# Linear Geometric Homogeneity for Control and Estimation in Finite Time

Andrey Polyakov

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Andrey Polyakov. Linear Geometric Homogeneity for Control and Estimation in Finite Time. Automatic Control Engineering. Université de Lille, 2018. tel-04426543

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Université
de Lille
1
SCIENCES
ET TECHNOLOGIES

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<figure><img src="image_4.png" /></figure>

Mémoire
présenté pour obtenir le diplome
d'Habilitation à diriger des recherches

Spécialité: Génie informatique, automatique et traitement du signal

Date de soutenance: le 29/11/2018

# Homogénéité géométrique linéaire pour le contrôle et l'estimation en temps fini

Andrey POLYAKOV

Non-A POST, Inria Lille

Jury

Reporters:

Yacine CHITOUR, Professor, Université Paris-Sud

Christophe PRIEUR, Director of Research at CNRS, GIPSA-Lab, Grenoble

Isabelle QUEINNEC, Director of Research at CNRS, LAAS, Toulouse

Jury Members:

Jean-Michel CORON, Professor, Université Pierre et Marie Curie

Francoise LAMNABHI-LAGARRIGUE, Director of Research at CNRS, L2S

Jaime Alberto MORENO PÉREZ, Professor, UNAM, Mexico

Wilfrid PERRUQUETTI, Professor, Centrale Lille

Jean-Pierre RICHARD, Professor, Centrale Lille
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Summary

The thesis is presented in order to obtain the "Habilitation à diriger des recherches" degree in automation and industrial informatics.

The author received the Ph.D. degree in System Analysis, Control and In-
formation Processing from Voronezh State University, Russia in 2005. From 2004
to 2007 and from 2008 to 2010 he was with the same university as lecturer and
associate professor, respectively. From 2007-2008 he was research assistant (post-
doc) with CINVESTAV-IPN, Mexico. From 2010 to 2012 he has served as a leader
researcher at the Institute of Control Sciences of Russian Academy of Sciences
in Moscow. In 2012 he joined NON-A team of Inria Lille, France. His main re-
search interests include sliding mode control, time delay system, set-theoretic and
homogeneity-based methods in control. During the scientific carrier he produced
more than 150 publication.

The thesis deals with one of the main research direction of the applicant in
the last 5 years: design of finite/fixed-time controllers and observers by means of
a canonical homogeneous norm induced by linear dilations. Such a norm simplifies
the analysis and design of homogeneous control systems, and allows a liner control
and state estimation algorithms to be extended easily to some non-linear cases.
Canonical homogeneous norms appear in the control and systems theory in the
form of controllability functions and implicit Lyapunov functions.

The first chapter of the thesis presents the scientific background and the
experience of the author. In the second chapter, the concept of homogeneity (sym-
metry) with respect to linear dilations (i.e. linear geometric homogeneity) is in-
troduced and some related notions such as the canonical homogeneous norm are
studied. Solutions to several control and estimation problems for linear and non-
linear systems in a Banach space $\mathbb{B}$ and in the Euclidean space $\mathbb{R}^n$ are presented.
In the third chapter, the future directions of the research of the author are dis-
cussed. Finally, the list of publications of the author and the texts of four selected
papers, which are closely related with the topic of the thesis but not surveyed in
the chapter 2, are presented in Appendix.
Acknowledgments

I am grateful to *my family* for their patience and the personal sacrifices which they have given to my work. I also acknowledge *my colleagues* and *friends* from France, Russia, Mexico, Israel, UK, Ireland and other counties for their strong support, for fruitful ideas, suggestions, criticism and always interesting discussions.
# Contents

<table><tr><td>Notation</td><td>5</td></tr><tr><td>1 Scientific Background and Experience</td><td>7</td></tr><tr><td>1.1 Diplomas and Grades</td><td>7</td></tr><tr><td>1.2 Positions</td><td>7</td></tr><tr><td>1.3 Research and Publication Activities</td><td>8</td></tr><tr><td>1.4 Research projects</td><td>10</td></tr><tr><td>1.5 Supervision</td><td>12</td></tr><tr><td>1.6 Teaching</td><td>13</td></tr><tr><td>1.7 Prizes and Distinctions</td><td>13</td></tr><tr><td>2 Linear Geometric Homogeneity in Systems and Control</td><td>15</td></tr><tr><td>2.1 Introduction</td><td>15</td></tr><tr><td>2.2 Homogeneous Systems</td><td>23</td></tr><tr><td>2.2.1 Linear Dilations</td><td>23</td></tr><tr><td>2.2.2 Homogeneous Mappings and Equations</td><td>34</td></tr><tr><td>2.3 Control Design and State Estimation Using Linear Dilations</td><td>40</td></tr><tr><td>2.3.1 Homogeneous Stabilization in a Hilbert Space</td><td>40</td></tr><tr><td>2.3.2 Quadratic Stabilizability of Homogeneous Systems in ℝ<sup>n</sup></td><td>52</td></tr><tr><td>2.3.3 Homogeneous Observers in ℝ<sup>n</sup></td><td>61</td></tr><tr><td>2.4 Conclusion remarks</td><td>65</td></tr><tr><td>3 Future Directions of Research</td><td>69</td></tr><tr><td>3.1 Fundamental research</td><td>69</td></tr><tr><td>3.1.1 Short term perspective</td><td>69</td></tr><tr><td>3.1.2 Long term perspective</td><td>70</td></tr><tr><td>3.2 Applications</td><td>71</td></tr><tr><td>3.3 ControlHub Platform</td><td>71</td></tr><tr><td>Bibliography</td><td>72</td></tr><tr><td>Appendix</td><td>79</td></tr></table>
Notation

