#  LaTeX Author Guidelines for ICCV Proceedings

First Author
Institution1
Institution1 address
<EMAIL_ADDRESS>Second Author
Institution2
First line of institution2 address
<EMAIL_ADDRESS>
###### Abstract

The ABSTRACT is to be in fully-justified italicized text, at the top of the
left-hand column, below the author and affiliation information. Use the word
“Abstract” as the title, in 12-point Times, boldface type, centered relative
to the column, initially capitalized. The abstract is to be in 10-point,
single-spaced type. Leave two blank lines after the Abstract, then begin the
main text. Look at previous ICCV abstracts to get a feel for style and length.

##  1 Introduction

Please follow the steps outlined below when submitting your manuscript to the
IEEE Computer Society Press. This style guide now has several important
modifications (for example, you are no longer warned against the use of sticky
tape to attach your artwork to the paper), so all authors should read this new
version.

###  1.1 Language

All manuscripts must be in English.

###  1.2 Dual submission

Please refer to the author guidelines on the ICCV 2023 web page for a
discussion of the policy on dual submissions.

###  1.3 Paper length

Papers, excluding the references section, must be no longer than eight pages
in length. The references section will not be included in the page count, and
there is no limit on the length of the references section. For example, a
paper of eight pages with two pages of references would have a total length of
10 pages. There will be no extra page charges for ICCV 2023.

Overlength papers will simply not be reviewed. This includes papers where the
margins and formatting are deemed to have been significantly altered from
those laid down by this style guide. Note that this LaTeX guide already sets
figure captions and references in a smaller font. The reason such papers will
not be reviewed is that there is no provision for supervised revisions of
manuscripts. The reviewing process cannot determine the suitability of the
paper for presentation in eight pages if it is reviewed in eleven.

###  1.4 The ruler

The LaTeX style defines a printed ruler which should be present in the version
submitted for review. The ruler is provided in order that reviewers may
comment on particular lines in the paper without circumlocution. If you are
preparing a document using a non-LaTeX document preparation system, please
arrange for an equivalent ruler to appear on the final output pages. The
presence or absence of the ruler should not change the appearance of any other
content on the page. The camera-ready copy should not contain a ruler. (LaTeX
users may uncomment the `\iccvfinalcopy` command in the document preamble.)
Reviewers: note that the ruler measurements do not align well with the lines
in the paper — this turns out to be very difficult to do well when the paper
contains many figures and equations, and, when done, looks ugly. Just use
fractional references (e.g. this line is $095.5$), although in most cases one
would expect that the approximate location will be adequate.

###  1.5 Mathematics

Please number all of your sections and displayed equations. It is important
for readers to be able to refer to any particular equation. Just because you
didn’t refer to it in the text doesn’t mean some future readers might not need
to refer to it. It is cumbersome to have to use circumlocutions like “the
equation second from the top of page 3 column 1”. (Note that the ruler will
not be present in the final copy, so is not an alternative to equation
numbers). All authors will benefit from reading Mermin’s description of how to
write mathematics: http://www.pamitc.org/documents/mermin.pdf.

###  1.6 Blind review

Many authors misunderstand the concept of anonymizing for blind review. Blind
review does not mean that one must remove citations to one’s own work — in
fact, it is often impossible to review a paper unless the previous citations
are known and available.

Blind review means that you do not use the words “my” or “our” when citing
previous work. That is all. (But see below for tech reports.)

Saying “this builds on the work of Lucy Smith [1]” does not say that you are
Lucy Smith; it says that you are building on her work. If you are Smith and
Jones, do not say “as we show in [7]”, say “as Smith and Jones show in [7]”
and at the end of the paper, include reference 7 as you would any other cited
work.

An example of a bad paper just asking to be rejected:

> An analysis of the frobnicatable foo filter.
>
> In this paper, we present a performance analysis of our previous paper [1]
> and show it to be inferior to all previously known methods. Why the previous
> paper was accepted without this analysis is beyond me.
>
> [1] Removed for blind review

An example of an acceptable paper:

> An analysis of the frobnicatable foo filter.
>
> In this paper, we present a performance analysis of the paper of Smith _et
> al_. [1] and show it to be inferior to all previously known methods. Why the
> previous paper was accepted without this analysis is beyond me.
>
> [1] Smith, L and Jones, C. “The frobnicatable foo filter, a fundamental
> contribution to human knowledge”. Nature 381(12), 1-213.

If you are making a submission to another conference at the same time, which
covers similar or overlapping material, you may need to refer to that
submission in order to explain the differences, just as you would if you had
previously published related work. In such cases, include the anonymized
parallel submission [Authors14] as additional material and cite it as

> [1] Authors. “The frobnicatable foo filter”, F&G 2014 Submission ID 324,
> Supplied as additional material fg324.pdf.

Finally, you may feel you need to tell the reader that more details can be
found elsewhere, and refer them to a technical report. For conference
submissions, the paper must stand on its own, and not require the reviewer to
go to a tech report for further details. Thus, you may say in the body of the
paper “further details may be found in [Authors14b]”. Then submit the tech
report as additional material. Again, you may not assume the reviewers will
read this material.

Sometimes your paper is about a problem that you tested using a tool that is
widely known to be restricted to a single institution. For example, let’s say
it’s 1969, you have solved a key problem on the Apollo lander, and you believe
that the ICCV70 audience would like to hear about your solution. The work is a
development of your celebrated 1968 paper entitled “Zero-g frobnication: How
being the only people in the world with access to the Apollo lander source
code makes us a wow at parties”, by Zeus _et al_.

You can handle this paper like any other. Don’t write “We show how to improve
our previous work [Anonymous, 1968]. This time we tested the algorithm on a
lunar lander [name of lander removed for blind review]”. That would be silly,
and would immediately identify the authors. Instead, write the following:

> We describe a system for zero-g frobnication. This system is new because it
> handles the following cases: A, B. Previous systems [Zeus et al. 1968]
> didn’t handle case B properly. Ours handles it by including a foo term in
> the bar integral.
>
> …
>
> The proposed system was integrated with the Apollo lunar lander, and went
> all the way to the moon, don’t you know. It displayed the following
> behaviors which shows how well we solved cases A and B: …

As you can see, the above text follows standard scientific conventions, reads
better than the first version and does not explicitly name you as the authors.
A reviewer might think it likely that the new paper was written by Zeus _et
al_., but cannot make any decision based on that guess. He or she would have
to be sure that no other authors could have been contracted to solve problem
B.

FAQ
Q: Are acknowledgements OK?
A: No. Leave them for the final copy.
Q: How do I cite my results reported in open challenges? A: To conform with
the double-blind review policy, you can report the results of other challenge
participants together with your results in your paper. For your results,
however, you should not identify yourself and should not mention your
participation in the challenge. Instead, present your results referring to the
method proposed in your paper and draw conclusions based on the experimental
comparison to other results.

Figure 1: Example of a caption. It is set in Roman so mathematics (always set
in Roman: $B\sin A=A\sin B$) may be included without an ugly clash.

###  1.7 Miscellaneous

Compare the following:
`$conf_a$` $conf_{a}$ `$\mathit{conf}_a$` $\mathit{conf}_{a}$
See The TeXbook, p165.

The space after _e.g_., meaning “for example”, should not be a sentence-ending
space. So _e.g_. is correct, e.g. is not. The provided `\eg` macro takes care
of this.

When citing a multi-author paper, you may save space by using “et alia”,
shortened to “ _et al_.” (not “et. al.” as “et” is a complete word.) However,
use it only when there are three or more authors. Thus, the following is
correct: “ Frobnication has been trendy lately. It was introduced by Alpher
[Alpher02], and subsequently developed by Alpher and Fotheringham-Smythe
[Alpher03], and Alpher _et al_. [Alpher04].”

This is incorrect: “… subsequently developed by Alpher _et al_. [Alpher03] …”
because reference [Alpher03] has just two authors. If you use the `\etal`
macro provided, then you need not worry about double periods when used at the
end of a sentence as in Alpher _et al_.

For this citation style, keep multiple citations in numerical (not
chronological) order, so prefer [Alpher03, Alpher02, Authors14] to [Alpher02,
Alpher03, Authors14].

Figure 2: Example of a short caption, which should be centered.

##  2 Formatting your paper

All text must be in a two-column format. The total allowable width of the text
area is $6\frac{7}{8}$ inches (17.5 cm) wide by $8\frac{7}{8}$ inches (22.54
cm) high. Columns are to be $3\frac{1}{4}$ inches (8.25 cm) wide, with a
$\frac{5}{16}$ inch (0.8 cm) space between them. The main title (on the first
page) should begin 1.0 inch (2.54 cm) from the top edge of the page. The
second and following pages should begin 1.0 inch (2.54 cm) from the top edge.
On all pages, the bottom margin should be 1-1/8 inches (2.86 cm) from the
bottom edge of the page for $8.5\times 11$-inch paper; for A4 paper,
approximately 1-5/8 inches (4.13 cm) from the bottom edge of the page.

###  2.1 Margins and page numbering

All printed material, including text, illustrations, and charts, must be kept
within a print area 6-7/8 inches (17.5 cm) wide by 8-7/8 inches (22.54 cm)
high.

Page numbers should be included for review submissions but not for the final
paper. Review submissions papers should have page numbers in the footer with
numbers centered and .75 inches (1.905 cm) from the bottom of the page and
start on the first page with the number 1.

Page numbers will be added by the publisher to all camera-ready papers prior
to including them in the proceedings and before submitting the papers to IEEE
Xplore. As such, your camera-ready submission should not include any page
numbers. Page numbers should automatically be removed by uncommenting (if it’s
not already) the line

    % \iccvfinalcopy

near the beginning of the .tex file.

###  2.2 Type-style and fonts

Wherever Times is specified, Times Roman may also be used. If neither is
available on your word processor, please use the font closest in appearance to
Times to which you have access.

MAIN TITLE. Center the title 1-3/8 inches (3.49 cm) from the top edge of the
first page. The title should be in Times 14-point, boldface type. Capitalize
the first letter of nouns, pronouns, verbs, adjectives, and adverbs; do not
capitalize articles, coordinate conjunctions, or prepositions (unless the
title begins with such a word). Leave two blank lines after the title.

AUTHOR NAME(s) and AFFILIATION(s) are to be centered beneath the title and
printed in Times 12-point, non-boldface type. This information is to be
followed by two blank lines.

The ABSTRACT and MAIN TEXT are to be in a two-column format.

MAIN TEXT. Type main text in 10-point Times, single-spaced. Do NOT use double-
spacing. All paragraphs should be indented 1 pica (approx. 1/6 inch or 0.422
cm). Make sure your text is fully justified—that is, flush left and flush
right. Please do not place any additional blank lines between paragraphs.

Figure and table captions should be 9-point Roman type as in Figures 1 and 2.
Short captions should be centered.

Callouts should be 9-point Helvetica, non-boldface type. Initially capitalize
only the first word of section titles and first-, second-, and third-order
headings.

FIRST-ORDER HEADINGS. (For example, 1\. Introduction) should be Times 12-point
boldface, initially capitalized, flush left, with one blank line before, and
one blank line after.

SECOND-ORDER HEADINGS. (For example, 1.1. Database elements) should be Times
11-point boldface, initially capitalized, flush left, with one blank line
before, and one after. If you require a third-order heading (we discourage
it), use 10-point Times, boldface, initially capitalized, flush left, preceded
by one blank line, followed by a period and your text on the same line.

###  2.3 Footnotes

Please use footnotes111This is what a footnote looks like. It often distracts
the reader from the main flow of the argument. sparingly. Indeed, try to avoid
footnotes altogether and include necessary peripheral observations in the text
(within parentheses, if you prefer, as in this sentence). If you wish to use a
footnote, place it at the bottom of the column on the page on which it is
referenced. Use Times 8-point type, single-spaced.

###  2.4 References

List and number all bibliographical references in 9-point Times, single-
spaced, at the end of your paper. When referenced in the text, enclose the
citation number in square brackets, for example [Authors14]. Where
appropriate, include the name(s) of editors of referenced books.

Method | Frobnability
---|---
Theirs | Frumpy
Yours | Frobbly
Ours | Makes one’s heart Frob
Table 1: Results. Ours is better.

###  2.5 Illustrations, graphs, and photographs

All graphics should be centered. Please ensure that any point you wish to make
is resolvable in a printed copy of the paper. Resize fonts in figures to match
the font in the body text, and choose line widths that render effectively in
print. Many readers (and reviewers), even of an electronic copy, will choose
to print your paper in order to read it. You cannot insist that they do
otherwise, and therefore must not assume that they can zoom in to see tiny
details on a graphic.

When placing figures in LaTeX, it’s almost always best to use
`\includegraphics`, and to specify the figure width as a multiple of the line
width as in the example below

       \usepackage[dvips]{graphicx} ...
       \includegraphics[width=0.8\linewidth]
                       {myfile.eps}

###  2.6 Color

Please refer to the author guidelines on the ICCV 2023 web page for a
discussion of the use of color in your document.

