  * [45] Xu, Y., Xu, Y., Lv, T., Cui, L., Wei, F., Wang, G., Lu, Y., Florencio, D., Zhang, C., Che, W., Zhang, M., Zhou, L.: LayoutLMv2: Multi-modal pre-training for visually-rich document understanding. In: Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers). pp. 2579–2591. Association for Computational Linguistics, Online (Aug 2021). https://doi.org/10.18653/v1/2021.acl-long.201, https://aclanthology.org/2021.acl-long.201
  * [46] Xu, Y., Li, M., Cui, L., Huang, S., Wei, F., Zhou, M.: Layoutlm: Pre-training of text and layout for document image understanding. In: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. pp. 1192–1200 (2020)
  * [47] Xu, Y., Lv, T., Cui, L., Wang, G., Lu, Y., Florencio, D., Zhang, C., Wei, F.: Layoutxlm: Multimodal pre-training for multilingual visually-rich document understanding. arXiv preprint arXiv:2104.08836 (2021)
  * [48] Yang, X., Yumer, E., Asente, P., Kraley, M., Kifer, D., Lee Giles, C.: Learning to extract semantic structure from documents using multimodal fully convolutional neural networks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 5315–5324 (2017)


# The exponential tracking and disturbance rejection for the unstable Burgers’
equation with general references and disturbances

Weijiu Liu
Department of Mathematics
University of Central Arkansas
201 Donaghey Avenue, Conway, AR 72035, USA
Corresponding author. Email<EMAIL_ADDRESS>Phone: 1-501-450-5661, Fax:
1-501-450-5662

###### Abstract

In solving the problem of asymptotic tracking and disturbance rejection, it
has been long always assumed that the reference to be tracked and the
disturbance to be rejected must be generated by an exosystem such as a finite
dimensional exosystem with pure imaginary eigenvalues. The objective of this
paper is to solve such a tracking problem for the unstable Burgers’ equation
without this assumption. Our treatment of this problem is straightforward.
Using the method of variable transform, the tracking problem is split into two
separate problems: a simple Neumann boundary stabilization problem and a
dynamical Neumann boundary regulator problem. Unlike the existing literature
where the regulator problem is always kept independent, the stabilization
problem here is simplified to an independent linear diffusion equation by
moving the instability term and the nonlinear term to the dynamical regulator
problem, whereas the dynamical regulator problem does depend on the
stabilization problem. Thus we can first easily handle the stabilization
problem and then solve the dynamical regulator problem by using the
fundamental theory of partial differential equations. The boundary feedforward
controller is explicitly constructed by using the reference, the disturbance
and the solution of the stabilization problem while the boundary feedback
controller is easily designed for the linear diffusion equation without using
a complex method such as the backstepping method. It is proved that, under the
designed feedback and feedforward controllers, the tracking error converges to
zero exponentially. This theoretical result is confirmed by a numerical
example.

Key Words: Burgers’ equation, Feedback and feedforward boundary control,
Dynamical regulator equation, Exponential tracking, Disturbance rejection.

##  1  Introduction

Asymptotic tracking and disturbance rejection is one of fundamental problems
in control theory. In solving this problem, it has been long always assumed
that the reference to be tracked and the disturbance to be rejected must be
generated by an exosystem such as a finite dimensional exosystem with pure
imaginary eigenvalues (see, e.g., [2, 9, 10, 11, 12, 14, 19, 25, 26, 27, 28,
29, 31, 32, 33, 34, 35, 36, 37, 39, 41]). In fact, this assumption is
sufficient, but not necessary for making the problem solvable.

The objective of this paper is to solve the tracking problem for the unstable
Burgers’ equation without this assumption:

$\displaystyle\frac{\partial u}{\partial t}$ $\displaystyle=$
$\displaystyle\nu\frac{\partial^{2}u}{\partial x^{2}}-u\frac{\partial
u}{\partial x}+au+u_{d},$ (1) $\displaystyle\nu\frac{\partial u}{\partial
x}(0,t)$ $\displaystyle=$ $\displaystyle f_{0},\quad\nu\frac{\partial
u}{\partial x}(1,t)=f_{1},$ (2) $\displaystyle u(x,0)$ $\displaystyle=$
$\displaystyle u_{0}(x).$ (3)

In the above equations, $\nu>0$ is a viscosity parameter, $a=a(x,t)$ is a
continuous function, $u_{d}=u_{d}(x,t)$ is a source disturbance, $f_{0}$ and
$f_{1}$ are control inputs, and the function $u_{0}=u_{0}(x)$ is an initial
state in an appropriate function space. When $a$ is positively large enough,
the equilibrium $0$ of the uncontrolled and undisturbed Burgers’ equation is
unstable.