• $\mathbb{N}$ is the set of natural numbers; $\mathbb{Z}$ is the set of integers; $\mathbb{R}$ is the set of reals; $\bar{\mathbb{R}} = \mathbb{R} \cup \{-\infty\} \cup \{+\infty\}$ and $\mathbb{R}_+ = [0, +\infty)$; $\mathbb{C}$ is the set of complex numbers.

• $\mathcal{I}$ denotes one of the following intervals: $[a, b]$, $(a, b)$, $[a, b)$ or $(a, b]$, where $a < b$, $a, b \in \mathbb{R}$ or $a = -\infty$, and/or $b \in +\infty$.

• The Cartesian product of sets is denoted by $\times$.

• $\mathbb{B}$ is a real Banach space with a norm $\|\cdot\|$ and $\mathbb{H}$ is a real Hilbert space with an inner product $\langle \cdot, \cdot \rangle$.

• $S = \{x \in \mathbb{B} : \|x\| = 1\}$ is the unit sphere in $\mathbb{B}$.

• $\mathcal{L}(X, Y)$ is the space of linear bounded operators $X \to Y$, where $X$ and $Y$ are Banach spaces, and

$$ \|A\|_{\mathcal{L}(X,Y)} = \sup_{u \neq 0} \frac{\|Au\|_Y}{\|u\|_X} \quad \text{and} \quad \lfloor A \rfloor_{\mathcal{L}(X,Y)} = \inf_{u \neq 0} \frac{\|Au\|_Y}{\|u\|_X}, \quad A \in \mathcal{L}(X,Y). $$

If $X = Y$ we use notations $\|A\|$ and $\lfloor A \rfloor$ for shortness.

• If $P = P^\top \in \mathbb{R}^{n \times n}$ then $P > 0$ means that the matrix $P$ is positive definite.

• $\lambda_{\min}(P)$ and $\lambda_{\max}(P)$ denotes minimum and maximum eigenvalues of symmetric matrix $P = P^\top \in \mathbb{R}^n$.

• $\bar{X}$ denotes the closure of the set $X$ of a metric space.

• $\partial\Omega$ is the boundary set of a $\Omega \subset \mathbb{R}^n$.

• $C_0^\infty(\Omega, \mathbb{R}^m)$ is the set of infinitely differentiable (smooth) functions having compact support in $\Omega$, where $\Omega \subset \mathbb{R}^m$ is an open set.

• $H^n(\Omega, \mathbb{R}^m) = W^{n,2}(\Omega, \mathbb{R}^m)$ is the Sobolev space of functions $\Omega \subset \mathbb{R}^k \to \mathbb{R}^m$.

• $H_0^n(\Omega, \mathbb{R}^m)$ is the closure of $C_0^\infty(\Omega, \mathbb{R}^m)$ with respect to the norm in $H^n$.

• $L^2(\Omega, \mathbb{R}^m) = H^0(\Omega, \mathbb{R}^m)$ is the Lebesgue space of quadratically integrable functions $\Omega \to \mathbb{R}^m$.

• The set consisting of elements $x_1, x_2, \dots, x_n$ is denoted by $\{x_1, x_2, \dots, x_n\}$.
• The sign function is defined by

$$ \operatorname{sign}_{\sigma}(\rho) := \begin{cases} 1 & \text{if } \rho > 0, \\ -1 & \text{if } \rho < 0, \\ \sigma & \text{if } \rho = 0, \end{cases} \qquad (1) $$

where $\sigma \in \mathbb{R}: -1 \le \sigma \le 1$. If a concrete value of $\sigma$ is not important for considerations, we use the notation $\operatorname{sign}(\rho)$.