##  3 Final copy

You must include your signed IEEE copyright release form when you submit your
finished paper. We MUST have this form before your paper can be published in
the proceedings.


# Understanding DNS Query Composition at B-Root

Jacob Ginesin, Northeastern University, Jelena Mirkovic, USC/ISI

###### Abstract

The Domain Name System (DNS) is part of critical internet infrastructure, as
DNS is invoked whenever a remote server is accessed (an URL is visited, an API
request is made, etc.) by any application. DNS queries are served in
hierarchical manner, with most queries served locally from cached data, and a
small fraction propagating to the top of the hierarchy – DNS root name
servers. Our research aims to provide a comprehensive, longitudinal
characterization of DNS queries received at B-Root over ten years. We sampled
and analyzed a 28-billion-query large dataset from the ten annual “Day in the
Life of the Internet (DITL)” experiments from 2013 through 2022. We sought to
identify and quantify unexpected DNS queries, establish longitudinal trends,
and compare our findings with published results of others. We found that
unexpected query traffic increased from 39.57% in 2013 to 67.91% in 2022, with
36.55% of queries being priming queries. We also observed growth and decline
of Chromium-initiated, random DNS queries. Finally, we analyzed the largest
DNS query senders and established that most of their traffic consists of
unexpected queries.

##  1 Introduction

The Domain Name System (DNS) is the Internet’s system for mapping alphanumeric
resource names (e.g. the name of a web or a mail server) to their respective
values (in most cases an IP address). This system is critical for basic
functioning of the Internet. DNS queries are issued whenever a remote server
is accessed by a client application, usually to obtain an IP address for a
given server’s name. A missing or an incorrect reply to such queries can halt
all communication between the server and the client.

In many cases DNS queries are answered by a caching resolver on the client’s
local network, which oftentimes has the full response in its cache. If the
local caching resolver does not know the answer to a query, it will interact
with other DNS participants to obtain, cache, and return the answer to the
client. The DNS system is organized as a hierarchy of DNS name servers, with
servers at the higher levels of the hierarchy containing information about
servers one level lower than itself. At the top of the hierarchy resides 13
DNS roots. Most DNS queries are answered by lower levels of the DNS hierarchy,
but some queries propagate to DNS roots.

We consider queries that do not follow DNS server naming conventions or occur
too frequently with respect to historal data to be unexpected queries.
Previous work analyzing DNS traces data has revealed a surprising amount of
unexpected queries hitting the root servers [10, 11, 19]. Yet, none of the
previous work provides a full characterization of unexpected and expected
traffic into disjoint and meaningful categories. Such a classification would
help us better understand the root causes and severity of different types of
unexpected traffic. Our research aims to develop a comprehensive
classification of DNS queries, and use it to study trends in DNS query traffic
at B-root over the past ten years

We make the following contributions in this paper:

  1. 1.

We propose a detailed, comprehensive DNS query classification scheme to cover
main root causes of unexpected DNS traffic.

  2. 2.

We quantify unexpected DNS query traffic at B-root, one of 13 DNS roots, both
in aggregate and per class of interest. We study longitudinal trends in
unexpected DNS queries over the course of ten years, using annually collected
“Day in the Life of the Internet (DITL)” data [4]. We find an increase in
unexpected traffic — from 39.57% in 2013 to 67.91% in 2022. We additionally
find 36.55% of traffic in 2022 is due to empty queries.

  3. 3.

We identify top senders of DNS queries to B-root, then classify the traffic
coming from each sender. We reveal most traffic from top senders consists of
unexpected queries.

##  2 Background and Related Work

In this section we provide more details about DNS hierarchy and query
resolution, as well as discuss prior work characterizing traffic at DNS query
resolvers.

###  2.1 DNS Hierarchy

DNS queries are issued by applications and operating systems whenever a
connection is established with a remote server. For example, if a user types
the URL www.example.com into their browser, a DNS query containing the
aforementioned name is sent to discover the corresponding IP address. The
query is first sent to a caching resolver – usually a server in the same local
network as the query sender. The caching resolver (resolver for short) will
attempt to respond by searching for the query’s answer in its cache. If the
full answer is not in the cache, the resolver will interact with different
authoritative name servers to try to determine the full answer. Such an answer
will be returned to the client, then saved in cache to respond to potential
future queries.

The DNS utilizes a distributed, hierarchical zoning system in order to
designate authority, ensure the system’s robustness, and effectively
distribute query traffic across servers. Each name, e.g. www.example.com, can
be viewed as collection of name segments separated by dots [2]. Each name
segment resides in a separate name space, each of which has a designated
authoritative name server. Such servers will answer queries about names within
that name space, either by providing the full answer or by directing the query
sender to an authoritative name server for a subset of the given name space.
In our example, the caching resolver trying to find the IP address for
www.example.com may have in its cache the name and IP address of the name
server authoritative for .com top-level domain (TLD). The resolver may repeat
its query to the TLD name server, and receive back the name and IP address of
the name server authoritative for example.com second-level domain (sTLD). The
resolver will cache this new information, then repeat its query to this sTLD
name server and receive a full response, which will be cached and returned to
the client.

If the resolver from our example does not have information about the relevant
TLD server (e.g., .com), it will send its query to one of DNS root name
servers. The names and IP addresses of root name servers are often hard-coded
in operating system releases, thus a resolver always knows how to reach a DNS
root. The root zone is served by 13 logical root name servers (13 root server
names, A–M) and hundreds of physical servers.111The number of root servers
used to be capped at 13 due to the size limitation of UDP reply packets. After
the introduction and rollout of anycast, the number of physical servers per
DNS root has increased. The root name server will provide the name and IP
address of the relevant TLD server, which will be cached by the caching
resolver. The rest of name resolution continues as described in the previous
paragraph.

Responses from DNS replies carry a “time-to-live (TTL)” value, a number of
seconds that the authoritative server suggests they should be cached. A
resolver can decide to cache a DNS record for a shorter time than the
recommended TTL. DNS records for names higher in the hierarchy, such as sTLD
and TLD servers, usually have TTL values in hours or even days. Caching should
ensure that a resolver can quickly reply to most client DNS queries, and that
higher levels of DNS hierarchy do not receive too frequent queries from any
resolver.

Two recent extensions to DNS protocol introduce changes to the query
resolution process. Query minimization (QMIN) RFC 7816 [9] instructs DNS
resolvers to protect their clients’ privacy by only asking each authoritative
name server for the name segment that the resolver is currently trying to
resolve. In our example, the resolver would not send the full www.example.com
query to each authoritative name server. Instead, it would send a com query to
a root name server, a example.com query to the TLD name server, and the full
query to the sTLD name server. RFC 8109 [13] introduces priming queries, i.e.,
queries of type nameserver (NS) for the root zone “.”. Such queries can be
sent to any root, and the reply should specify all root server names and IP
addresses. Priming queries can help a resolver learn a new IP address for a
root name server. In practice, the mapping between root server names and IP
addresses has been stable enough as to not require additional root servers to
be introduced [12].

Finally, while the most popular DNS query maps a DNS name into an IP address
(query type A for IPv4 address, query type AAAA for IPv6 address), there are
other types of DNS queries. A NS query returns the name and often the IP
address of the authoritative name server for the specified query name. A
pointer record (PTR) query asks for reverse mapping from an IP address into a
DNS name. A start-of-authority (SOA) query requests some metadata about the
name, such as the email address of the administrator, when the domain was last
updated, and how long the server should wait between refreshes. Mail exchanger
(MX) queries are for mail servers serving a given name. There exists several
other, less frequent, query types [1].

###  2.2 DNS Query Classification

Castro et al. [10] analyzed DITL datasets at eight root servers from years
2006 through 2008. They uncovered very high volumes of unexpected traffic
across the root zone — almost 98% of queries were identified as unexpected.
Unexpected queries were specified to fall into any one of the following
categories: invalid query class (a field in a DNS query with five valid
values), A or AAAA queries where the query name is an address, queries with
invalid TLDs, queries with non printable characters or underscores, PTR
queries for a private IP address, identical queries (same class, type, name
and ID), repeated queries (same class, type, and name but different ID), and
queries where referral records (TLD and sTLD) have not been properly cached.
Our study is in a sense a modern sequel to this previous work. We extend the
Castro et al. study in a few ways: (1) we dive deeper into invalid query
categories, characterizing them by their root cause, (2) our analysis covers
newer DNS query trends, like query minimization and priming queries, (3) our
analysis spans ten years of DITL data albeit at only one root, and (4) we
analyze top senders in several invalid query categories to investigate if
there are any commonalities between them that would explain their querying
behavior.

###  2.3 DNS Sender Analysis

A recent study measured centralization in senders to B-Root, with a specific
focus on tracking the 5 top cloud providers [14]. This study reveals that in
2020, more than 30% of all queries to two TLD name servers and B-root were
sent from five large cloud providers: Google, Amazon, Microsoft, Facebook, and
Cloudflare. We extend upon this work by examining and ranking all senders at
B-root instead of just cloud providers. We find senders which often send at
rates that are too high often send predominantly queries with invalid TLDs. A
similar trend was observed by Castro et al. [10] in 2006–2008.

##  3 Dataset

Each year, most root servers and several TLD servers collect and publish all
their query traffic on a specific, predetermined day. This effort is known as
“Day in the Life of the Internet” or DITL, and is undertaken to produce useful
data for research [4]. Although DITL data has been collected at other root
name servers since 2006, B-Root joined the experiment in 2013; likewise, our
sample covers just ten years of traffic (2013–2022) [16]. Ideally, we would
have analyzed all roots’ data from DITL collection. However, this data is only
available on OARC servers, which have limited computational power. For this
reason, we started with B-root data, which was available locally at our
servers, and we plan to extend our analysis to other roots’ data in the
future.

To speed up our analysis, we analyzed samples from B-Root’s DITL data. For
each year of DITL experiment data, we utilized the sample function from
Python’s random package [17] to generate ten year-specific subsets of data.
For each of our 10 subsets, we generate 4 additional subsets each denoting one
of four time segments: 12-1am, 6-7am, 12-1pm, and 6-7pm. Table 1 shows the
details of the dataset we analyze in this paper. In total, we study 28 billion
DNS queries spread over 10 years, one day per year.

Date (Y/M/D) | Main | 12-1am | 6-7am | 12-1pm | 6-7pm
---|---|---|---|---|---
2013/05/28 | 1.00B | 19.96M | 37.68M | 69.44M | 48.53M
2014/04/28 | 0.98B | 255.10M | 61.24M | 11.19M | 46.80M
2015/04/13 | 0.96B | 35.87M | 42.82M | 63.72M | 29.55M
2016/04/05 | 4.22B | 148.62M | 154.42M | 344.51M | 172.56M
2017/04/11 | 3.50B | 134.81M | 128.26M | 183.75M | 172.21M
2018/04/10 | 3.90B | 97.01M | 121.28M | 255.69M | 181.40M
2019/04/08 | 3.63B | 93.38M | 157.30M | 295.90M | 103.52M
2020/05/05 | 3.52B | 67.86M | 144.47M | 267.39M | 116.75M
2021/04/13 | 2.07B | 79.78M | 69.06M | 112.40M | 93.40M
2022/04/12 | 4.11B | 92.40M | 105.70M | 326.66M | 87.48M
Table 1: Evaluated Datasets

##  4 Methodology

In this section we describe our methodology to determine query classes, and
how we implemented our approach.

###  4.1 Classification Goals

While previous works have primarily focused on opportunistically measuring
some aspects of unexpected queries, our goal is to provide a more
comprehensive, general classification. We seek to define a method to allow us
to stratify DNS queries into sections denoting different root causes of
unexpected traffic. To do this, we consider two qualities to be of interest
when creating our classification method: full-coverage and mutual exclusivity.
A method that has full-coverage places every single query of a given dataset
into a single, defined category at each level of classification. A method that
is mutually exclusive ensures there’s no overlap between query categories at
the same classification level.

We opt to primarily classify queries based on query names; yet, we also
recognize it’s possible to classify queries by other features (e.g. query
types, cached/uncached queries, repeated queries) and we hope to do so in the
future. In developing our name-based method that fulfills the aforementioned
qualities, we consider the DNS zoning hierarchy as defined by RFC 1035 [2]. In
moving down the zoning hierarchy from the root zone (right to left in the
context of a textual DNS query), we recognize three mutually exclusive
possibilities: the query is empty (“.”), the query ends following some text
(e.g. “foo.”), or the query continues (e.g. “[more query].foo.”). This method
is recursive on the latter query case, which is relevant for processing
queries whose subdomains are multi-level [2].