Let $r(t)$ be a desired reference. We introduce the tracking error

$e(t)=\int_{0}^{1}u(x,t)dx-r(t).$ (4)

Then the problem of exponential tracking and disturbance rejection for the
Burgers’ equation is to design a feedback and feedforward controllers
$f_{0},f_{1}$ such that

$|e(t)|\leq Ce^{-\lambda t}$ (5)

where $C$ and $\lambda$ are positive constants.

Mathematical theory on the tracking problem has been well developed for finite
dimensional control systems (see, e.g., [14]) and recently extended to partial
differential equations (PDEs). Aulisa et al. [2], Byrnes et al. [6], and
Natarajan et al. [37] developed an abstract theory on the problem in Hilbert
spaces and applied it to partial differential equations (PDEs) such as the
heat equation and the wave equation. The flatness method was developed to
handle the trajectory planning and feedforward control design (see, e.g., [12,
32, 33, 34, 35, 38, 39, 40, 41]). Wagner et al. [40] studied the tracking
problem for the one-dimensional semilinear wave equation by using the flatness
method. The asymptotic tracking and disturbance rejection of the one-
dimensional parabolic partial differential equations were studied by Deutscher
[9, 10] and the case with a long time delay was investigated by Gu et al.
[13]. The same problem for the one-dimensional hyperbolic partial differential
equations and Schrödinger equation were studied by Deutscher [11] and Zhou et
al [41], respectively. In all of these important researches, the references
and disturbances were assumed to be governed by a finite dimensional exosystem
with pure imaginary eigenvalues.

Although the problem of feedback stabilization and tracking for the Burgers’
equation has received extensive attention (see, e.g., [4, 5, 7, 8, 15, 16, 18,
19, 23, 24, 30]), to my knowledge, the tracking problem has not been studied
yet in the case of general references and disturbances, which are not required
to be generated by an exosystem. Our treatment of this problem is
straightforward. Using the method of variable transform used in [3, 14, 25,
26, 27, 28, 29], the tracking problem is split into two separate problems: a
simple Neumann boundary stabilization problem and a dynamical Neumann boundary
regulator problem. Unlike the existing literature where the regulator problem
is always kept independent (see, e.g., [2, 14]), the stabilization problem
here is simplified to an independent linear diffusion equation by moving the
instability term and the nonlinear term to the dynamical regulator problem,
whereas the dynamical regulator problem does depend on the stabilization
problem. Thus we can first easily handle the stabilization problem and then
solve the dynamical regulator problem by using the fundamental theory of
partial differential equations (see, e.g., [21]). The feedforward controller
is explicitly constructed by using the reference $r$, the disturbance $u_{d}$
and the solution of the stabilization problem while the feedback controller is
easily designed for the linear diffusion equation without using a complex
method such as the backstepping method. It is proved that, under the designed
feedback and feedforward controllers, the tracking error converges to zero
exponentially. This theoretical result is confirmed by a numerical example.

##  2 Exponential tracking

In what follows, $H^{s}(0,1)$ denotes the usual Sobolev space (see [1, 22])
for any $s\in\mathbb{R}$. For $s\geq 0$, $H^{s}_{0}(0,1)$ denotes the
completion of $C_{0}^{\infty}(0,1)$ in $H^{s}(0,1)$, where
$C_{0}^{\infty}(0,1)$ denotes the space of all infinitely differentiable
functions on $(0,1)$ with compact support in $(0,1)$. We use the following
$H^{1}$ norm of $H^{1}(0,1)$

$\|u\|_{H^{1}}=\left[u(0)^{2}+\int_{0}^{1}\left(\frac{\partial u}{\partial
x}\right)^{2}dx\right]^{1/2},\;u\in H^{1}(0,1),$

which is equivalent to the usual one. The norm on $L^{2}(0,1)$ is denoted by
$\|\cdot\|$. It is easy to see that

$\|u\|^{2}\leq 2\|u\|_{H^{1}}^{2}.$ (6)

Let $X$ be a Banach space and $T>0$. We denote by $C^{n}([0,T];X)$ the space
of $n$ times continuously differentiable functions defined on $[0,T]$ with
values in $X$, and write $C([0,T];X)$ for $C^{0}([0,T];X)$. In what follows,
for simplicity, we omit the indication of the varying range of $x$ and $t$ in
equations and we understand that $x$ varies from $0$ to $1$ and $t$ from $0$
to $\infty$.