• If $V(\cdot) \in C^1(\mathbb{R}^n, \mathbb{R})$ then $\nabla V(x) := \left(\frac{\partial V}{\partial x_1}, \dots, \frac{\partial V}{\partial x_n}\right)^T$. If $s \in C^1(\mathbb{R}^n, \mathbb{R}^m)$, $s(\cdot) = (s_1(\cdot), \dots, s_m(\cdot))^T$ then $\nabla s(x)$ is the matrix $\mathbb{R}^{n \times m}$ of the partial derivatives $\frac{\partial s_j}{\partial x_i}$, $i = 1, 2, \dots, m, j = 1, 2, \dots, n$.

• Let $\mathbb{F}_d(\mathbb{R}^n)$ (resp. $\mathbb{H}_d(\mathbb{R}^n)$) be the set of continuous vector fields $\mathbb{R}^n \setminus \{0\} \to \mathbb{R}^n$ (resp. functions $\mathbb{R}^n \setminus \{0\} \to \mathbb{R}$) homogeneous with respect to the dilation $\mathbf{d}$.

• Let $\deg_{\mathbb{F}_d}(f)$ (resp. $\deg_{\mathbb{H}_d}(h)$) denote the homogeneity degree of $f \in \mathbb{F}_d(\mathbb{R}^n)$ (resp. $h \in \mathbb{H}_d(\mathbb{R}^n)$).
# Chapter 1

# Scientific Background and Experience

In this chapter, my scientific background, the main achievements and scientific carrier are expressed shortly. Next, in the second chapter, the main results obtained in the field of homogeneity-based control and estimation are presented with more details. Some planned directions of research are briefly described in the third chapter.

## 1.1 Diplomas and Grades

Both my degrees I have obtained from Voronezh State University, Russia:

* December 2005: PhD in Physics and Mathematics (specialization in *System Analysis, Control and Information Processing*)
  Title: Control of Neutral and Unstable Plants Using Relay Delayed Feedback
  Supervisor: Prof. Vadim Strygin
  Reporters: Prof. A. Kobrin and Prof. G. Lozgachev

* July 2003: Masters Degree in Applied Mathematics and Informatics
  Supervisor: Prof. V. Strygin

## 1.2 Positions

I was working in 4 scientific centers (3 different countries) during my career:

* 2012-present: France, Inria Lille - Nord Europe, Non-A team project, presently Non-A Post. Position: Researcher (CR) - permanent (since 2013)
* 2010-2012: Russia, Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia; Lab.7 "Robust and Adaptive Control"; Position: Leader Researcher - permanent

* 2008-2010 (and 2004-2007): Russia, Voronezh State University, Faculty of Applied Mathematics and Mechanics; Position: Associate Professor (Lecturer in 2004-2007) - permanent

* 2007-2008: Mexico, CINVESTAV-IPN, Department of Automatic Control. Position: Postdoc - temporary.

## 1.3 Research and Publication Activities

### Research topics

My research activities in the field of control theory can be split in four topics

* *Homogeneity in Systems and Control* (since 2012).

Homogeneity is a sort of a symmetry allowing a lot of useful properties to be discovered in the context of systems theory and control design. For example, asymptotic stability of homogeneous system ensures its robustness (in the sense of input to state stability) with respect to a rather large class of perturbations and it implies finite-time stability if the homogeneity degree is negative. Last years, I have dealt with control and observer design based on homogeneity technique as well as with both quantitative and qualitative analysis of homogeneous systems. In particular, the methods of implicit Lyapunov functions, linear matrices inequalities and consistent discretizations have been elaborated for homogeneous systems. Some homogeneity based methods have been successfully extended to infinite dimensional systems. A few recent results are reported in the next chapter.

* *Set-Theoretic Methods in Control and Estimation* (since 2007).

In practice, asymptotic (or finite-time) stability and as well as exact estimation are hard to be realized due to technical nonidealities of actuators and sensors, uncertainties in system models, etc. In this case set-theoretic methods (e.g. stability of sets) become proper tools for quantitative control system analysis and design. Attractive ellipsoids method is one of the most efficient tools in this context. It allows the linear control/observer design approaches to be applied for a class of quasi-Lipschitz nonlinear models. This methodology is summarized in the book (co-authored by me) "Attractive Ellipsoids in Robust Control", Bikhauser, 2014. In the last years, I have also dealt with minimax and interval observers techniques as well as observer-based control design for both finite and infinite dimensional systems.
* *Sliding Mode Control* (since 2000).

Being introduced in the early 60s, sliding mode is one of the first methods of robust control design. However, a mathematically rigorous analysis of the sliding mode control system is always complicated due to the discontinuous nature of the control laws. In this context I have proposed a unified approach to Lyapunov function-based stability analysis of the so-called second order sliding mode systems. I have also dealt with extension of attractive ellipsoids and homogeneity methods to sliding mode systems and with application of sliding mode control design methodology in flow control systems.