In accordance with our method, we stratified our data into three all-
encompassing, mutually exclusive categories - empty (“.”), has-TLD (e.g.
“example.com”), and one-word (e.g. “foobar.”). Next we split the has-TLD
category into valid-TLD and invalid-TLD by comparing the TLD of each query to
IANA’s maintained valid TLD list [3]. Within the one-word category we also
attempt to detect presence of valid TLDs, which can occur due to query
minimization [9]. We further stratified the valid-TLD category by categorizing
valid TLDs by frequency.

Before categorizing invalid-TLD queries by TLD frequency, we separated
classifications of queries we deemed interesting. We quantified queries that
contained top-level domains consisting of entirely numbers because they’re
deemed invalid by RFC1034 [1]. We quantified queries from Appletalk, a
discontinued proprietary suite of networking protocols for Apple products, as
it could potentially indicate legacy Apple product usage [8], leaking private
data into the public Internet. We quantified queries with TLDs containing “bad
encoding” (ASCII depicted as “\xxx\xxx”) because of its high frequency in DITL
data. Because Chromium-initiated queries are known to occasionally contain an
invalid-TLD [15], we quantified those as well (the importance of Chromium-
initiated queries is discussed further in Section 5.5).

We separated Chromium-initiated queries from within the one-word category due
to their overabundance in certain years. Chromium-initiated queries are
discussed further in Section 5.5. We quantified minimized queries (minimized
queries at root servers look like one-word queries whose content is a valid
top-level domain) in our collections after the technique’s introduction in
March of 2016 [9]. Minimized queries and their importance are discussed
further in Section 5.6.

Our implementation of our classification method involves use of dictionary-
based matching and regular expressions. We achieve exclusivity by enforcing
the order in which we apply classification criteria within a Python program.

##  5 Results

In this Section we present our results. We show the breakdown of DNS query
traffic in 2013 and 2022 in Section 5.1. We analyze trends in query types in
Section 5.2. We analyze longitudinal trends in Section 5.3. We explore top
senders of queries to B-Root in 5.4. We specifically explore Chromium-
initiated queries in Section 5.5. We quantify the increasing presence of
minimized queries in Section 5.6. We explore empty queries in Section 5.7.

###  5.1 2013 & 2022 B-Root Traffic Breakdown and Comparison

Figure 1: Classification of 1.00 billion DNS traces at B-Root in 2013 Figure
2: Classification of 4.11 billion DNS traces at B-Root in 2022

We applied our classification method to DITL collections of B-Root DNS traces
from 2013 and 2022 with the intent of revealing trends over the past ten years
(see Figures 1 and 2 respectively). After splitting our data into the four
previously designated categories - empty, valid-TLD, invalid-TLD, and one-word
- we did additional work to further stratify each category. Within valid-TLD,
we quantified the highest frequency valid top-level domains. Within invalid-
TLD, we quantified Appletalk queries [8], queries with top-level domains that
are incorrectly encoded (e.g. ”[query].\xxx\xxx\xxx”), queries with all-number
top-level domains222All-number top-level domains are explicitly specified as
invalid by RFC1034., and Chromium-initiated queries (see Section 5.5). Beyond
these specific categories within invalid-TLD, we quantified the highest
frequency unique invalid top-level domains. Within the One-Word category, we
quantified Chromium-resulting queries. Because minimized queries were
introduced in 2016, we characterized them only in our 2022 dataset.

Between 2013 and 2022, we see a 34% increase in empty queries, a 36% reduction
in valid-TLD queries, a 10% reduction in invalid-TLD queries, and a 10%
increase in one-word queries. The large increase in empty queries is
significant and could be due to priming queries, as discussed in Section 5.7.

Within valid-TLD, we see a sharp reduction in the percentage of .com queries
(21.43% to 5.67%), .net queries (13.83% to 3.74%), and .org queries (2.77% to
0.60%). Surprisingly, .arpa queries stay at approximately the same percentage
across the 10 year gap (2.66% to 2.87%).

Within invalid-TLD, we see a small increase in .internal queries despite an
overall reduction in the category—this is potentially indicative of a
persistent, growing leak. Appletalk queries decrease from 1.13% to .57%, which
is expected given Appletalk is long defunct [8].

###  5.2 Query Types

Figure 3 shows the distribution of queries by type at B-Root from years 2013
through 2022. For all years, A-Type queries, used to request an IPv4 address
for a given query name, are the most common (60% of the total previous to
2022). AAAA-type queries, used to request IPv6 an address, are generally the
second most common query type (15% of the total previous to 2022). In 2022, we
measure a large reduction in A and AAAA-type queries and a large increase in
NS-type queries. The increase in NS-type queries is associated with the
implementation of resolver priming [13]—priming queries are further discussed
in Section 5.6.

2013201420152016201720182019202020212022$0$%$20$%$40$%$60$%$80$%$100$%Percent
Of All QueriesAAAAAPTRNSSOASRVMXTXTDS Figure 3: Breakdown of DNS query types
at B-Root from 2013 through 2022

###  5.3 Longitudinal Trends

We applied our classification method to each of our collections of B-Root DNS
traces from 2013 through 2022 with the intent of discovering longitudinal
trends. Figure 4 shows the breakdown of empty, one-word, invalid-TLD, and
valid-TLD queries for each year 2013–2022. Valid-TLD queries consistently
decline from 57.82% in 2013 to 22.84% in 2022. invalid-TLD queries stay
approximately constant through the 10 year sample, hovering between 20% and
30% of all queries. One-word queries see a steady increase from 8.26% in 2013
to 68.45% in 2020, followed by a sharp decline to 18.87% in 2020. This rise
and fall is largely due to Chromium-resulting queries, as further discussed in
section 5.5. Empty queries hovered around 3% until jumping to 37.42% in 2022.
This sudden increase is thought to be a result of excessive priming queries,
as discussed in section 5.7.

2013201420152016201720182019202020212022$0$%$20$%$40$%$60$%$80$%$100$%Percent
Of All QueriesEmptyOne-WordInvalid-TLDValid-TLD Figure 4: Breakdown of
longitudinal trends from 2013 through 2022 in DITL datasets at B-Root

###  5.4 Top Senders

We identified resolvers that are top senders in DITL dataset from 2022, and
show them and their query composition in Figure 5. Amazon Web Services (AWS)
accounts for the 1st, 2nd, 3rd, 8th, and 10th highest IP host groups and
account for approximately 14% of all queries to B-Root. AWS sends almost
entirely invalid-TLD and one-word queries to B-Root. Microsoft Azure, another
cloud computing platform, has a similar query classification breakdown to AWS.
This is potentially indicative of rented cloud machines being misconfigured or
used for malicious purposes. Charter and Compudyn, both internet service
providers, account for 3.43% of all traffic to B-Root. Both providers
primarily send invalid-TLD queries, potentially indicating a misconfiguration.
Additionally, empty and valid-TLD queries aren’t present in significant
quantities from these large senders.

AWS (44.242.0.0)AWS (34.223.0.0)AWS (35.80.0.0)Microsoft (40.112.0.0)Compudyn
(198.235.0.0)Charter (66.75.0.0)Charter (65.24.0.0)AWS (13.52.0.0)Charter
(68.114.0.0)AWS (34.218.0.0)$0$%$2$%$4$%$6$%Percent Of All QueriesInvalid-
TLDOne-WordValid-TLDEmpty Figure 5: Top query senders in 2022 DITL dataset at
B-root

###  5.5 Chromium-Resulting Queries

Chromium is an open-source web browser project primarily maintained by Google.
In addition to Google Chrome, several other major web browsers including
Microsoft Edge, Opera, Brave, Samsung Internet, and Amazon Silk are based on
the Chromium codebase. In total, approximately 75% of the web-browser market
share is Chromium-based [7].

Chromium includes a feature titled Omnibox, which allows users to enter
website names, URLs, or search terms. Chromium then decides if the entered
term is a URL or a search term by performing a DNS query. A URL will result in
a valid response, while a search term will not — Chromium can then supply
search results from Google. However, a user’s machine may be behind a captive
portal (e.g., in a hotel), which intercepts each DNS query and responds with
either the correct response (e.g., in the URL case) or with a redirect to an
internal Web site (e.g., in the search term case). This situation would
interfere with the Chromium’s response to user input. For this reason each
Chromium browser attempts to detect presence of captive portals by sending
three randomly generated query names [6] [5]. These queries contain 7–15,
lowercase alphabetic characters (e.g., “daozjwend.”).

As a consequence of this feature combined with Chromium’s high market share,
root zone name servers have reported a very high quantity of Chromium-
originating queries. Our findings at B-Root agree with the findings of
previous work quantifying these queries [15]. We see a gradual increase in
Chromium-originating queries from 2013 through 2020, followed by a sharp
decline after 2020 following a change to Omnibox’s probing process [18]. This
trend is shown in Figure 6. Because Chromium-resulting queries have been known
to appear both with and without a TLD [15], we quantify both types.

2013201420152016201720182019202020212022$0$%$20$%$40$%Percent Of All
QueriesChromiumChromium+TLD Figure 6: Chromium-initiated queries from 2013
through 2022 in DITL datasets at B-Root

###  5.6 DNS Query Name Minimization

2016201720182019202020212022$0$%$2$%$4$%$6$%$8$%Percent Of All
Queriesothercomnetorg Figure 7: Breakdown of minimized DNS queries at B-Root
from 2016 through 2022

We seek to quantify the presence of minimized queries (these appear at B-Root
as one word valid TLDs, e.g. com) since the inception of the query
minimization specification in March of 2016 [9]. Because the DNS is highly
distributed and controlled by hundreds of different organizations, a change in
the DNS protocol takes time to propagate. Thus, studying the quantity of
minimized queries hitting the root zone could provide insight into the speed
at which DNS protocol changes propagate throughout the entire DNS network,
potentially aiding in future DNS specification development. In our data, we
find a generally steady increase in minimized queries after 2016, as shown in
Figure 7. In accordance with the frequency of valid TLDs found in Section 5.1,
the distribution of minimized queries is expected.

###  5.7 Empty Queries

One of the most notable outliers we discovered in our data is the
overabundance of empty queries in 2022—37.42% of queries in 2022 to B-Root are
empty. Before 2022, empty queries only ever occupied as much as 4% of all
traces. Upon investigation, we find empty query top senders only account for a
small fraction of all empty queries, as shwon in Figure 8. Similarly, we find
each sender, on average, sends 2.8 empty queries to B-Root. We also find
97.20% of empty queries sent to B-Root in 2022 are type NS. The
decentralization and query type indicate the majority of the empty queries
hitting B-Root in 2022 are Priming Queries, a standard introduced in March of
2017 [13]. In comparison with the gradual growth of minized query traffic
after the introduction of RFC7816, as shown in Figure 4 the growth of empty
queries was nonexistent, then sudden. To the best of our knowledge we are the
first to identify this pattern, and in the future we hope to track down the
root cause.

EDFC1-RIPE (163.114.0.0)InfoMagnetics (198.163.0.0)AWS (54.240.0.0)Atl.
Network (205.251.0.0)AWS (44.242.0.0)Charter (71.8.0.0)Wave Broadband
(76.14.0.0)CWC-CSIRT (162.246.0.0)Cable Choice (24.227.0.0)Charter
(24.199.0.0)$0$%$0.2$%$0.4$%$0.6$%Percent Of All Empty QueriesNSDNSKEYSOAA
Figure 8: Top senders of empty queries in 2022 DITL dataset at B-Root

##  6 Conclusion and Further Directions

This investigation into B-Root’s DNS traces collected from the annual DITL
experiment over ten years characterized longitudinal trends, as well as modern
issues, such as a high volume of priming queries. Future work involves
characterizing valid TLD traffic at B-root and identifying unexpected queries
in that category. We would also like to analyze other root’s TLD data and see
if trends identified at B-root apply to other roots. We encourage other DNS
operators to implement our classification method. We also hope to extend our
classification approach with more categories in the future.

##  7 Acknowledgement

This work was performed as a part of a Research Experience for Undergraduates
(REU) program, supported by National Science Foundation (NSF) grant #2051101.