To split the tracking problem into a stabilization problem and a dynamical
regulator problem, we introduce the variable transform

$u=\hat{u}+U,\quad f_{0}=\hat{f}_{0}+F_{0},\quad f_{1}=\hat{f}_{1}+F_{1}.$ (7)

Subtracting this transform into the problem (1) - (3) and the tracking error
equation (4), we obtain

$\displaystyle\frac{\partial\hat{u}}{\partial t}+\frac{\partial U}{\partial
t}$ $\displaystyle=$ $\displaystyle\nu\frac{\partial^{2}\hat{u}}{\partial
x^{2}}+\nu\frac{\partial^{2}U}{\partial x^{2}}+a(\hat{u}+U)$
$\displaystyle-(\hat{u}+U)\left(\frac{\partial\hat{u}}{\partial
x}+\frac{\partial U}{\partial x}\right)+u_{d},$
$\displaystyle\nu\frac{\partial\hat{u}}{\partial x}+\nu\frac{\partial
U}{\partial x}(0,t)$ $\displaystyle=$ $\displaystyle\hat{f}_{0}+F_{0},$
$\displaystyle\nu\frac{\partial\hat{u}}{\partial x}+\nu\frac{\partial
U}{\partial x}(1,t)$ $\displaystyle=$ $\displaystyle\hat{f}_{1}+F_{1},$
$\displaystyle\hat{u}(x,0)+U(x,0)$ $\displaystyle=$ $\displaystyle u_{0}(x),$
$\displaystyle e(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}[\hat{u}(x,t)+U(x,t)]dx-r(t).$

This problem can be split into a stabilization problem

$\displaystyle\frac{\partial\hat{u}}{\partial t}$ $\displaystyle=$
$\displaystyle\nu\frac{\partial^{2}\hat{u}}{\partial
x^{2}}-\hat{u}\frac{\partial\hat{u}}{\partial x},$ (8)
$\displaystyle\nu\frac{\partial\hat{u}}{\partial x}(0,t)$ $\displaystyle=$
$\displaystyle\hat{f}_{0},\quad\nu\frac{\partial\hat{u}}{\partial
x}(1,t)=\hat{f}_{1},$ (9) $\displaystyle\hat{u}(x,0)$ $\displaystyle=$
$\displaystyle u_{0}(x)-r(0),$ (10) $\displaystyle e(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\hat{u}(x,t)dx,$ (11)

and a dynamical regulator problem

$\displaystyle\frac{\partial U}{\partial t}$ $\displaystyle=$
$\displaystyle\nu\frac{\partial^{2}U}{\partial x^{2}}-U\frac{\partial
U}{\partial x}+a(\hat{u}+U)$ (12) $\displaystyle-\left(\hat{u}\frac{\partial
U}{\partial x}+U\frac{\partial\hat{u}}{\partial x}\right)+u_{d},$
$\displaystyle\nu\frac{\partial U}{\partial x}(0,t)$ $\displaystyle=$
$\displaystyle F_{0},\quad\nu\frac{\partial U}{\partial x}(1,t)=F_{1}$ (13)
$\displaystyle U(x,0)$ $\displaystyle=$ $\displaystyle r(0),$ (14)
$\displaystyle\int_{0}^{1}U(x,t)dx$ $\displaystyle=$ $\displaystyle r(t).$
(15)

In the control design (see, e.g., [2, 14]), the terms $a(\hat{u}+U)$ and
$\hat{u}\frac{\partial U}{\partial x}+U\frac{\partial\hat{u}}{\partial x}$ are
usually put in the equation (8) in the stabilization problem such that the
regulator problem is independent from the stabilization problem. However, if
we do so for the Burgers’ equation, then the feedback controllers
$\hat{f}_{0}$ and $\hat{f}_{1}$ for the stabilization problem (8) - (10) is
difficult or impossible to design. In fact, we can further simplify the
stabilization problem by moving the term
$\hat{u}\frac{\partial\hat{u}}{\partial x}$ from the stabilization problem to
the regulator problem and then obtain the following linear stabilization
problem and the regulator problem:

$\displaystyle\frac{\partial\hat{u}}{\partial t}$ $\displaystyle=$
$\displaystyle\nu\frac{\partial^{2}\hat{u}}{\partial x^{2}},$ (16)
$\displaystyle\nu\frac{\partial\hat{u}}{\partial x}(0,t)$ $\displaystyle=$
$\displaystyle\hat{f}_{0},\quad\nu\frac{\partial\hat{u}}{\partial
x}(1,t)=\hat{f}_{1},$ (17) $\displaystyle\hat{u}(x,0)$ $\displaystyle=$
$\displaystyle u_{0}(x)-r(0),$ (18) $\displaystyle e(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\hat{u}(x,t)dx,$ (19)

and

$\displaystyle\frac{\partial U}{\partial t}$ $\displaystyle=$
$\displaystyle\nu\frac{\partial^{2}U}{\partial x^{2}}-U\frac{\partial
U}{\partial x}+a(\hat{u}+U)$ (20)
$\displaystyle-\hat{u}\frac{\partial\hat{u}}{\partial
x}-\left(\hat{u}\frac{\partial U}{\partial x}+U\frac{\partial\hat{u}}{\partial
x}\right)+u_{d},$ $\displaystyle\nu\frac{\partial U}{\partial x}(0,t)$
$\displaystyle=$ $\displaystyle F_{0},\quad\nu\frac{\partial U}{\partial
x}(1,t)=F_{1}$ (21) $\displaystyle U(x,0)$ $\displaystyle=$ $\displaystyle
r(0),$ (22) $\displaystyle\int_{0}^{1}U(x,t)dx$ $\displaystyle=$
$\displaystyle r(t).$ (23)

Because the regulator problem (12) - (15) of (20) - (23) is not dissipative,
its well-posedness is challenging and open. Since $\int_{0}^{1}U(x,t)dx=r(t)$
exists for all times, we could conjecture that it has a unique global
solution. We will use this conjecture in the following theorem.