* *Time Delay Systems* (since 2000).

Delay effects accompany any real control systems. Analysis and design of control systems in the case of input and state delays is important for control practice. In this context, I was involved in development of relay and sliding mode control design for systems with delays. Analysis of robustness of homogeneous systems with respect to delays. Control design for bi-linear time switched systems with delays. Interval state estimation and interval control design for uncertain delay systems.

## Structure of publications

The complete list of publications is given in Appendix A. This section gives just general overview of my publication activity. I have more than 150 scientific publications:

* *International Journal Papers:* more than **50**
(including 20 journal papers in the top-ranked control journals: Automatica and IEEE Transactions on Automatic Control)

* *International Conference Papers:* more than **80**
(including about 35 papers in top-ranked control conferences: IEEE Conference on Decision and Control and IFAC World Congress)

* *Books:* 1
(A. Poznyak, A. Polyakov, V. Azhmyakov "Attractive Ellipsoids in Robust Control", Birkhauser, 2014)

* *Book chapters:* 4

* *National Journals and Conferences* (in Russian): more than **10**

My publication activity in the research topics is summarized in the next table.

<table><thead><tr><th>Research Topic</th><th>International Journals</th><th>International Conferences</th><th>Books and Chapters</th><th>National Journals and Conferences</th><th>Total</th></tr></thead><tbody><tr><td>Homogeneity</td><td>19</td><td>31</td><td>1</td><td>3</td><td>54</td></tr><tr><td>Set-Theoretic</td><td>11</td><td>19</td><td>2</td><td>0</td><td>32</td></tr><tr><td>Sliding Mode</td><td>16</td><td>17</td><td>1</td><td>1</td><td>35</td></tr><tr><td>Time Delays</td><td>13</td><td>15</td><td>1</td><td>7</td><td>36</td></tr></tbody></table>
## Patent

* M. Feingesicht, A.Polyakov F. Kerherve, J.P. Richard, 'Dispositif de controle actif du recollement d'un ecoulement sur un profil', Patent (brevet) INPI deposit date June 15th, 2017, Nb 1755440, Patent (brevet)

## 1.4 Research projects

The research projects I have been involved in the last 5 years (in France):

* **ANR PRC DIGITSLID (2018-2021)**
  Subject: "Digital set-valued and homogeneous sliding mode control and differentiators: the implicit approach"
  Coordinator: Bernard Brogliato
  Partners: Inria Grenoble, Inria Lille, CNRS LS2N (Nantes)

The project is devoted to the development of consistent discretization schemes for sliding mode observers and controllers. The methodology of the implicit discretization for some classes of sliding mode systems are planned to be developed. I am responsible for research activities of NON-A Post(future Valse) team of Inria in the context of this project. We will develop an innovative schemes for consistent discretization of homogeneous control and estimation algorithms.

* **ANR PRC Finite4SoS (2015-2020)**
  Subject: "Finite-time Control and Estimation for Systems of Systems"
  Coordinator: Wilfrid Perruquetti
  Partners: Inria Lille, Université Pierre et Marie Curie, MINES ParisTech

The project is aimed at a creation of a unified framework for analysis and design of controlled system of systems based on the finite-time stability concept. The key feature of the project is heterogeneity of the inter-connected subsystems modeled by differential law of different types (ODEs, PDEs, time-delay systems, differential inclusions). In this project, I have developed a homogeneity concept for operators in Banach spaces and a universal controller for finite-time stabilization of an evolution equations. Homogeneous observers for ODEs have been developed by PhD Student Francisco Lopez under my supervision.

* **ANR JCJC ROCC-SYS (2014-2018)**
  Subject: "Robust control of cyber-physical systems"
  Coordinator: Laurentiu Hetel
  People involved: Andrey Polyakov, Christophe Fiter, Alexandre Kruszewski

The project deals with control problems for plants governed by digital controllers. The sampled, switched and networked nature of control devices
arise some specific issues studied in this project. In particular, I have devel-
oped algorithms of control design for relay systems. The methods of convex
embedding, implicit Lyapunov function and attractive ellipsoids have been
utilized for successful solving of the robust stabilization problem for such
control systems.

* ANR PRC Chaslim (2011-2015)

Subject: "Chattering-free sliding modes"

Coordinator: Bernard Brogliato

Partners: Inria Grenoble, Inria Lille, CNRS IRCCyN

In this project some methods of chattering reduction of sliding mode
control were developed. They include novel discretization schemes for the
first order and the second order systems as well as homogeneity-based design
of high order sliding mode algorithms allowing chattering reduction. In this
project, I have developed the implicit Lyapunov function method for high
order sliding mode control design and proposed an efficient scheme of a digital
control implementation allowing completely remove the chattering.