## References

  * [1] Domain names - concepts and facilities.  RFC 1034, November 1987.
  * [2] Domain names - implementation and specification.  RFC 1035, November 1987.
  * [3] List of Top-Level Domains - ICANN — icann.org.  https://www.icann.org/resources/pages/tlds-2012-02-25-en, 2012.
  * [4] DITL Traces and Analysis — DNS-OARC — dns-oarc.net.  https://www.dns-oarc.net/oarc/data/ditl, 2022.
  * [5] Network Stack Use in Chromium — chromium.org.  https://www.chromium.org/developers/design-documents/network-stack/network-stack-use-in-chromium/#networkchangenotifier, 2022\.
  * [6] Omnibox: History Provider — chromium.org.  https://www.chromium.org/omnibox-history-provider/, 2022.
  * [7] W3Counter: Web Browser Market Share Trends — w3counter.com.  https://www.w3counter.com/trends, 2022.
  * [8] Apple.  Mac OS X v10.6: Mac 101 - Printing — web.archive.org.  http://support.apple.co/kb/HT3771, 2010.
  * [9] Stéphane Bortzmeyer.  DNS Query Name Minimisation to Improve Privacy.  RFC 7816, March 2016.
  * [10] Sebastian Castro, Duane Wessels, Marina Fomenkov, and Kimberly Claffy.  A day at the root of the internet.  ACM SIGCOMM Computer Communication Review, 38, 2008.
  * [11] Peter B. Danzig, Katia Obraczka, and Anant Kumar.  Analysis of wide-area name server traffic a study of the internet domain name system.  1992\.
  * [12] IANA.  Root Servers — iana.org.  https://www.iana.org/domains/root/servers, 2022.
  * [13] Peter Koch, Matt Larson, and Paul E. Hoffman.  Initializing a DNS Resolver with Priming Queries.  RFC 8109, March 2017.
  * [14] Giovane C.M. Moura, Sebastian Castro, Wes Hardaker, Maarten Wullink, and Cristian Hesselman.  Clouding up the internet: How centralized is dns traffic becoming?  2020\.
  * [15] Matthew Thomas.  Chromium’s impact on root DNS traffic — APNIC Blog — blog.apnic.net.  https://blog.apnic.net/2020/08/21/chromiums-impact-on-root-dns-traffic/, 2020\.
  * [16] USC/ISI.  ANT Datasets — ant.isi.edu.  https://ant.isi.edu/datasets/all.html.
  * [17] Guido Van Rossum.  The Python Library Reference, release 3.8.2.  Python Software Foundation, 2020.
  * [18] Verisign.  Chromium’s Reduction of Root DNS Traffic — blog.verisign.com.  https://blog.verisign.com/domain-names/chromiums-reduction-of-root-dns-traffic/, 2021\.
  * [19] D Wessels and M Fomenkov.  Wow, that’s a lot of packets.  In Passive and Active Network Measurement Workshop (PAM), 04 2003\.

##  8 Appendix

###  8.1 Code

The code used for processing B-Root’s DITL data is publicly available at
https://github.com/STEELISI/DITL-Analysis-Scripts

###  8.2 2013-2022 Traffic Breakdown

In this section we provide additional diagrams resulting from the application
of the classification method described in Section 4 to all 10 years of DITL
data. Note the addition of the ”minimized queries” category from 2016 and
forward.

Figure 9: Classification of 1.00 billion DNS traces at B-Root in 2013. Figure
10: Classification of 0.98 billion DNS traces at B-Root in 2014. Figure 11:
Classification of 0.96 billion DNS traces at B-Root in 2015. Figure 12:
Classification of 4.22 billion DNS traces at B-Root in 2016.  Figure 13:
Classification of 3.50 billion DNS traces at B-Root in 2017. Figure 14:
Classification of 3.90 billion DNS traces at B-Root in 2018. Figure 15:
Classification of 3.63 billion DNS traces at B-Root in 2019. Figure 16:
Classification of 3.52 billion DNS traces at B-Root in 2020. Figure 17:
Classification of 2.07 billion DNS traces at B-Root in 2021. Figure 18:
Classification of 100 million DNS traces at B-Root in 2022.


# Lipschitz regularity of controls and inversion mapping for a class of smooth
extremization problems

Vincenzo Basco<EMAIL_ADDRESS>Thales Alenia Space, Via
Saccomuro, 24 Roma Italy

###### Abstract

In the contest of optimal control problems, regularity results for optima are
known when addressing fiber-strictly convex Lagrangian. For infinite time
horizons, or for settings with infinite dimensional dynamics, the equivalence
between minima/maxima and extremals could break down. Commonly, this is due to
a loss of convexity/concavity of the cost functional or to a presence of state
constraints, in which further controllability assumptions are needed. For many
science applications, this a trend is not required, as in energy saving
problems. In the present paper, we deal with the set of a functional’s
extremals subject to end-point restrictions. We consider an affine control
system and a cost functional associated to an autonomous Lagrangian. The
dynamics is smooth, satisfying the Lie bracket condition, and the functional
is assumed merely Fréchet differentiable. Here we provide a regularity result
for controls in the context of constrained extremization problems, under
weaker conditions on Lagrangian than the not met classical ones. More
precisely, we show a characterization for the Lipschitz regularity of controls
associated with the extremal trajectories steering two fixed points, assuming
the absence of singular controls. As main application, we construct a locally
Lipschitz inversion mapping from the ambient space to the set of constrained
extremals.

###### keywords:

Extremization Problems; Lie Algebra of Vector Fields; Singularities.

##  1 Introduction

Classical control theory takes into account minimization – or maximization –
problems of functionals along trajectories given by a dynamical system (cfr.
[3], [16], [19]). These optimization problems are generally well defined in
the presence of convex – or concave – data or for non-negative Lagrangians.
For many science applications, this a trend is not required for energy saving
problems, where non-convex energy functionals and the presence of
singularities are commonly an indication of instabilities, so that the direct
methods in classical optimization cannot be applied (see e.g. [11],[12],[15]).
In this case, it is required that a system evolves along a trajectory that
makes the functional action stationary, i.e. its first variation is zero.
Since any critical point of a smooth convex – or concave – function is a
minimum – or a maximum, the extremization problems reduce so to classical
optimization problems when the functional meets these requirements and the
constraints are linear. Hence, the seek of extremals – or stationary points –
rather than minima – or maxima – reveal to be necessary, the concept of
extremization is a natural generalization of optimization to the non-convex
case. This formulation of control problems in terms of stationary optimization
has recently proved in [5] to be very useful for the investigation of boundary
constrained problems.

The point of view of extremization problems appears useful in engineering
applications, where the effects of small perturbations are not negligible. As
in the case of classical optimization problems, a fundamental role for
regularity investigations of feedback controls is played by the absence of
singularity (see Section 2 below). Singular points may occur in control
problems: the importance of singular paths – namely associated to a singular
control – of affine control system was investigated since a long time in the
in calculus of variations (cfr. [6],[7],[8]). Singular paths are candidates as
minimizers for time optimal control problems (cfr. [3],[18]) and they are
strictly related to the singularities of the end-point map. It is a well known
fact that for control unconstrained optimization problems associated to
running cost functional, the solutions of the abnormal maximum principle
projects onto the set of singular trajectories. However, such set and the
solutions of the normal maximum principle does not trivially intersect
generally: this could happen when considering degenerate affine control
systems with end-point constraints (e.g., the geodesics of the manifold
spanned by the three-dimensional Martinet distribution).

The optimal synthesis of control problems is quite challenging while
continuity of the cost functional, sufficient and necessary conditions, and
sensitivity relations in absence of singular paths are known (cfr.
[1],[3],[19],[4],[9]) and duality connections, involving the Legendre
transform, are investigated (see [13]). The classical theory on the regularity
of controls in the absence of singularities has been exhaustively invested for
Lagrangian two times differentiable and strictly convex in the control
variable ([6] and the literature therein). Nevertheless, the absence of
singularity does not fit for the optimal synthesis in many investigations in
applied science. To deal with it, in the case of optimization problems, well
known and succesfull results have been developed by using continuation
arguments for indirect algorithms methods or convergently probability-one
homotopy techniques (cfr. [20],[2]).

Although the assumption of absence of singular trajectories could be strong,
results to ensure regularity of feedback controls under less restrictive
assumptions on the Lagrangian are needed when addressing extremization under
end-point constraints. We emphasize, in optimization context, singular time
optimal paths imply the failure of the Lipschitz continuity of feedback
controls. This can be see by [4, Proposition 5.3- (ii)] which implies, along a
singular time optimal paths, the horizontal proximal gradient of time minimal
function containing a non-zero vector. A necessary condition for the Lipschitz
continuity of feedback controls at a given end-point is the triviality of
horizontal proximal gradient of time minimal function at such point. So,
without making additional assumptions – such as the absence of singular
minimizing controls like in [4] – one cannot hope for a Lipschitz regularity.

In this paper we relax the classical assumptions, requiring at most the well
definition of the problem. The main concern is to provide a regularity
characterization for controls associated with extremal trajectories of a given
cost functional. More exactly, we show that the set of controls related to
end-point constrained extremals is equi-Lipschitz continuous, in absence of
singular paths. The trajectories are assumed to solve an affine dynamics and
the Lagrangian is possibly unbounded and not strictly convex in the fiber. We
provide further an application of this result constructing a Lipschitz
inversion mapping for constrained extremals.

The outline of this paper is as follows. In Section 2 we give some basic
definitions, notations, and the general background. Section 3 indicates the
problem statement and the main assumptions. In Section 4 we state the main
result of this paper showing a regularity characterization for controls
associated with constrained extremal trajectories. The Section 5 is devoted to
an application case.

##  2 Background

For any $u,w\in\textit{I}\\!R^{n}$ we denote by $|u|$ and $u\cdot w$ the
Euclidean norm of $u$ and the scalar product between $u$ and $w$,
respectively. $\textit{I}\\!B_{\delta}(x)$ is the closed ball in
$\textit{I}\\!R^{m}$ centered at $x$ of radius $\delta>0$, and
$\textit{I}\\!B:=\textit{I}\\!B_{1}(0)$,
$\textit{I}\\!B_{\delta}:=\textit{I}\\!B_{\delta}(0)$. For any $p\geq 1$,
$\mathscr{L}^{p}(a,b;\textit{I}\\!R^{m})$ denotes the space of all Lebesgue
measurable functions on $[a,b]$ that are $p$-integrable, endowed with the norm
$\left\|u\right\|_{p}^{p}=\int_{a}^{b}|u(s)|^{p}ds$. The Hilbert space
$\mathscr{L}^{2}(a,b;\textit{I}\\!R^{m})$ is equipped with the scalar product
defined by $\langle\,u,w\rangle=\int_{a}^{b}u(s)\cdot w(s)ds$. For any subset
$A\subset\mathscr{L}^{2}(a,b;\textit{I}\\!R^{m})$, we denote by
$\textnormal{cl}_{w}A$ the weak closure of $A$. Consider
$(V,\left\|.\right\|_{V})$ and $(B,\left\|.\right\|_{B})$ be two Banach spaces
and let $\mathscr{V}\subset V$ be an open subset. Consider a mapping
$\Phi:\mathscr{V}\rightarrow B$. If $\Phi$ is linear, we denote by
$\Phi^{\star}$ the adjoint (or transpose) operator. We denote by
$\text{d}\Phi$ the Fréchet derivative of $\Phi$ (whenever it exists). High
order Fréchet derivatives (whenever they exist) of $\Phi$ at $u$ shall written
$\text{d}^{j}\Phi(u)$ or $\text{d}^{j}_{u}\Phi(u)$. If $\text{d}^{j}\Phi$ is a
continuous map on $\mathscr{V}$ (wrt the operator norm topology) we say that
$\Phi$ is of class $C^{j}$. Let $\mathscr{F}$ be a family of smooth vector
fields on $\textit{I}\\!R^{n}$. We denote by $Lie_{\mathscr{F}}(x)$ the Lie
algebra of vector fields generated by $\mathscr{F}$ at
$x\in\textit{I}\\!R^{n}$ constructed as follows. Let
$Lie^{1}_{\mathscr{F}}(x)$ the set defined by $\textnormal{span
}\\{X(x)\,|\,X\in\mathscr{F}$ and define recursively, for all
$i\in\textit{I}\\!N^{+}$, the set $Lie^{i+1}_{\mathscr{F}}(x)$ defined by the
spanning of $Lie^{i}_{\mathscr{F}}(x)\cup\\{[X,Y](x)\,|\,X\in
Lie^{1}_{\mathscr{F}}(x),\,Y\in Lie^{i}_{\mathscr{F}}(x)\\}$ where
$[X,Y](x):=\text{d}Y(x)(X(x))-\text{d}X(x)(Y(x))$ is the Lie bracket. Then the
Lie algebra is defined by
$Lie_{\mathscr{F}}(x)=\bigcup_{i\in\textit{I}\\!N}Lie^{i+1}_{\mathscr{F}}(x)$.