Using the stabilization problem (8) - (10) and the regulator problem (12) -
(15), we can design the feedback and feedforward controllers as stated in the
following theorem.

######  Theorem 2.1.

Assume that $k>1/6$ and the initial condition $u_{0}\in H^{2}(0,1)$. Suppose
that $a(x,t)$ and $u_{d}(x,t)$ are continuous and $r(t)$ is continuously
differentiable. Let $\hat{u}$ be the solution of the stabilization problem (8)
- (10) and assume that the regulator problem (12) - (15) has a unique global
classical solution. Then, under the feedback and feedforward controllers:

$\displaystyle\hat{f}_{0}(t)$ $\displaystyle=$ $\displaystyle
k\left[\hat{u}(0,t)+[\hat{u}(0,t)]^{3}\right],$ (24)
$\displaystyle\hat{f}_{1}(t)$ $\displaystyle=$
$\displaystyle-k\left[\hat{u}(1,t)+[\hat{u}(1,t)]^{3}\right],$ (25)
$\displaystyle F_{0}(t)$ $\displaystyle=$
$\displaystyle\hat{u}(0,t)U(0,t)+\frac{1}{2}[U(0,t)]^{2},$ (26) $\displaystyle
F_{1}(t)$ $\displaystyle=$
$\displaystyle\hat{u}(1,t)U(1,t)+\frac{1}{2}[U(1,t)]^{2}+r^{\prime}(t)$ (27)
$\displaystyle-\int_{0}^{1}[a(x,t)(\hat{u}(x,t)+U(x,t))+u_{d}(x,t)]dx.$

the problem (1) - (3) has a unique solution satisfying

$|e(t)|\leq\|u_{0}-r(0)\|e^{-\lambda t/2},$ (28)

where $\lambda=\min\left(\nu,k-\frac{1}{6}\right)$.

_Proof_. If $k>1/6$ and the initial condition $u_{0}\in H^{2}(0,1)$, then it
was proved in [18, 23] that the stabilization problem (8) - (10) with the
feedback controllers (24) and (25) has a unique solution satisfying

$\hat{u}\in C([0,\infty);H^{2}(0,1)).$

Moreover, multiplying the equation (8) by $\hat{u}$ and integrating it from
$0$ to $1$, we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{0}^{1}[\hat{u}(x,t)]^{2}dx$
$\displaystyle=$
$\displaystyle-k\left[\hat{u}(1,t)+[\hat{u}(1,t)]^{3}\right]\hat{u}(1,t)$
$\displaystyle-k\left[\hat{u}(0,t)+[\hat{u}(0,t)]^{3}\right]\hat{u}(0,t)$
$\displaystyle-\nu\int_{0}^{1}\left[\frac{\partial\hat{u}}{\partial
x}(x,t)\right]^{2}dx-\frac{1}{3}\left([\hat{u}(1,t)]^{3}-[\hat{u}(0,t)]^{3}\right)$
$\displaystyle\leq$
$\displaystyle-\left(k-\frac{1}{6}\right)\left([\hat{u}(0,t)]^{2}+[\hat{u}(1,t)]^{2}\right.$
$\displaystyle\left.+[\hat{u}(0,t)]^{4}+[\hat{u}(1,t)]^{4}\right)-\nu\int_{0}^{1}\left[\frac{\partial\hat{u}}{\partial
x}(x,t)\right]^{2}dx$ $\displaystyle\leq$
$\displaystyle-\left(k-\frac{1}{6}\right)[\hat{u}(0,t)]^{2}-\nu\int_{0}^{1}\left[\frac{\partial\hat{u}}{\partial
x}(x,t)\right]^{2}dx$ $\displaystyle\leq$
$\displaystyle-\lambda\left([\hat{u}(0,t)]^{2}+\int_{0}^{1}\left[\frac{\partial\hat{u}}{\partial
x}(x,t)\right]^{2}dx\right)$ $\displaystyle\leq$
$\displaystyle-\frac{\lambda}{2}\int_{0}^{1}\left[\hat{u}(x,t)\right]^{2}dx.\quad(\mbox{use
}\eqref{l2-h1})$

Solving this inequality, we obtain

$\|\hat{u}(t)\|\leq\|u_{0}-r(0)\|e^{-\lambda t/2}.$ (29)