* CPER DATA, project "ControlHub" (2016-Now)

Coordinator: Andrey Polyakov

Participants: Dagher Roudy, Dherbomez Gerald, Jean-Pierre Richard, Denis

Efimov

ControlHub is the platform for a rapid prototyping of control algorithms for cyber-physical systems. It allows the End-User to have a remote access to real experimental setups in order to validate mathematical control algorithms. ControlHub is under construction by Non-A Post(future Valse) team of Inria Lille Nord Europe.

* CPER ELSAT2020, project "Contratech" (2015-Now)

Coordinator: Jean-Marc Foucaut

Partners: Univ. Vallenciennes, Centrale Lille, LMFL, CNRS LAMIH, ON-
ERA, Inria Lille, CNRS CRISTAL

The Contratech platform joins research groups of different institutions
around the important and challenging problem of active flow control.

* "RECO_T" North-European Inria Associate team with IBM Research in Dublin,
Ireland

Subject: "Robust Estimation and Control with Time Constraints"

Coordinator: Andrey Polyakov (Inria)

The project bridges the research team of NON-A Post (future Valse),
Inria Lille, France and IBM Research Lab in Dublin. It is aimed at develop-
ment of minimax control and estimation algorithms based on homogeneity
technique.
* "HoTSMoCE" Inria Associate team with UNAM, Mexico
  Subject: "Homogeneity Tools for Sliding Mode Control and Estimation"
  Coordinator: Denis Efimov (Inria)

The project "HoTSMoCE" bridges the research team of NON-A Post
(future Valse), Inria Lille, France and Sliding Mode Control Lab of National
Autonomous University of Mexico (UNAM).

1.5 Supervision

I was also involved (or I am still involved) in supervision of the following Postdocs
and PhD students in France:

Postdocs

* Nicolas Espitia (10.2017-now; co-supervisors: Prof. Wilfrid Perruquetti and Dr. Denis Efimov)

* Tonametl Sanchez (10.2017-now; co-supervisor: Prof. Jean-Pierre Richard)

* Manuel Mera (15.07.2014-15.07.2015)

PhD Students

* Maxime Feingesicht (01.01.2015-11.12.2017; Co-supervisors: Jean-Pierre Richard, Andrey Polyakov and Franck Kerherve)

* Francisco Lopez Ramirez (10.2015-19.11.2018; Co-supervisors: Wilfrid Perruquetti, Andrey Polyakov, Denis Efimov,)

* Youness Braidiz (since 10.2017; Co-supervisors: Wilfrid Perruquetti, Denis Efimov, Andrey Polyakov)

* Siyuan Wang (since 10.2017; Co-supervisors: Gang Zheng, Andrey Polyakov)

* Alex Dos Reis de Souza (since 10.2018; Co-supervisors: Denis Efimov, Jean-Luc Gouze, Andrey Polyakov)

I was also involved (or I am still involved) in supervision of **PhD students** in Mexico and Russia, e.g. Nadhynee Martinez Fonesca (since 12.2015; UPIBI-IPN, Mexico; Co-supervisor: Dr. Isaac Chairez); Konstantin Zimenko (10.2015-22.11.2018, ITMO University, Russia; Supervisor: Artem Kremlev) Tatyana Kharkovskaya (since 10.2016, ITMO University and Centralle Lille in co-tutelle; Co-supervisors: Jean-Pierre Richard, Artem Kremlev).

Masters Students

* Erwin Jezequel (Internship, 2018, Centrale Lille)

* Marc Sanmillan (Internship, 2018, Centrale Lille)

* Dmitrii Kondratovich (2012; Institute of Control Sciences of Russian Academy of Sciences)
* Elena Maslennikova (2009, Voronezh State University)

* Aleksej Baushev (2009, Voronezh State University)

* Andrey Solov'ev (2009, Voronezh State University)

* Artem Batuev (2009, Voronezh State University)

* Evgenii Orazaev (2009, Voronezh State University)

* Sergej Kosinov (2007, Voronezh State University)

* Leonid Kukushkin (2007, Voronezh State University)

* Konstantin Chernyshev (2007, Voronezh State University)

## 1.6 Teaching

The courses were given in Voronezh State University (2004-2007 and 2008-2010; more than 3000 hours):

* Algebra & Analytical Geometry (for bachelors, 1st year);

* Programming Languages and Algorithms (for bachelors, 1st and 2nd years);

* Algorithms for Sparse Matrices (for bachelors, 3rd year);

* Mathematical Software (for bachelors, 3rd year);

* Algorithms of Computer Graphics (for bachelors, 4th year);