###  2.1 Extremization problems

Let $X$ be an Hilbert space and consider a family of subset
$\\{\mathscr{V}(x)\\}_{x}\subset X\times X\times$ $\textit{I}\\!{R}$. We
denote by

$\displaystyle\begin{split}\delta\mathscr{V}(x)=0\end{split}$

the problem of determining all couples $(u,z)\in X\times\textit{I}\\!{R}$ such
that

$\displaystyle\begin{split}(u,0,z)\in\mathscr{V}(x)\text{. }\end{split}$ (1)

The set of all $z\in X$ such that there exists $u\in X$, called extremal,
satisfying (1) is denoted by $\\{\delta\mathscr{V}(x)=0\\}$ and we refer to it
as the extremization (or staticization) problem associated to
$\mathscr{V}(x)$. It occurs e.g. when $\mathscr{V}(x)$ is the Lagrangian sub-
manifold associated with some function $\Phi:X\rightarrow\textit{I}\\!R$ that
is $C^{1}$ (in the Fréchet sense)
$\mathscr{V}(x)=\left\\{\left(u,\text{d}\Phi(u),\Phi(u)\right)\mid u\in
X\right\\}$ and $\text{d}\Phi$ stands for the Fréchet derivative. In that case
(1) becomes $\text{d}\Phi(u)=0$ and $z=\Phi(u),$ so that
$\\{\delta\mathscr{V}(x)=0\\}$ is simply the unconstrained smooth
extremization problem of determining the critical points and values of $\Phi$.
We consider the particular case when
$\\{\mathscr{V}(x)\\}_{x\in\textit{I}\\!R^{n}}$ is the family of manifolds

$\displaystyle\begin{split}&\\{\,(u,\lambda_{0}\text{d}\Phi(u)-\sum_{j=1}^{n}\lambda_{j}\text{d}_{u}\mathcal{C}_{j}(u;x),\Phi(u))\text{
such that}\\\
&\;\mathcal{C}(u;x)=0,(\lambda_{0},\lambda)\in\textit{I}\\!{R}^{1+n}-\\{0\\}\,\\}\end{split}$

where $\mathcal{C}(x)=(\mathcal{C}_{j}(.;x))_{j}$ and
$f,\mathcal{C}_{j}(.;x):X\rightarrow\textit{I}\\!R$ are $C^{1}$ functions for
any $x$. If $\\{\text{d}_{u}\mathcal{C}_{j}(u;x),1\leq j\leq n\\}$ is a set of
linearly independent vectors for any

$\displaystyle\begin{split}u\in S(x):=\\{\mathcal{C}(.;x)=0\\}\end{split}$

it follows that $u\in\\{\delta\mathscr{V}(x)=0\\}$ iff $u$ is a critical point
of $\left.\Phi\right|_{S(x)}$, the restriction of $\Phi$ to $S(x)$. In that
case, we write $\delta\mathscr{V}(x)=0$ as

$\displaystyle\begin{split}\delta_{S(x)}\Phi=0\end{split}$ (2)

and call it a constrained smooth extremization problem. Whenever the set
$\\{\delta_{S(x)}\Phi=0\\}$ is non-empty, we define the set-valued function

$\displaystyle\begin{split}x\mapsto\langle\delta\rangle_{S(x)}\Phi:=\\{\Phi(u)\,|\,u\in\\{\delta_{S(x)}\Phi=0\\}\\}.\end{split}$

If $S(x)$ is the whole space, we shall write $\langle\delta\rangle_{u}\Phi$ in
place of $\langle\delta\rangle_{S(x)}\Phi$.

###  2.2 Lack of convexity: a motivational example

It is of great interest to consider extremization problems rather that
minima/maxima for a large family of functional as those expressed by energy-
driven (or saving) systems. Such class of functionals are represented by

$\displaystyle\begin{split}&\Phi(u)=\int_{I}V(\omega,u)+W(\omega,\nabla
u)d\omega,\quad I\subset\textit{I}\\!R^{n}\end{split}$

where $V$ and $W$ are continuous function and possibly non-convex. As example
consider the family of Ginzburg–Landau type energy functionals

$\displaystyle\begin{split}\Phi_{GL}(u)=\int_{I}|u^{2}-\alpha|^{2}+q^{2}||\nabla
u|^{2}-\beta|d\omega,\quad\alpha,\beta,q\geq 0\end{split}$

arising in many frameworks, including convective pattern formation and
magnetic thin films. It is straingforward to show that end-point constraints
reduce the functional $\Phi_{GL}$ to be generally non-convex. To see that,
consider the one-dimentional case $I=[-1,1]$ with parameters
$\alpha=\beta=q=1$, control law

$\displaystyle\begin{split}u^{\prime}=v,\quad
v\in\mathscr{L}^{2}(-1,1;\textit{I}\\!R)\end{split}$

and subject to end-points constraints

$\displaystyle\begin{split}&u(-1)=u(1)=0.\end{split}$

We have, $\Phi_{GL}(0)=4$ and $\Phi_{GL}(\xi_{\pm})=16/15$ where
$\xi_{\pm}(\omega)=\pm(1-|\omega|)$.

###  2.3 Singular points

Consider $X_{1},...,X_{m}$ smooth vector fields on $\textit{I}\\!R^{n}$ and
the solution $\xi(.)$ associated with
$u\in\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m})$ of the affine control system

$\displaystyle\begin{split}&\xi^{\prime}(s)=\sum_{i=1}^{m}u_{i}(s)X_{i}(\xi(s)),\;{s\in[0,T]}\textnormal{
a.e.}\\\ &\xi(0)=x_{0}\in\textit{I}\\!R^{n}\end{split}$ (3)

with the end-point constraints

$\displaystyle\begin{split}\mathcal{C}(u;x)=\xi(T)-x.\end{split}$ (4)

A point $u\in S(x)$ is said to be a singular point for $S(x)$ if

$\displaystyle\begin{split}\text{d}_{u}\mathcal{C}(u;x)\text{ is not
surjective.}\end{split}$

We denote by $S^{\prime}(x)$ the set of all singular points of $S(x)$.
Singular points may occurs in the set of extremization problem
$\delta_{S(x)}\Phi=0$, e.g., in sub-Riemannian structures. The importance of
singular paths (namely associated to a singular control, i.e. a singular point
of $S(x)$) of the above affine control system was investigated since a long
time in the in calculus of variations (cfr. [6],[7],[8]). The singular paths
are candidates as minimizers for time optimal control problems (cfr. [3],[18]
and the literature therein) and they are strictly related to the singularities
of the end-point map (cfr. Section 2). Indeed, for control problems minimizing
a running cost functional associated to $L$, bounded from below in the state,
such trajectories are solutions of the abnormal ($\lambda_{0}=0$, see below)
maximum principle corresponding to the associated Hamiltonian

$\displaystyle\begin{split}\sup_{u\in\textit{I}\\!R^{m}}(\sum_{i=1}^{m}u_{i}p\cdot
X_{i}(x)+\lambda_{0}L(x,u)).\end{split}$

When $m=n$, as known, extremals are not abnormal, i.e. $\lambda_{0}\neq 0$.
The optimal synthesis of such control problems is quite challenging although
continuity of the cost functional, necessary conditions, and sensitivity
relations in absence of singular paths are known (cfr. [1],[3],[19]) and
duality connections, involving the Legendre transform, are investigated (see
[13]).

Nevertheless, in the case $m<n$, there exist extremals that are normal but
singular (cfr. [6]). Indeed, in this setting, there are privileged and
prohibited paths, and the singular trajectories turn out to be the
singularities of the set of curves satisfying the constraint (4). In
particular, when final end-point conditions are imposed, the sub-Riemannian
manifold generated by the vector fields $X_{i}$ may have points for which
there are not curves steering them. To deal with it, Hörmander in [14]
introduced a condition on the Lie algebra associated to the distribution
spanned by $X_{i}$’s (see Section 2).

##  3 Problem statement and main assumptions

Let $T>0$ and $X_{1},...,X_{m}$ a set of vector fields on
$\textit{I}\\!R^{n}$. We consider the following

######  Assumption 1.

$X_{i}$’s are smooth ($C^{\infty}$ or analytic) bounded vector fields,
linearly independent any $x\in\textit{I}\\!R^{n}$, and
$Lie_{\left\\{X_{1},...,X_{m}\right\\}}(x)=\textit{I}\\!R^{n}$ for all
$x\in\textit{I}\\!R^{n}$.

######  Remark 3.1.

The mapping $x\mapsto\textnormal{span}\left\\{X_{1}(x),...,X_{m}(x)\right\\}$
is called a smooth distribution of rank $m$ on $\textit{I}\\!R^{n}$ and the
condition on the Lie algebra in Assumption 1 is called Hörmander’s condition.
From the Chow-Rashevsky Theorem (see [10],[17]), for any
$x,y\in\textit{I}\\!R^{n}$ there exists an absolutely continuous arc
$\xi:[0,T]\rightarrow\textit{I}\\!R^{n}$, with square integrable derivative,
such that

$\displaystyle\begin{split}&\xi^{\prime}(s)\in\textnormal{span}\left\\{X_{i}(\xi(s))\right\\}_{i}\quad
s\in[0,T]\textnormal{ a.e.}\\\ &\xi(0)=x,\,\xi(T)=y.\end{split}$

Furthermore, for any arc $\xi(.)$ steering $x\in\textit{I}\\!R^{n}$ to
$y\in\textit{I}\\!R^{n}$ in time $T$, there exists a unique
$u_{\xi}\in\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m})$ satisfying

$\displaystyle\begin{split}\xi^{\prime}(s)=\sum_{i=1}^{m}(u_{\xi}(s))_{i}X_{i}(\xi(s)),\;s\in[0,T]\textnormal{
a.e.}\end{split}$

For any $x_{0}\in\textit{I}\\!R^{n}$ consider
$\mathscr{V}^{T,x_{0}}\subset\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m})$ be such
that for every control $u\in\mathscr{V}^{T,x_{0}}$ the associated solution
$\xi_{x_{0},u}(.)$ of the affine system (3), with starting point
$\xi_{x_{0},u}(0)=x_{0}$, is well defined on $[0,T]$. Since $X_{i}$’s are
smooth, then $\mathscr{V}^{T,x_{0}}$ can be chosen to be open (see [6, 16]).
The end-point map
$E_{T,x_{0}}:\mathscr{V}^{T,x_{0}}\rightarrow\textit{I}\\!R^{n}$ is defined by

$\displaystyle\begin{split}E_{T,x_{0}}(u)=\xi_{x_{0},u}(T).\end{split}$

Notice that, under assumptions Assumption 1, for any
$x_{0}\in\textit{I}\\!R^{n}$ we can choose

$\displaystyle\begin{split}\mathscr{V}^{T,x_{0}}=\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m}).\end{split}$

Let $L:\textit{I}\\!R^{n}\times\textit{I}\\!R^{m}\rightarrow\textit{I}\\!R$ be
a two time continuously differentiable Lagrangian. In the following, we fix
$T>0$, $x_{0},x\in\textit{I}\\!R^{n}$, and denote by
$\Phi:\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m})\rightarrow\textit{I}\\!R$ the
functional defined by

$\displaystyle\begin{split}\Phi(u)=\int_{0}^{T}L(\xi_{x_{0},u}(s),u(s))\,ds\end{split}$

and by
$\mathcal{C}:\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m})\rightarrow\textit{I}\\!R^{n}$
the map

$\displaystyle\begin{split}\mathcal{C}(u)=E_{T,x_{0}}(u)-x.\end{split}$

Throught this paper, in order to carrying out our analysis, we consider the
basic conditions

$\displaystyle\begin{split}\Phi\in
C^{1}\quad\&\quad\\{\delta_{S(x)}\Phi=0\\}\neq\emptyset\end{split}$ (5)

and the further assumptions

######  Assumption 2.

for any $x\in\textit{I}\\!R^{n}$, $\text{d}_{u}L(x,.)$ is a
$C^{1}$-diffeomorphism, i.e. bijective with continuously differentiable
inverse.

######  Assumption 3.

$\exists\theta,\,\psi,\,\varphi:\textit{I}\\!R^{+}\rightarrow\textit{I}\\!R^{+}$
continuous satisfying

$\displaystyle\begin{split}&\liminf_{|u|\rightarrow\infty}\frac{\theta(|u|)}{|u|^{2}}>0\\\
&\limsup_{|x|\rightarrow\infty}\frac{\psi(|x|)}{|x|^{2}}<+\infty\end{split}$

such that

$\displaystyle\begin{split}&L(x,u)\geq\theta(|u|)-\psi(|x|),\,\forall
x\in\textit{I}\\!R^{n}\,\forall u\in\textit{I}\\!R^{m}\\\
&|\text{d}_{x}L(x,u)|\leqslant\varphi(r)\left(|u|^{2}+1\right),\,\forall
x\in\textit{I}\\!B_{r}\,\forall u\in\textit{I}\\!{R}^{m}.\end{split}$

######  Assumption 4.

$S(x)^{\prime}\cap\,\\{\delta_{S(x)}\Phi=0\\}=\emptyset$.

It follows some remarks.

######  Remark 3.2.

  1. (a)

It is well known that, under the Assumption 1, $E_{T,x_{0}}$ turn out to be of
class $C^{1}$ and $\text{d}E_{T,x_{0}}$ is weakly continuous (see [6]). The
Hörmander’s condition on the vector fields $X_{i}$’s needs to be imposed to
consider controllability for end-point constraints. If $\textnormal{span
}\\{X_{i}(x)\\}_{i}$ is the whole space, e.g. as in mass-spring configurations
systems, then it is satisfied.