Integrating the equation (12) over $[0,1]$ , we obtain

$\displaystyle\frac{d}{dt}\int_{0}^{1}U(x,t)dx=\nu\frac{\partial U}{\partial
x}(1,t)-\nu\frac{\partial U}{\partial x}(0,t)$
$\displaystyle+\hat{u}(0,t)U(0,t)+\frac{1}{2}U^{2}(0,t)$
$\displaystyle-\hat{u}(1,t)U(1,t)-\frac{1}{2}U^{2}(1,t)+\int_{0}^{1}u_{d}(x,t)dx$
$\displaystyle+\int_{0}^{1}a(x,t)[\hat{u}(x,t)+U(x,t)]dx.$

It then follows from the boundary condition (13) and the feedforward
controllers (26) and (27) that

$\frac{d}{dt}\int_{0}^{1}U(x,t)dx=r^{\prime}(t).$

Integrating this equation and using the initial condition (14), we obtain

$\int_{0}^{1}U(x,t)dx=r(t).$ (30)

So $U$ satisfies the equation (15). Since $\int_{0}^{1}U(x,t)dx$ exists for
all $t\geq 0$, the problem (12) - (14) has a unique solution $U$ for all
$t>0$.

Finally it follows from the equations (11), (15), and (29) that

$\displaystyle|e(t)|$ $\displaystyle=$
$\displaystyle\left|\int_{0}^{1}u(x,t)dx-r(t)\right|$ $\displaystyle=$
$\displaystyle\left|\int_{0}^{1}u(x,t)dx-\int_{0}^{1}U(x,t)dx\right|$
$\displaystyle=$ $\displaystyle\left|\int_{0}^{1}\hat{u}(x,t)dx\right|$
$\displaystyle\leq$
$\displaystyle\left[\int_{0}^{1}dx\right]^{1/2}\left[\int_{0}^{1}[\hat{u}(x,t)]^{2}dx\right]^{1/2}$
$\displaystyle\leq$ $\displaystyle\|u_{0}-r(0)\|e^{-\lambda t/2}.$

This completes the proof.

Using the stabilization problem (16) - (18) and the regulator problem (20) -
(23), we can design the feedback and feedforward controllers as stated in the
following theorem.

######  Theorem 2.2.

Assume that $k>0$ and the initial condition $u_{0}\in H^{2}(0,1)$. Suppose
that $a(x,t)$ and $u_{d}(x,t)$ are continuous and $r(t)$ is continuously
differentiable. Let $\hat{u}$ be the solution of the stabilization problem
(16) - (18) and assume that the regulator problem (20) - (23) has a unique
global classical solution. Then, under the feedback and feedforward
controllers:

$\displaystyle\hat{f}_{0}(t)$ $\displaystyle=$ $\displaystyle k\hat{u}(0,t),$
(31) $\displaystyle\hat{f}_{1}(t)$ $\displaystyle=$
$\displaystyle-k\hat{u}(1,t),$ (32) $\displaystyle F_{0}(t)$ $\displaystyle=$
$\displaystyle\hat{u}(0,t)U(0,t)+\frac{1}{2}\left([\hat{u}(0,t)]^{2}+[U(0,t)]^{2}\right),$
(33) $\displaystyle F_{1}(t)$ $\displaystyle=$
$\displaystyle\hat{u}(1,t)U(1,t)+\frac{1}{2}\left([\hat{u}(1,t)]^{2}+[U(1,t)]^{2}\right)+r^{\prime}(t)$
(34) $\displaystyle-\int_{0}^{1}[a(x,t)(\hat{u}(x,t)+U(x,t))+u_{d}(x,t)]dx.$

the problem (1) - (3) has a unique solution satisfying

$|e(t)|\leq\|u_{0}-r(0)\|e^{-\lambda t/2},$ (35)

where $\lambda=\min\left(\nu,k\right)$.

The proof of this theorem is the same as the proof of Theorem 2.1.

##  3 A numerical example

We conduct a numerical simulation to confirm the above theoretical result. In
the numerical computations, we take $\nu=5$, $a=20$, $k=15$, $u_{0}(x)=0$,
$r(t)=2+4\cos(\pi t)-3\sin(\pi t)$, and $u_{d}(x,t)=3+5\cos(\pi x)\sin(\pi
t)-2\sin(\pi x)\cos(\pi t)$. Then the problem (1) - (3) and the problem (8) -
(10) are solved numerically by the difference method. The Figure 1 shows that
the average of $u$, $u_{a}(t)=\int_{0}^{1}u(x,t)dx$, quickly tracks the
reference $r(t)=2+\cos(\pi t)-3\sin(\pi t)$ under the feedback and feedforward
controllers either (24) - (27) or (31) - (34).