* Relational Database Management Systems (for bachelors, 4th year);

* Computational Mathematics (for bachelors, 3rd and 4th years);

* Internet Programming (for masters, 1st year);

* Linear Matrix Inequalities (for masters, 2nd year);

## 1.7 Prizes and Distinctions

### Prizes

* Letov Prize 2011 (Institute of Control Sciences, Russian Academy of Sciences)

### Membership of Editorial Boards

* International Journal of Robust and Nonlinear Control (2014-2018)

* Journal of The Franklin Institute (2014-2016)

* Journal of Optimization Theory and Applications (2014-2018)

* Automation and Remote Control (2015-now)
## Invited Talks

### Conferences:

* "Active Flow Control: Sliding Mode Control Approach", Workshop on Active Drag Reduction Aachen, Germany, 15-16 March 2018

* "Sliding mode control of flow separation", 2nd Workshop on Machine Learning Control, Valenciennes, France, July 5th-6th, 2017

* "Robust Stabilization with Time Constraints: Implicit Lyapunov Function Approach", International Conference "Optimization And Applications in Control and Data Science" on the occasion of Boris Polyak's 80th birthday, Moscow, Russia, 13-15 May 2015

### Tutorial courses:

* Tutorial course for 6th IEEE International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE-2009), November 10-13, 2009, Toluca, Mexico. (6 hours, title: "Lyapunov Function Method for Nonlinear and Discontinuous Systems")

* Tutorial course for Traditional Whole-Russian School "Control, Information and Optimization" in 2011 (2 hours; title: "Lyapunov Function Design for High Order Sliding Modes") and in 2015 (4 hours; "Stability Notions and Lyapunov Functions for Sliding Mode Control Systems")

* Tutorial course "Stability Notions and Lyapunov Functions for Sliding Mode Control Systems" (Seminar of NON-A/SyNeR teams, Inria, 20 and 21 October 2013, 2 hours)

I gave talks at *seminars* of several research institutions: CNRS L2S (2017), Inria (2014), Institute of Control Sciences of Russian Academy of Sciences (2013, 2009), CINVESTAV-IPN, Mexico (2012, 2010, 2008), UNAM, Mexico (2012), Institute for Problems in Mechanics of Russian Academy of Sciences (2010, 2005).

## Reviewing activity

* Outstanding reviewer of Automatica (2016).

* I also serve as a reviewer in high ranked journals/conferences : IEEE Transactions of Automatic Control, Systems & Control Letters, International Journal of Robust and Nonlinear, Conference on Decision and Control etc.
# Chapter 2

# Linear Geometric Homogeneity in Systems and Control

## 2.1 Introduction

### Dilation Symmetry

Symmetry is a certain invariance when some characteristics of an object do not change under a set of transformations. Symmetry occurs in many branches of mathematics. The simplest examples of the symmetry can be found in the geometry, for example, as an invariance of a figure with respect to such transformations as rotation, translation, reflection and dilation. It is well-known that the rotation remains invariant both size and shape of the figure, while the dilation changes its size leaving invariant only the shape.

The symmetry with respect to the dilation can also be discovered for the level set of the so-called homogeneous function. In mathematical analysis, symmetry of a function $f$ with respect to the uniform dilation of its argument

$$x \to \lambda x,$$

where $\lambda > 0$ is the scaling factor, is known as homogeneity. In the simplest case, it is defined as follows

$$f(\lambda x) = \lambda f(x), \quad \lambda > 0.$$

In other words, homogeneity is a dilation symmetry. The latter always needed for linearity, which additionally asks for the central symmetry $f(-x) = -f(x)$ and the additivity $f(x + y) = f(x) + f(y)$. Thus all linear functions are homogeneous with respect to the uniform dilation. The class of homogeneous mappings is placed somewhere in-between essentially non-linear and linear ones.

In 18th century the homogeneity with respect to the uniform dilation (standard homogeneity) has been studied by Leonhard Euler. His notion of homogeneity
is also well known in the context of polynomial functions.

**Definition.** A function $f: \mathbb{R} \to \mathbb{R}$ is said to be homogeneous if there exists a number $\nu \in \mathbb{R}$ such that $f(\lambda x) = \lambda^{\nu}f(x), \forall \lambda > 0$. The number $\nu$ is called homogeneity degree of the function $f$.

According to this definition, any linear function has homogeneity degree 1, but, the quadratic one

$$x = (x_1, x_2)^{\top} \xrightarrow{f} x_1^2 + x_1x_2 + x_2^2$$

is homogeneous of degree 2. Its level sets are ellipsoids centered at the origin. Obviously, they are symmetric with respect to the uniform dilation.