  2. (b)

From the regularity of $(\Phi,\mathcal{C})$, Remark 3.2-(a), and the
definition of extremal given in the previous section, it straightforward to
show that the set $\\{\delta_{S(x)}\Phi=0\\}$ is weakly closed wrt the
$\mathscr{L}^{2}$ topology. Furthermore, (5), Assumption 4, and Lagrange’s
Multiplier Theorem leads to consider such set be non-empty.

  3. (c)

Applying the Inverse Mapping Theorem, Assumption 2 is satisfied for the class
of strictly convex Lagrangian in the control, i.e. $\text{d}_{u}^{2}L>0$. In
such a setting, the condition (5) is naturally met for minimizing control
problems. Moreover, under the Assumption 2 and 4, and by using the
Pontryagin’s maximum principle, the regularity characterization of minimizing
controls are known (cfr. [7],[8]).

  4. (d)

From Assumption 4, we would like to remark that the set of Lagrange
multipliers $(\lambda_{0},\lambda)\neq(0,0)$ reduces to consider
$\lambda_{0}\neq 0$.

##  4 Regularity of constrained extremals

Next, we state and prove the main result of this paper under the Assumptions
1-4. We provide a characterization of regularity of controls associated with
constrained extremals in the space of uniformly Lipschitz continuous
functions, under the assumption of absence of singular controls.

######  Theorem 5 (Regularity characterization).

Consider Assumptions 1-4. Then the following statements are equivalent:

  1. (i)

$\Phi|_{\\{\delta_{S(x)}\Phi=0\\}}<+\infty$.

  2. (ii)

there exists $K>0$ such that

$\displaystyle\begin{split}|u(s)|\leq K\quad\forall s\in[0,T]\,\forall
u\in\\{\delta_{S(x)}\Phi=0\\}.\end{split}$

  3. (iii)

there exists $K>0$ such that

$\displaystyle\begin{split}&|u(T)|\leq K,\quad|u(s)-u(t)|\leq K|s-t|\\\
&\forall s,t\in[0,T]\,\forall u\in\\{\delta_{S(x)}\Phi=0\\}.\end{split}$

###  4.1 Some preparatory results

In the following we define the set-valued Hamiltonian
$H:\textit{I}\\!{R}^{n}\times\textit{I}\\!{R}^{n}\multimap\textit{I}\\!{R}$ by

$\displaystyle\begin{split}H(x,p)=\langle\delta\rangle_{u}(\,\sum_{i=1}^{m}u_{i}p\cdot
X_{i}(x)-L(x,u)\,)\end{split}$

and, for all $x,p\in\textit{I}\\!R^{n}$ and $u\in\textit{I}\\!R^{m}$, we put

$\displaystyle\begin{split}&h(x,p,u):=\sum_{i=1}^{m}u_{i}p\cdot
X_{i}(x)-L(x,u)\\\ &Z(x,p):=(p\cdot X_{1}(x),...,p\cdot
X_{m}(x))^{\star}.\end{split}$

Furthermore, for any $u\in\\{\delta_{S(x)}\Phi\\}$, we define the (non-empty)
set

$\displaystyle\begin{split}\Lambda(u;x):=\\{\lambda\in\textit{I}\\!R^{n}\,|\,\text{d}\Phi(u)=\lambda\text{d}_{u}\mathcal{C}(u;x)\\}.\end{split}$

######  Lemma 6.

The set-valued Hamiltonian is single-valued and there exists a locally
Lipschitz continuous function
${w}:\textit{I}\\!{R}^{n}\times\textit{I}\\!{R}^{m}\rightarrow\textit{I}\\!{R}^{m}$,
with ${w}(x,.)$ a $C^{1}$-diffeomorphism for any $x\in\textit{I}\\!R^{n}$,
satisfying:

  1. (i)

${w}(x,.)=(\text{d}_{u}L(x,.))^{-1}$ for all $x\in\textit{I}\\!R^{n}.$

  2. (ii)

$H(x,p)={w}(x,Z(x,p))\cdot Z(x,p)-L(x,{w}(x,Z(x,p)))$ for all
$x,p\in\textit{I}\\!R^{n}$.

From Assumption 2 and the regularity of $L$, there exists a locally Lipschitz
map
${w}:\textit{I}\\!{R}^{n}\times\textit{I}\\!{R}^{m}\rightarrow\textit{I}\\!{R}^{m}$
such that ${w}(x,.)$ is a $C^{1}$-diffeomorphism for all
$x\in\textit{I}\\!R^{n}$ and satisfying the statement (i). Moreover, for all
$(x,p,u)\in\textit{I}\\!{R}^{n}\times\textit{I}\\!{R}^{n}\times\textit{I}\\!{R}^{m}$

$\displaystyle\begin{cases}\text{d}_{x}h(x,p,u)=\sum_{i=1}^{m}u_{i}\text{d}X_{i}(x)^{\star}p-\text{d}_{x}L(x,u)\\\
{\text{d}_{p}h(x,p,u)=\sum_{i=1}^{m}u_{i}X_{i}(x)}\\\
{\text{d}_{u_{i}}h(x,p,u)=p\cdot X_{i}(x)-\text{d}_{u_{i}}L(x,u)},\;\forall
i=1,...,m\end{cases}$ (6)

which implies that, for any
$(x,p)\in\textit{I}\\!{R}^{n}\times\textit{I}\\!{R}^{n}$,
$\text{d}_{u}h(x,p,u)=0$ if and only if

$\displaystyle\begin{split}\text{d}_{u}L(x,u)=(p\cdot X_{1}(x),...,p\cdot
X_{m}(x))^{\star}.\end{split}$

So, for any
$(x,p,u)\in\textit{I}\\!{R}^{n}\times\textit{I}\\!{R}^{n}\times\textit{I}\\!{R}^{m}$,
we have that $\text{d}_{u}h(x,p,u)=0$ if and only if $u={w}(x,Z(x,p))$. Hence,
the Hmiltonian is single-valued and for all $x,p\in\textit{I}\\!R^{n}$

$\displaystyle\begin{split}&H(x,p)\\\ &=h(x,p,{w}(x,Z(x,p))\\\
&={w}(x,Z(x,p))\cdot Z(x,p)-L(x,{w}(x,Z(x,p)))\end{split}$ (7)

and the statement (ii) follows.

######  Proposition 7.

Let $u\in\\{\delta_{S(x)}\Phi\\}$.

Then, for any $\lambda\in\Lambda(u;x)$, there exists an absolutely continuous
arc $p=p_{u}:[0,T]\rightarrow\textit{I}\\!{R}^{n}$ solving the Hamiltonian
system

$\displaystyle\begin{split}\begin{cases}\xi^{\prime}(s)=\text{d}_{p}H(\xi(s),p(s))\\\
-p^{\prime}(s)=\text{d}_{x}H(\xi(s),p(s))\end{cases}\quad
s\in[0,T]\textnormal{ a.e.}\end{split}$ (8)

with final condition

$\displaystyle\begin{split}p(T)=\lambda\end{split}$ (9)

which satisfies the relation

$\displaystyle\begin{split}Z(\xi(s),p(s))=\text{d}_{u}L(\xi(s),p(s))\quad\forall
s\in[0,T].\end{split}$ (10)

In particular, the control $u(.)$ satisfies

$\displaystyle\begin{split}u(s)={w}\left(\xi(s),Z\left(\xi(s),p(s)\right)\right)\quad\forall
s\in[0,T].\end{split}$ (11)

Let $v\in\mathscr{L}^{2}\left(0,T;\textit{I}\\!{R}^{m}\right)$ and consider
$\lambda\in\Lambda(u;x)$. We have

$\displaystyle\begin{split}&\langle\,\text{d}\Phi(u),v\rangle\\\
&=\int_{0}^{T}\text{d}_{x}L(\xi(s),u(s))\cdot\text{d}E_{s,x_{0}}(u)(v)\\\
&\qquad\qquad\qquad\qquad+\text{d}_{u}L\left(\xi(s),u(s)\right)\cdot
v(s)\,ds.\end{split}$

Let us denote by $\Psi(.)$ the fundamental solution of

$\displaystyle\begin{cases}\Psi^{\prime}(s)=A(s)\Psi(s)&s\in[0,T]\textnormal{
a.e.}\\\ \Psi(0)=I\end{cases}$

where for all $s\geq 0$

$\displaystyle\begin{split}&A(s):=\sum_{i=1}^{m}u_{i}(s)\text{d}X_{i}(\xi(s))\\\
&B(s):=(X_{1}(\xi(s)),...,X_{m}(\xi(s))).\end{split}$

Then, by the known expression of $\text{d}E_{s,x_{0}}(u)$ (cfr. [18,
Proposition 1.8]),

$\displaystyle\begin{split}&\int_{0}^{T}\text{d}_{x}L(\xi(s),u(s))\cdot\text{d}E_{s,x_{0}}(u)(v)ds\\\
&=\int_{0}^{T}(\text{d}_{x}L(\xi(s),u(s))\cdot\int_{0}^{s}\Psi(s)\Psi(r)^{-1}B(r)v(r)dr)ds\\\
&=\int_{0}^{T}\int_{0}^{s}\text{d}_{x}L(\xi(s),u(s))\cdot\Psi(s)\Psi(r)^{-1}B(r)v(r)drds.\end{split}$

Using the Fubini theorem, we have

$\displaystyle\begin{split}&\int_{0}^{T}\int_{0}^{s}\text{d}_{x}L(\xi(s),u(s))\cdot\Psi(s)\Psi(r)^{-1}B(r)v(r)drds\\\
&=\int_{0}^{T}\int_{r}^{T}\text{d}_{x}L(\xi(s),u(s))\cdot\Psi(s)\Psi(r)^{-1}B(r)v(r)dsdr.\end{split}$

Hence,

$\displaystyle\begin{split}&\int_{0}^{T}\text{d}_{x}L(\xi(s),u(s))\cdot\text{d}E_{s,x_{0}}(u)(v)ds\\\
&=\int_{0}^{T}(\int_{s}^{T}\left(\Psi(r)\Psi(s)^{-1}B(s)\right)^{\star}\\\
&\qquad\qquad\qquad\cdot\text{d}_{x}L(\xi(r),u(r))dr\cdot v(s))ds\\\
&=\int_{0}^{T}(B(s)^{\star}\left(\Psi(s)^{-1}\right)^{\star}\\\
&\qquad\qquad\qquad\cdot\int_{s}^{T}\Psi(r)^{\star}\text{d}_{x}L(\xi(r),u(r))dr\cdot
v(s))ds.\end{split}$

Furthermore, for all
$v\in\mathscr{L}^{2}\left(0,T;\textit{I}\\!{R}^{m}\right)$

$\displaystyle\begin{split}&\lambda\cdot\text{d}E_{T,x_{0}}(u)(v)\\\
&=\lambda\cdot\int_{0}^{T}\Psi(T)\Psi(s)^{-1}B(s)v(s)ds\\\
&=\int_{0}^{T}\lambda\cdot\Psi(T)\Psi(s)^{-1}B(s)v(s)ds\\\
&=\int_{0}^{T}B(s)^{\star}\left(\Psi(s)^{-1}\right)^{\star}\Psi(T)^{\star}\lambda\cdot
v(s)ds.\end{split}$

Let us set for every $s\in[0,T]$

$\displaystyle\begin{split}&p(s):=\left(\Psi(s)^{-1}\right)^{\star}\Psi(T)^{\star}\lambda-\left(\Psi(s)^{-1}\right)^{\star}\\\
&\qquad\qquad\qquad\cdot\int_{s}^{T}\Psi(r)^{\star}\text{d}_{x}L(\xi(r),u(r))dr.\end{split}$

Then, since $v$ is arbitrary, it follows that

$B(s)^{\star}p(s)=\text{d}_{u}L\left(\xi(s),u(s)\right)\quad\forall s\in[0,T]$

and so (10) holds. Now, the relation in (11) follows from (10) and Lemma
6-(i). Also,

$-{p}^{\prime}(s)=A(s)^{\star}p(s)-\text{d}_{x}L(\xi(s),u(s))\quad
s\in[0,T]\text{ a.e.}$

So, recalling (7) and from (11), we have for a.e. $s\in[0,T]$

$\displaystyle\begin{cases}\text{d}_{x}H(\xi(s),p(s))=\text{d}_{x}h\,[s]\\\
\text{d}_{p}H(\xi(s),p(s))=\text{d}_{p}h\,[s],\end{cases}$

where we put $[s]:=(\xi(s),p(s),{w}(\xi(s),Z(\xi(s),p(s))))$ for all
$s\in[0,T]$. Then, recalling (6), the pair $(\xi(.),p(.))$ solves the
Hamiltonian system (8) with final condition (9).

###  4.2 Proof of Theorem 5

The implication (ii)$\Longrightarrow$(i) immeditely follows from the
equibounded property of trajectories and from the continuity of the
Lagrangian. We need only to show (ii)$\Longrightarrow$(iii) and
(i)$\Longrightarrow$(ii).