Figure 1: The average of $u$, $u_{a}(t)=\int_{0}^{1}u(x,t)dx$, quickly tracks
the reference $r(t)=2+4\cos(\pi t)-3\sin(\pi t)$ under the feedback and
feedforward controllers either (24) - (27) (left figure) or (31) - (34) (right
figure).

##  4 Discussion

Other tracking errors can be considered. Let $r_{0}(t)$ and $r_{1}(t)$ be two
desired references and define the tracking error by

$e_{1}(t)=u(0,t)-r_{0}(t)+u(1,t)-r_{1}(t).$ (36)

Using the variable transform (7) and this tracking error, we can obtain the
stabilization problem

$\displaystyle\frac{\partial\hat{u}}{\partial t}$ $\displaystyle=$
$\displaystyle\nu\frac{\partial^{2}\hat{u}}{\partial
x^{2}}-\hat{u}\frac{\partial\hat{u}}{\partial x},$ (37)
$\displaystyle\nu\frac{\partial\hat{u}}{\partial x}(0,t)$ $\displaystyle=$
$\displaystyle\hat{f}_{0},\quad\nu\frac{\partial\hat{u}}{\partial
x}(1,t)=\hat{f}_{1},$ (38) $\displaystyle\hat{u}(x,0)$ $\displaystyle=$
$\displaystyle u_{0}(x)-r_{0}(0)(1-x)-r_{1}(0)x,$ (39) $\displaystyle
e_{1}(t)$ $\displaystyle=$ $\displaystyle\hat{u}(0,t)+\hat{u}(1,t),$ (40)

and the dynamical regulator problem

$\displaystyle\frac{\partial U}{\partial t}$ $\displaystyle=$
$\displaystyle\nu\frac{\partial^{2}U}{\partial x^{2}}-U\frac{\partial
U}{\partial x}+a(\hat{u}+U)$ (41) $\displaystyle-\left(\hat{u}\frac{\partial
U}{\partial x}+U\frac{\partial\hat{u}}{\partial x}\right)+u_{d},$
$\displaystyle U(0,t)$ $\displaystyle=$ $\displaystyle r_{0}(t),\quad
U(1,t)=r_{1}(t),$ (42) $\displaystyle U(x,0)$ $\displaystyle=$ $\displaystyle
r_{0}(0)(1-x)+r_{1}(0)x,$ (43) $\displaystyle\nu\frac{\partial U}{\partial
x}(0,t)$ $\displaystyle=$ $\displaystyle F_{0},\quad\nu\frac{\partial
U}{\partial x}(1,t)=F_{1},$ (44)

If the problem (41) - (43) has a global solution, then the feedforward
controllers are given by the equation (44). However, it seems challenging to
show it has a global solution even though it has a unique classical solution
within some time from $0$ to $T$ (see, e.g., [21]).

We can also consider the tracking error

$e_{2}(t)=u(0,t)-r_{0}(t)+\int_{0}^{1}u(x,t)dx-r(t).$ (45)

Using the variable transform (7) and this tracking error, we can obtain the
stabilization problem

$\displaystyle\frac{\partial\hat{u}}{\partial t}$ $\displaystyle=$
$\displaystyle\nu\frac{\partial^{2}\hat{u}}{\partial
x^{2}}-\hat{u}\frac{\partial\hat{u}}{\partial x},$ (46)
$\displaystyle\nu\frac{\partial\hat{u}}{\partial x}(0,t)$ $\displaystyle=$
$\displaystyle\hat{f}_{0},\quad\nu\frac{\partial\hat{u}}{\partial
x}(1,t)=\hat{f}_{1},$ (47) $\displaystyle\hat{u}(x,0)$ $\displaystyle=$
$\displaystyle u_{0}(x)-u_{r}(0),$ (48) $\displaystyle e_{2}(t)$
$\displaystyle=$ $\displaystyle\hat{u}(0,t)+\int_{0}^{1}\hat{u}(x,t)dx,$ (49)

and the dynamical regulator problem

$\displaystyle\frac{\partial U}{\partial t}$ $\displaystyle=$
$\displaystyle\nu\frac{\partial^{2}U}{\partial x^{2}}-U\frac{\partial
U}{\partial x}+a(\hat{u}+U)$ (50) $\displaystyle-\left(\hat{u}\frac{\partial
U}{\partial x}+U\frac{\partial\hat{u}}{\partial x}\right)+u_{d},$
$\displaystyle U(0,t)$ $\displaystyle=$ $\displaystyle
r_{0}(t),\quad\nu\frac{\partial U}{\partial x}(1,t)=F_{1},$ (51)
$\displaystyle U(x,0)$ $\displaystyle=$ $\displaystyle r(0),$ (52)
$\displaystyle\int_{0}^{1}U(x,t)dx$ $\displaystyle=$ $\displaystyle r(t),$
(53) $\displaystyle\nu\frac{\partial U}{\partial x}(0,t)$ $\displaystyle=$
$\displaystyle F_{0}.$ (54)