Euler's *Homogeneous Function Theorem* is one of the famous results underlying the modern homogeneity theory:

**Theorem.** A continuously differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ is homogeneous of degree $k$ if and only if $\frac{\partial f}{\partial x} = k f(x), x \in \mathbb{R}^n$.

From this theorem, in particular, we conclude that homogeneity of a function is inherited by its derivatives. In fact, the dilation symmetry can be inherited by other mathematical objects induced by homogeneous functions. In particular, this chapter deals with a symmetry of non-linear generalized homogeneous differential equations and some related properties important for control systems design.

## Homogeneous Differential Equations

Let us consider the simplest scalar ordinary differential equation (ODE) with the homogeneous right-hand side

$$\dot{x} = -x^{\nu}, \quad x \in \mathbb{R},$$

where $\nu = p/q$, $p$ is an odd integer and $q$ is an even natural number. Its solution with the initial condition $x(0) = x_0$ is given by

$$x(t, x_0) = \frac{x_0}{(1 + (\nu - 1)x_0^{\nu - 1}t)^{1/(\nu - 1)}}.$$

Hence, we easily derive a symmetry of solutions with respect to simultaneous dilation of the initial condition and the time variable $t$:

$$x(\lambda^{1-\nu}t, \lambda x_0) = \lambda x(t, x_0), \quad \lambda > 0.$$

The time scaling factor depends on homogeneity degree $\nu$. In fact, such a symmetry of solutions holds for any homogeneous differential equation [78].
**Theorem.** Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuous homogeneous vector field of degree $\nu \in \mathbb{R}$ such that the Cauchy problem

$$ \dot{x} = f(x), \quad x(0) = x_0 \in \mathbb{R}^n $$

admits a solution $x(t, x_0)$ defined for all $t > 0$. Then

$$ x(\lambda^{1-\nu}t, \lambda x_0) = \lambda x(t, x_0), \quad \lambda > 0, $$

where $x(\cdot, \lambda x_0)$ is a solution to the same problem with the scaled initial condition $x(0) = \lambda x_0$.

Homogeneity simplifies qualitative analysis and design of control systems, since homogeneous dynamical systems behave similarly (in a certain sense) to linear ones. For example, $x = 0$ is always an equilibrium of continuous homogeneous system. If there exists another equilibrium $x^* \neq 0$ then the set $M = \{x \in \mathbb{R}^n : x = \lambda x^*, \lambda > 0\}$ is invariant and $f(x) = 0$ for all $x \in M$. From the latter theorem we immediately conclude that local property (e.g. stability of the origin) always implies the global one. Robustness of homogeneous control systems can be granted by its stability in the disturbance-free case [69], [4], [8]. In other words, homogeneous systems have similar properties to linear ODE models (see Table 2.1 and, for example, [79], [67], [34], [69], [9], [46], [55] for proofs of the mentioned properties of homogeneous systems).

## Homogeneity and Convergence Rate

Homogeneity degree allows an additional qualitative analysis to be provided easily. For example, finite-time or fixed-time stability (see e.g. [68], [10], [48], [4], [54], [63] for more details) can be derived from homogeneity degree. Indeed, if $\nu = 1/3$ then the solution of the scalar homogeneous system $\dot{x} = -x^\nu$ is given by

$$ x(t, x_0) = \begin{cases} \left(|x_0|^{\frac{2}{3}} - \frac{2}{3}t\right)^{\frac{3}{2}} & \text{if } t \in \left[0, \frac{3}{2}|y_0|^{\frac{2}{3}}\right), \\ 0 & \text{if } t \ge \frac{3}{2}|x_0|^{\frac{2}{3}}. \end{cases} $$