(ii)$\Longrightarrow$(iii): From the embedding
$\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m})\subset\mathscr{L}^{1}(0,T;\textit{I}\\!R^{m})$
and applying the Gronwall Lemma, we can choose $\hat{K}>0$ such that

$\displaystyle\begin{split}&\left\|u\right\|_{1}+\left\|u\right\|_{2}\leq\hat{K},\quad\xi_{x_{0},u}(.)\subset\textit{I}\\!B_{\hat{K}}\\\
&\;\forall u\in\\{\delta_{S(x)}\Phi=0\\}.\end{split}$ (12)

First, we claim that the family of co-states $p_{u}\textnormal{ solving
}(\ref{Ham_system})$

$\displaystyle\begin{split}&\\{\,p_{u}:[0,T]\rightarrow\textit{I}\\!R^{n}\,|\,p_{u}(T)=\lambda\in\Lambda(u;x),\\\
&\;u\in\\{\delta_{S(x)}\Phi=0\\}\,\\}\end{split}$ (13)

is equibounded. We have that

$\displaystyle\begin{split}\exists c>0\,:\,\left|\lambda\right|\leqslant
c\quad\forall\lambda\in\Lambda(u;x)\,\forall
u\in\\{\delta_{S(x)}\Phi=0\\}.\end{split}$

Otherwise, suppose, by contradiction, that there exist two sequences
$\left\\{u_{k}\right\\}_{k}\subset\\{\delta_{S(x)}\Phi=0\\}$ and
$\lambda_{k}\in\Lambda(u_{k};x)$ such that
$\lim_{k}\left|\lambda_{k}\right|=+\infty$. We may assume that
$\lambda_{k}\neq 0$ for all $k\in\textit{I}\\!N$. Thus, for every
$k\in\textit{I}\\!N$ and $v\in\mathscr{L}^{2}(0,T;\textit{I}\\!{R}^{m})$

$\displaystyle\begin{split}\frac{\lambda_{k}}{\left|\lambda_{k}\right|}\cdot\text{d}_{u}\mathcal{C}\left(u_{k};x\right)(v)=\frac{1}{\left|\lambda_{k}\right|}\langle\,\text{d}\Phi\left(u_{k}\right),v\rangle.\end{split}$
(14)

Passing to subsequences and keeping the same notation, we may assume that
there exist $\lambda\in\textit{I}\\!{R}^{n}$ and
$u\in\textnormal{cl}_{w}\,\\{\delta_{S(x)}\Phi=0\\}$, with $|\lambda|=1$, such
that $\frac{\lambda_{k}}{\left|\lambda_{k}\right|}\rightarrow\lambda$ and
$u_{k}\rightarrow u$ weakly. Recalling Remark 3.2 and passing to the limit as
$k\rightarrow\infty$ in (14) we have

$\displaystyle\begin{split}\lambda\cdot\text{d}_{u}\mathcal{C}(u;x)=0\end{split}$

and, from our assumptions, a contradiction follows. Now, from the smoothness
of the vector fields $X_{i}$, let $M>0$ be such that
$|\text{d}X_{i}(x)^{\star}\xi|\leqslant M|\xi|$ for all
$\xi\in\textit{I}\\!R^{n}$, $x\in\textit{I}\\!B_{\hat{K}}$, and all
$i=1,\ldots,m$. Applying Proposition 7, we have that for any
$u\in\\{\delta_{S(x)}\Phi=0\\}$ and any $\lambda\in\Lambda(u;x)$ there is an
absolutely continuous arc $p_{u}:[0,T]\rightarrow\textit{I}\\!{R}^{n}$ such
that for a.e. $s\in[0,T]$

$\displaystyle\begin{cases}{p}_{u}^{\prime}(s)=\sum_{i=1}^{m}u_{i}(s)\text{d}X_{i}\left(\xi_{x_{0},u}(s)\right)^{\star}p_{u}(s)\\\
\qquad\qquad\qquad\qquad\qquad-\text{d}_{x}L\left(\xi_{x_{0},u}(s),u(s)\right)\\\
p_{u}(T)=\lambda.\end{cases}$

By Assumption 3, we can find a constant $\tilde{K}=\varphi({\hat{K}})>0$ such
that for every $u\in\\{\delta_{S(x)}\Phi=0\\}$ and for a.e. $s\in[0,T]$

$\displaystyle\begin{split}\left|{p}_{u}^{\prime}(s)\right|\leqslant
M\sqrt{m}|u(s)|\left|p_{u}(s)\right|+\tilde{K}\left(|u(s)|^{2}+1\right).\end{split}$

Thus, for all $u\in\\{\delta_{S(x)}\Phi=0\\}$ and all $s\in[0,T]$

$\displaystyle\begin{split}&\left|p_{u}(s)\right|\\\
&\leq\left|p_{u}(T)\right|+\int_{s}^{T}\tilde{K}\left(|u(r)|^{2}+1\right)dr\\\
&\qquad+\int_{s}^{T}M\sqrt{m}|u(r)|\left|p_{u}(r)\right|dr\\\
&\leq(c+\tilde{K}(\hat{K}^{2}+T))+\int_{s}^{T}M\sqrt{m}|u(r)|\left|p_{u}(r)\right|dr.\end{split}$

Applying Gronwall’s Lemma and recalling (12), it follows the claim (13). Thus,
from the claim and the Gronwall’s Lemma again, we have that there exists
$R\geq K$ such that for any $u\in\\{\delta_{S(x)}\Phi=0\\}$ and any co-states
$p_{u}(.)$ in (13)

$\displaystyle\begin{split}\xi_{x_{0},u}(.)\subset\textit{I}\\!B_{R},\quad
p_{u}(.)\subset\textit{I}\\!B_{R}.\end{split}$ (15)

Hence, using (11) and the locally Lipschitz continuity of $\mathfrak{u}(.,.)$,
we get (iii).

(i)$\Longrightarrow$(ii): Fix $u\in\\{\delta_{S(x)}\Phi=0\\}$ and, from
Assumpation 4, let $C>0$ and $\alpha>0,\hat{\alpha}>0$ such that
$\theta(r)\geq\alpha r^{2}$ and $\psi(r)\leq\hat{\alpha}r^{2}$ for all $r>C$.
For all $t\in[0,T]$, we have

$\displaystyle\begin{split}\int_{0}^{t}\left|u(s)\right|^{2}ds&=\int_{[0,t]\cap\left\\{s:\left|u(s)\right|>C\right\\}}\left|u(s)\right|^{2}\,ds\\\
&\qquad+\int_{[0,t]\cap\left\\{s:\left|u(s)\right|\leq
C\right\\}}\left|u(s)\right|^{2}\,ds\\\
&\leq\int_{[0,t]\cap\left\\{s:\left|u(s)\right|>C\right\\}}\left|u(s)\right|^{2}\,ds+C^{2}T.\end{split}$
(16)

Now, from (i) and Assumption 3, there exists $\hat{M}>0$, not depending on
$u(.)$, such that

$\displaystyle\begin{split}&\int_{[0,t]\cap\left\\{s:\left|u(s)\right|>C\right\\}}\left|u(s)\right|^{2}\,ds\\\
&\leq\alpha^{-1}\int_{[0,t]\cap\left\\{s:\left|u(s)\right|>C\right\\}}\theta(\left|u(s)\right|)\,ds\\\
&\leq\alpha^{-1}\int_{0}^{t}\theta(\left|u(s)\right|)\,ds\\\
&\leq\alpha^{-1}\int_{0}^{t}L(\xi_{x_{0},u}(s),u(s))\,ds\\\
&\qquad+\alpha^{-1}\int_{0}^{t}\psi(|\xi_{x_{0},u}(s)|)\,ds\\\
&\leq\alpha^{-1}\hat{M}+\alpha^{-1}\int_{0}^{t}\psi(|\xi_{x_{0},u}(s)|)\,ds.\end{split}$
(17)

Putting $M=\max_{i}\sup_{x\in\textit{I}\\!R^{n}}|X_{i}(x)|<\infty$, we have
for all $s\in[0,T]$

$\displaystyle\begin{split}|\xi_{x_{0},u}(s)|&\leq|x_{0}|+\int_{0}^{s}\sum_{i=1}^{m}|X_{i}(\xi_{x_{0},u}(\tau))||u_{i}(\tau)|d\tau\\\
&\leq|x_{0}|+mM\int_{0}^{s}\sum_{i=1}^{m}|u_{i}(\tau)|d\tau.\end{split}$

Since $\psi(|\xi_{x_{0},u}(s)|)\leq\hat{\alpha}|\xi_{x_{0},u}(s)|^{2}$ for all
$s\in[0,T]$ and using that
$\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m})\subset\mathscr{L}^{1}(0,T;\textit{I}\\!R^{m})$,
we have that there exists a constant $\hat{c}>1$, not depending on $u(.)$,
such that for all $s\in[0,T]$

$\displaystyle\begin{split}&\psi(|\xi_{x_{0},u}(s)|)\\\
&\leq\hat{c}\;(\int_{0}^{s}|u(\tau)|^{2}d\tau+|x_{0}|({\int_{0}^{s}|u(\tau)|^{2}d\tau})^{1/2}+|x_{0}|^{2}).\end{split}$

Hence, for all $t\in[0,T]$

$\displaystyle\begin{split}&\int_{0}^{t}\psi(|\xi_{x_{0},u}(s)|)ds\\\ &\leq
2\hat{c}T|x_{0}|^{2}+2\hat{c}\int_{0}^{t}(\int_{0}^{s}|u(\tau)|^{2}d\tau)ds.\end{split}$
(18)

Now, from (16)-(18), the Gronwall’s Lemma, and since all the involved
constants does not depends on $u\in\\{\delta_{S(x)}\Phi=0\\}$, we conclude
that there exists $k>0$ such that $\left\|u\right\|^{2}\leq k$ for all
$u\in\\{\delta_{S(x)}\Phi=0\\}$. Consequently, applying again Gronwall’s
Lemma, we have that all trajectories $\xi_{x_{0},u}(.)$ associated with
controls $u\in\\{\delta_{S(x)}\Phi=0\\}$ are uniformly bounded, that is, there
exists $\tilde{K}>0$ such that for every $u\in\\{\delta_{S(x)}\Phi=0\\}$

$\displaystyle\begin{split}\xi_{x_{0},u}(.)\subset\textit{I}\\!B_{\tilde{K}}.\end{split}$
(19)

From Proposition 7 and replacing $\tilde{K}>0$ with a suitable greater
constant, we have that for any $u\in\\{\delta_{S(x)}\Phi=0\\}$ and any co-
states $p_{u}(.)$ in (13) satisfies

$\displaystyle\begin{split}p_{u}(.)\subset\textit{I}\\!B_{\tilde{K}}.\end{split}$
(20)

Hence, from the continuity of $H(.,.)$, we get

$\displaystyle\begin{split}N:=\sup_{(s,u)\in[0,T]\times\\{\delta_{S(x)}\Phi=0\\}}\;|H(\xi_{x_{0},u}(s),p_{u}(s))|<\infty.\end{split}$

Now, notice that, from Proposition 7, for all $u\in\\{\delta_{S(x)}\Phi=0\\}$
and all $s\in[0,T]$ we have

$\displaystyle\begin{split}&{w}\left(\xi_{x_{0},u}(s),Z\left(\xi_{x_{0},u}(s),p(s)\right)\right)\cdot
Z(\xi_{x_{0},u}(s),p(s))\\\
&=u(s)\cdot\text{d}_{u}L(\xi_{x_{0},u}(s),p(s)).\end{split}$

Hence, applying Lemma 6-(ii), for all $u\in\\{\delta_{S(x)}\Phi=0\\}$ and all
$s\in[0,T]$

$\displaystyle\begin{split}&H(\xi_{x_{0},u}(s),p_{u}(s))\\\
&=u(s)\cdot\text{d}_{u}L(\xi_{x_{0},u}(s),u(s))-L(\xi_{x_{0},u}(s),u(s))).\end{split}$
(21)

Now, fix $R>0$ and consider $\hat{N}>0$ be such that
$|L(x,u)|\leqslant\hat{N}$ for all
$x\in\textit{I}\\!B_{R},u\in\textit{I}\\!B.$ Since $L$ is smooth, for any
$x\in\textit{I}\\!B_{R}$ and any $r>0$ the function $L(x,.)+c(x,r)|\,.\,|^{2}$
is convex on $\textit{I}\\!B_{r}$ for a suitable $c(x,r)\geq 0$ (cfr. [3]).
Moreover, fix $r=r(R)>0$ such that

$\displaystyle\begin{split}\theta(|u|)>\psi(|x|)+2\hat{N}+N,\;\forall|u|>r\,\forall
x\in\textit{I}\\!B_{R}.\end{split}$ (22)