To find $F_{1}$, we integrate the equation (50) from $0$ to $1$ and use the
equation (53) to obtain

$\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle\nu\frac{\partial
U}{\partial x}(0,t)+\frac{1}{2}[(U(1,t))^{2}-(U(0,t))^{2}]$ (55)
$\displaystyle+\hat{u}(1,t)U(1,t)-\hat{u}(0,t)U(0,t)+r^{\prime}(t)$
$\displaystyle-\int_{0}^{1}[a(x,t)(\hat{u}(x,t)+U(x,t)+u_{d}(x,t)]dx.$

This results the following complex boundary value problem

$\displaystyle\frac{\partial U}{\partial t}$ $\displaystyle=$
$\displaystyle\nu\frac{\partial^{2}U}{\partial
x^{2}}-(\hat{u}+U)\frac{\partial U}{\partial
x}-U\frac{\partial\hat{u}}{\partial x}+u_{d},$ (56) $\displaystyle U(0,t)$
$\displaystyle=$ $\displaystyle r_{0}(t),$ (57)
$\displaystyle\nu\frac{\partial U}{\partial x}(1,t)$ $\displaystyle=$
$\displaystyle\nu\frac{\partial U}{\partial
x}(0,t)+\frac{1}{2}[(U(1,t))^{2}-(U(0,t))^{2}]$ (58)
$\displaystyle+\hat{u}(1,t)U(1,t)-\hat{u}(0,t)U(0,t)+r^{\prime}(t)$
$\displaystyle-\int_{0}^{1}[a(x,t)(\hat{u}(x,t)+U(x,t)+u_{d}(x,t)]dx,$
$\displaystyle U(x,0)$ $\displaystyle=$ $\displaystyle r(0).$ (59)

It seems that the proof of the solution existence of the problem is
challenging.

## References

  * [1] R. Adams, Sobolev Spaces. Academic Press, New York (1975).
  * [2] E. Aulisa and D. Gilliam, A Practical Guide to Geometric Regulation for Distributed Parameter Systems, Chapman and Hall/CRC, Boca Raton, FL, 2015.
  * [3] Brandon Ashley and Weijiu Liu, Asymptotic tracking and disturbance rejection of blood glucose regulation system, Mathematical Biosciences, 289, 2017, 78-88.
  * [4] A. Balogh and M. Krstić, Burgers’ Equation with Nonlinear Boundary Feedback: $H^{1}$ Stability, Well-Posedness and Simulation. Mathematical Problems in Engineering, vol. 6, 2000, Article ID 649242, https://doi.org/10.1155/S1024123X00001320
  * [5] J.A. Burns and S. Kang, A control problem for Burgers’ equation with bounded input/output. Nonlinear Dynamics 2 (1992) 235-262.
  * [6] C. I. Byrnes, I. G. Lauko, D. S. Gilliam, V. I. Shubov, Output regulation for linear distributed parameter systems, IEEE Trans. Autom. Control, 45, no. 12, pp. 2236-2252, 2000.
  * [7] C.I. Byrnes, D.S. Gilliam and V.I. Shubov, Boundary control for a viscous Burgers’ equation, in Identification Control for Systems Governed by Partial Differential Equations, H.T. Banks, R.H. Fabiano and K. Ito Eds., SIAM (1993) 171-185.
  * [8] H. Choi, R. Temam, P. Moin and J. Kim, Feedback control for unsteady flow and its application to the stochastic Burgers’ equation. J. Fluid Mech. 253 (1993) 509-543.
  * [9] J. Deutscher, A backstepping approach to the output regulation of boundary controlled parabolic PDEs, Automatica, 57 (2015), 56-64.
  * [10] J. Deutscher, Backstepping design of robust output feedback regulators for boundary controlled parabolic PDEs, IEEE Trans. Autom. Control, 61 (2016), 2288-2294.
  * [11] J. Deutscher, Finite-time output regulation for linear $2\times 2$ hyperbolic systems using backstepping, Automatica, 75 (2017), 54-62.
  * [12] G. Freudenthaler and T. Meurer, PDE-based multi-agent formation control using flatness and backstepping: Analysis, design and robot experiments, Automatica, Vol. 115 (2020), 108897
  * [13] J. Gu and J. Wang, Backstepping state feedback regulator design for an unstable reaction-diffusion PDE with long time delay, J. Dyn. Control Sys., 24(2018), 563-576.
  * [14] J. Huang, Nonlinear Output Regulation, Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2004)
  * [15] K. Ito and S. Kang, A dissipative feedback control for systems arising in fluid dynamics. SIAM J. Control Optim. 32 (1994) 831-854.
  * [16] K. Ito and Y. Yan, Viscous scalar conservation law with nonlinear flux feedback and global attractors. J. Math. Anal. Appl. 227 (1998) 271-299.
  * [17] H. K. Khalil, Nonlinear Systems. Prentice-Hall, Inc., New Jersey (1996).
  * [18] M. Krstić, On global stabilization of Burgers’ equation by boundary control. Systems & Control Letters 37 (1999) 123-141.
  * [19] M. Krstić, L. Magnis, and R. Vazquez, Nonlinear control of the viscous Burgers equation: trajectory generation, tracking, and observer design, Journal of Dynamic Systems, Measurement, and Control, vol. 131 (2009), 021012 -1-8.
  * [20] M. Krstić, I. Kanellakopoulos and P. Kokotović, Nonlinear and Adaptive Control Design. John Wiley & Sons, Inc., New York (1995).
  * [21] O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Society, Providence, Rhode Island, 1968.
  * [22] J.L. Lions and E. Magenes, Non-homogeneous Boundary value Problems and Applications, Vol.1. Springer-Verlag, Berlin (1972).
  * [23] W. Liu and M. Krstić, Backstepping Boundary Control of Burgers’ Equation with Actuator Dynamics. Systems and Control Letters, 41 (4) 2000, 291 - 303.
  * [24] W. Liu and M. Krstić, Adaptive control of Burgers’ equation with unknown viscosity. International Journal on Adaptive Control and Signal Processing  15(7), 2001, 745-766.
  * [25] W. Liu, Elementary Feedback Stabilization of the Linear Reaction Diffusion Equation and the Wave Equation, Mathematiques et Applications, Vol. 66, Springer, 2010.
  * [26] W. Liu, Boundary feedforward and feedback control for the exponential tracking of the unstable high-dimensional wave equation, Journal of Mathematical Analysis and Applications, vol 499, issue 1, July, 2021, https://doi.org/10.1016/j.jmaa.2021.125010
  * [27] W. Liu, Independence of convergence rate of the wave tracking error on structures of feedforward controllers, Automatica, https://doi.org/10.1016/j.automatica.2020.109264
  * [28] W. Liu, Feedforward boundary control for the regulation of a passive and diffusive scalar in 2-D unsteady flows, IEEE Transactions on Automatic Control, vol. 65, no. 11, pp. 4882 - 4886, 2020.
  * [29] W. Liu, A mathematical model for the robust blood glucose tracking. Mathematical Biosciences and Engineering, 16 (2), 2019, 759 - 781.
  * [30] H. V. Ly, K. D. Mease and E.S. Titi, Distributed and boundary control of the viscous Burgers’ equation. Numer. Funct. Anal. Optim. 18 (1997) 143-188.
  * [31] Florian Malchow and Oliver Sawodny, Feedforward Control of Inhomogeneous Linear First Order Distributed Parameter Systems. 2011 American Control Conference, San Francisco, CA, USA, 2011, 3597 - 3602.
  * [32] T. Meurer and M. Zeitz, Feedforward and Feedback Tracking Control of Nonlinear Diffusion-Convection-Reaction Systems Using Summability Methods. Ind. Eng. Chem. Res. 44, 2532 - 2548, 2005.
  * [33] Thomas Meurer and Andreas Kugi, Trajectory Planning and Feedforward Control Design for the Temperature Distribution in a Cuboid. Proc. Appl. Math. Mech. 6, 825 - 826, (2006)
  * [34] Thomas Meurer and M. Krstic, Finite-time multi-agent deployment: A nonlinear PDE motion planning approach, Automatica 47 (2011) 2534 - 2542.
  * [35] Thomas Meurer, Control of Higher–Dimensional PDEs: Flatness and Backstepping Designs. Springer, New York, 2013
  * [36] Thomas Meurer, Flatness-based motion planning and tracking. Lecture Notes for the Workshop ”New Trends in Control of Distributed Parameter Systems” at the 2016 IEEE CDC, Las Vegas (NV), USA.
  * [37] V. Natarajan, D. S. Gilliam, and G. Weiss, The State Feedback Regulator Problem for Regular Linear Systems, IEEE Trans. Autom. Control 59, pp. 2708-2722 (2014)
  * [38] A. Pisano, Y. Orlov, and E. Usai, Tracking control of the uncertain heat and wave equation via power-fractional and sliding-model techniques. SIAM J. Control. Optim. 49, No. 2, 2011, 363 - 382.
  * [39] Tilman Utz and Andreas Kugi, Flatness-based feedforward control design of a system of parabolic PDEs based on finite difference semi-discretization. Proc. Appl. Math. Mech. 12, 731 – 732 (2012).
  * [40] M. Wagner, T. Meurer, and A. Kugi, Feedforward control design for a semilinear wave equation, Proc. Appl. Math. Mech., 9, pp. 7-10, 2000.
  * [41] H.-C. Zhou and G. Weiss: Solving the regulator problem for a 1-D Schrödinger equation via backstepping, IFAC PapersOnLine, 50-1 (2017), 4516-4521.