Consider $x\in\textit{I}\\!B_{R},u\in\textit{I}\\!{R}^{m}\backslash\\{0\\},$
and put $v:=u/|u|$. Denote by
$g=g_{v}:\textit{I}\\!R\rightarrow\textit{I}\\!R$ the function
$g(s):=\text{d}_{u}L(x,sv)\cdot sv-L(x,sv)$. Then, from Assumption 2, the
function $g$ is continuously differentiable. So,
$g(s)=g(1)+\int_{1}^{s}g^{\prime}(\tau)d\tau$ for all $s\geq 1$. Now, for any
$\tau\geq 1$, we have $g^{\prime}(\tau)=\tau\text{d}^{2}_{u}L(x,\tau v)v\cdot
v$. Hence, for all $s\geq 1$, applying the Fundamental Theorem of Calculus we
have

$\displaystyle\begin{split}&\text{d}_{u}L(x,sv)\cdot sv-L(x,sv)\\\
&\geq\text{d}_{u}L(x,v)\cdot v-L(x,v)+\int_{1}^{s}\text{d}^{2}_{u}L(x,\tau
v)v\cdot v\;d\tau\\\ &=d_{u}L(x,v)\cdot v-L(x,v)+\tau d_{u}L(x,\tau v)\cdot
v|^{s=\tau}_{1=\tau}\\\ &\qquad-\int_{1}^{s}\text{d}_{u}^{2}L(x,\tau v)v\cdot
vd\tau\\\ &=\text{d}_{u}L(x,sv)\cdot(sv-v)+d_{u}L(x,v)-L(x,v).\end{split}$
(23)

Hence, since $|v|=1$, for any $\sigma>r$

$\displaystyle\begin{split}&\text{d}_{u}L(x,sv)\cdot(sv-v)\\\ &\geq
L(x,sv)-L(x,v)+c(x,\sigma)(s-1),\,\forall s\in[1,\sigma].\end{split}$ (24)

From (23) and (24), we have

$\displaystyle\begin{split}&\text{d}_{u}L(x,sv)\cdot sv-L(x,sv)\\\ &\geq
L(x,sv)+c(x,\sigma)(s-1)-2\hat{N},\,\forall s\in[1,\sigma].\end{split}$

So, for any $\sigma>r$ and putting $s=|u|$

$\displaystyle\begin{split}&\text{d}_{u}L(x,u)\cdot u-L(x,u)\\\
&\geq\theta(|u|)-\psi(|x|)+c(x,\sigma)(|u|-1)-2\hat{N},\,\forall|u|\in[1,\sigma].\end{split}$

Notice that $\sup_{x\in\textit{I}\\!B_{R}}c(x,\sigma)<+\infty$ for any
$\sigma>r$. Summing up and recalling (22), we can conclude that there exists a
constant $K>0$ (depending only on $\tilde{K}$) satisfying

$\displaystyle\begin{split}\text{d}_{u}L(x,u)\cdot u-L(x,u)>{N},\quad\forall
x\in\textit{I}\\!B_{\tilde{K}}\,\forall|u|>K.\end{split}$

By (21), we deduce that $|u(s)|\leqslant K$ for all $s\in[0,T]$ and all
$u\in\\{\delta_{S(x)}\Phi=0\\}$. The proof of Theorem 5 is now complete.

##  5 Application to Existence of Lipschitz Inversion Mappings

As main application of the Characterization Theorem 5, in this section we
provide an existence results of Lipschitz inversion mapping. In the following
we denote by $\mathscr{L}_{k}^{T}$ the class of all
$\textit{I}\\!R^{m}$-valued $k$-Lipschitz continuous functions on the time
interval $[0,T]$.

The proof of the following two lemmata are omitted since straightforward and
well known.

######  Lemma 5.1.

Let $X$ be a separable normed space and $\Phi:X\rightarrow\textit{I}\\!R^{n}$
be a linear, continuous, and surjective operator. Consider
$\left\\{x_{i}\right\\}_{i}$ dense in $X$. Then there exist linearly
independent vectors $x_{1},...,x_{n}$ such that
$\Phi:W\rightarrow\textit{I}\\!R^{n}$ is an isomorphism, where
$W=\textnormal{span}\,\left\\{x_{1},...,x_{n}\right\\}$.

######  Lemma 5.2 ([4],[18]).

Let $\tau>0$. Then the map

$[0,\tau]\times
L^{2}(0,\tau;\textit{I}\\!R^{m})\ni(s,u)\mapsto\textnormal{d}E_{s,x_{0}}(u)$

is continuous.

######  Theorem 8 (Inversion mapping).

If $\Phi|_{\\{\delta_{S(x)}\Phi=0\\}}<+\infty$, then there exist $k>0$, $r>0$,
and $\ell\geq 0$ such that for any
$(t,u)\in[0,T]\times\\{\delta_{S(x)}\Phi=0\\}$ we can find a map

$\displaystyle\begin{split}\mathfrak{u}_{t,u}:B_{r}(t)\times
B_{r}(x_{x_{0},u}(t))\rightarrow\\{k\text{-Lip. func. on
$[0,T]$}\\}\end{split}$

satisfying:

  1. (i)

$\mathfrak{u}_{t,u}\in C^{1}$.

  2. (ii)

$E_{s,x_{0}}(\mathfrak{u}_{t,u}(s,\beta))=\beta$ for all
$(s,\beta)\in\textit{I}\\!B_{r}(t)\times\textit{I}\\!B_{r}(x_{x_{0},u}(t))$.

  3. (iii)

$\textnormal{d}\mathfrak{u}_{t,u}$ is ${\ell}$-Lipschitz.

Let $(t_{0},u_{0})\in[0,T]\times\\{\delta_{S(x)}\Phi=0\\}$. We know that
$\textnormal{d}E_{t_{0},x_{0}}(u_{0})(\cdot)$ is surjective on
$\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m})$. Let $\mathscr{V}\subset
C^{1}(0,T;\textit{I}\\!R^{m})$ be a countable subset such that
$\overline{{\textnormal{span}\mathscr{V}}}=\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m})$.
Applying Lemma 5.1, there exist $n$ linearly independent vectors
$\left\\{v_{1}^{0},...,v_{n}^{0}\right\\}\subset\mathscr{V}$ such that the map

$\displaystyle\begin{split}(\alpha_{i})_{i}\mapsto\sum_{i=1}^{n}\alpha_{i}\textnormal{d}E_{t_{0},x_{0}}(u_{0})(v_{i}^{0})\end{split}$

is an isomorphism on $\textit{I}\\!R^{n}$. Define for any
$(t,u)\in(0,T)\times\mathscr{L}^{2}(0,T;\textit{I}\\!R^{m})$ the map
$\varphi^{0}_{t,u}:\textit{I}\\!R^{n}\rightarrow\textit{I}\\!R^{n}$ by
$\varphi^{0}_{t,u}((\alpha_{i})_{i})=\sum_{i=1}^{n}\alpha_{i}\textnormal{d}E_{t,x_{0}}(u)(v_{i}^{0})$.
By Lemma 5.2, there exist $\varrho_{0}>0,\,\mu_{0}>0$ such that for any
$(t,u)\in\mathcal{J}_{0}:=[t_{0}-\rho_{0},t_{0}+\rho_{0}]\times
B_{\mathscr{L}^{2}}(u_{0},\varrho_{0})$ the map
$\mathscr{E}_{t,u}^{0}:(0,T)\times\textit{I}\\!R^{n}\rightarrow(0,T)\times\textit{I}\\!R^{n}$,
defined by

$\displaystyle\begin{split}\mathscr{E}_{t,u}^{0}(s,(\alpha_{i})_{i})=(s,E_{s,x_{0}}(u+\sum_{i=1}^{m}\alpha_{i}v_{i}^{0}))\end{split}$

satisfies for all $(t,u)\in\mathcal{J}_{0}$

$\displaystyle\begin{split}|\det\left(\textnormal{d}\mathscr{E}_{t,u}^{0}(t,0)\right)|=\left|\det\varphi^{0}_{t,u}\right|\geq\mu_{0}\end{split}$

where $B_{\mathscr{L}^{2}}(u,r)$ stands for the closed ball in
$\mathscr{L}^{2}$ centered at $u$ of radius $r$. Now, from the Ascoli-Arzelà
theorem, the set $[0,T]\times\\{\delta_{S(x)}\Phi=0\\}$ is compact. Then there
exists $N\in\textit{I}\\!N^{+}$ such that, for all $j=1,...,N$, we can find
$\rho_{j}>0,\,\mu_{j}>0$,
$(t_{j},u_{j})\in[0,T]\times\\{\delta_{S(x)}\Phi=0\\}$, and linearly
independent $\\{v_{1}^{j},...,v_{n}^{j}\\}\subset\mathscr{V}$, such that

$\displaystyle\begin{split}&[0,T]\times\\{\delta_{S(x)}\Phi=0\\}\\\
&\subset\bigcup_{j=1,...,N}\,[t_{j}-\varrho_{j},t_{j}+\varrho_{j}]\times
B_{\mathscr{L}^{2}}(u_{j},\varrho_{j})\\\
&=:\bigcup_{j=1,...,N}\mathcal{J}_{j}.\end{split}$

Defining for any $(t,u)\in\mathcal{J}_{j}$ the maps
$\mathscr{E}_{t,u}^{j}:(0,T)\times\textit{I}\\!R^{n}\rightarrow(0,T)\times\textit{I}\\!R^{n}$
and $\varphi^{j}_{t,u}:\textit{I}\\!R^{n}\rightarrow\textit{I}\\!R^{n}$ by

$\displaystyle\begin{split}&\mathscr{E}_{t,u}^{j}(s,(\alpha_{i})_{i})=(s,E_{s,x_{0}}(u+\sum_{i=1}^{m}\alpha_{i}v_{i}^{j}))\\\
&\varphi^{j}_{t,u}((\alpha_{i})_{i})=\sum_{i=1}^{n}\alpha_{i}\textnormal{d}E_{t,x_{0}}(u)(v_{i}^{j})\end{split}$

we deduce that for all $(t,u)\in\mathcal{J}_{j}$ and $j=1,...,N$

$\displaystyle\begin{split}&|\det\left(\textnormal{d}\mathscr{E}_{t,u}^{j}(t,0)\right)|\\\
&=\left|\det\varphi^{j}_{t,u}\right|\geq\mu_{j}\geq\min\left\\{\mu_{1},...,\mu_{N}\right\\}>0.\end{split}$
(25)

Applying the Inverse Mapping Theorem to the map $\mathscr{E}_{t,u}^{j}$ and
using a compactness argument, we conclude that for each $j$ there exists
$r_{j}>0$ such that the set
$V_{j}(t,u):=(t-r_{j},t+r_{j})\times\textit{I}\\!B_{r_{j}}(E_{t}(u))$ is
isomorph to $({\mathscr{E}_{t,u}^{j}})^{-1}\left(V_{j}(t,u)\right)$ for any
$(t,u)\in\mathcal{J}_{j}$. Put $r=\min\left\\{r_{1},...,r_{N}\right\\}$ and
define for any $(t,u)\in\mathcal{J}_{j}$

$\displaystyle\begin{split}&\mathfrak{u}_{t,u}(t^{\prime},\beta)=u+\sum_{i=1}^{m}\alpha_{i}(t^{\prime},\beta)v_{i}^{j}\\\
&\forall(t^{\prime},\beta)\in V_{j}(t,u)\end{split}$

where
$(\mathscr{E}_{t,u}^{j})^{-1}(t^{\prime},\beta)=:(t^{\prime},\alpha(t^{\prime},\beta))$.
Notice that, since the coefficients ${\alpha_{i}}$ are bounded by a suitable
constant $M\geq 0$ and ${v_{i}^{j}}\in C^{1}(0,T;\textit{I}\\!R^{m})$, there
exists a constant $k\geq 0$ such that $\mathfrak{u}_{t,u}$ take values in
$\mathscr{L}_{k}^{T}$. Hence, $(i)$ and $(ii)$ follow. Moreover, from (25) and
the regularity of the end-point map, there exists a constant $\ell\geq 0$,
depending only on $k$, such that $\textnormal{d}\mathfrak{u}_{t,u}$ is
$\ell$-Lipschitz for all $(t,u)\in[0,T]\times\\{\delta_{S(x)}\Phi=0\\}$. So,
we get $(iii)$.

##  6 Conclusions

In this paper, we have addressed a class of extremization problems wrt end-
point constraints qualifications. We assumed the functional generating the
Lagrange sub-manifold merely Fréchet differentiable and the associated
Lagrangian possibly unbounded and not strictly convex in the fiber. In this
paper, with the use of recently investigated techniques for extremization
problems, we show that the controls associated with constrained extremals
enjoy of Lipschitz regularity whenever singular trajectory are absent.
Hörmander’s condition on the affine dynamics cannot be skipped in order to the
well-posedness of the problem. As main implication, we construct a locally
Lipschitz inversion mapping from the ambient space to the set of constrained
extremals, that turns out to be useful for the well definition of value
functions associated to extremization problems.

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