Obviously, it converges to zero in a finite-time. However, if $\nu > 1$ then any trajectory converges to a unit ball in a fixed time independently of the initial condition. Indeed, the formula for its solution (see the beginning of this section) implies that $|x(t, x_0)| \le 1$ for $t > \frac{1}{\nu-1}$ independently of $x_0 \in \mathbb{R}$. In fact, these properties can be established in a more general case. Therefore, homogeneity degree specifies the convergence rate of solutions to the origin. Namely, if the origin of homogeneous ODE is asymptotically stable then each trajectory vanishes in a finite-time time provided that $\nu < 1$ but if $\nu > 1$ then all trajectories convergence in a fixed-time to a neighborhood of the origin [46].
<table><caption>Table 2.1: Some properties of linear and homogeneous systems</caption><thead><tr><th></th><th>Linear System</th><th>Homogeneous System</th></tr></thead><tbody><tr><td></td><td>dot = Ax<br>A &isin; R<sup>n&times;n</sup></td><td>dot = f(x)<br>f(&lambda;x) = &lambda;'<sup>v</sup>f(x)</td></tr><tr><td>Trajectory Scaling</td><td>x(t, &lambda;x<sub>0</sub>) = &lambda;x(t, x<sub>0</sub>)</td><td>x(t, &lambda;x<sub>0</sub>) = &lambda;x(&lambda;<sup>v-1</sup>t, x<sub>0</sub>)</td></tr><tr><td>Local &harr; Global<br>(existence of solutions,<br>stability, etc)</td><td>&#10003;</td><td>&#10003;</td></tr><tr><td>Invariance &harr; Stability</td><td>&#10003;</td><td>&#10003;</td></tr><tr><td>Stability &rarr; Robustness<br>(Input-to-State Stability)</td><td>dot = Ax + Dw<br>w &isin; L<sup>&infin;</sup></td><td>dot = f(x, w)<br>w &isin; L<sup>&infin;</sup></td></tr><tr><td>Convergence Rate</td><td>Exponential</td><td>Finite (&nu; &lt; 1)<br>Exponential (&nu; = 1)<br>Fixed-time (&nu; &gt; 1)</td></tr><tr><td>Lyapunov Function</td><td>Quadratic<br>V = x<sup>&top;</sup>Px,<br>P &harr; 0</td><td>Homogeneous ("Quadratic-like")<br>V = x<sup>&top;</sup>&Xi;<sup>&top;</sup>(x)P&Xi;(x)x,<br>P &harr; 0, &Xi;(&lambda;x) = &Xi;(x) &isin; R<sup>n&times;n</sup>,<br>&part;&Xi;/&part;x<sub>i</sub>x = 0, det(&Xi;(x)) &ne; 0</td></tr></tbody></table>

## Locally Homogeneous Systems

An approximation by means of homogeneous functions (*homogeneous approximation* [29], [4]) can be utilized for local analysis of dynamical systems if, for example, linearization is non-informative (for stability analysis) or simply impossible. For example, the system $\dot{x} = -x(\sin^2(x) + x^2)$ is globally uniformly asymptotically stable. The corresponding homogeneous approximation at the origin $\dot{x} = -2x^3$ is also globally asymptotically stable, while its linearization $\dot{x} = 0$ is not.

Let us consider mechanical system consisting of a rigid body moving laterally on a contact surface and in a viscous environment (fluid). A simplest real-life example of such mechanical system is a car moving on a flat road. Let $z$ be the position of the center of mass of the body in an inertial frame. The equation describing motion of this system has the form

$$ \dot{z}(t) = v(t), \quad m\dot{v}(t) = F(t), \quad t > 0, \quad z(t) \in \mathbb{R}, $$

where $v(t)$ is the velocity, $m$ is the mass of the body, but $F$ is the sum of external forces.

We study only the deceleration phase for this mechanical system assuming that at the initial instant of time it has some non-zero velocity $\dot{z}(0) = v(0) \neq 0$. Dissipation of the energy may be caused by several external forces. To discover some fast (non-asymptotic) transition it is sufficient to consider only two of them:

* *drag force* (fluid resistance) is proportional to the square of the velocity [22]

$$ F_{drag}(t) = -k_{drag} v^2(t) \text{sign}(v(t)), $$
where $k_{drag} > 0$ is the coefficient of fluid (air) resistance and the sign function is given by

$$ \text{sign}(\rho) = \begin{cases} 1 & \text{if } \rho > 0, \\ 0 & \text{if } \rho = 0, \\ -1 & \text{if } \rho < 0; \end{cases} $$

* dry friction force is almost independent of a velocity magnitude and given by the model [5]

$$ F_{dry}(t) = -k_{dry} \text{sign}(v(t)), $$

where $k_{dry} > 0$ is the coefficient of dry friction.

The friction models also may contain some linear terms (proportional to velocity). We skip them for simplicity of analysis, since they will not effect the final conclusions about local homogeneity degree and convergence rate.

The sum of external forces $F(t)$ can be represented as follows

$$ F(t) = F_{drag}(t) + F_{dry}(t) = -(k_{dry} + k_{drag} v^2(t)) \text{sign}(v(t)). $$

and the differential equation describing evolution of the velocity of the body has the form:

$$ m\dot{v}(t) = -(k_{dry} + k_{drag} v^2(t)) \text{sign}(v(t)). $$

It is easy to see that $v = 0$ is the equilibrium of the last equation, which is globally stable, $v(t) \to 0$ as $t \to +\infty$. The solution can be found explicitly :

$$ v(t) = \tan \left( \arctan(|v(0)|) - \frac{\sqrt{k_{dry} k_{drag}}}{m} t \right) \text{sign}(v(0)). $$

This immediately implies $v(t) = 0$ for $t \ge \frac{m \arctan(|v(0)|)}{\sqrt{k_{dry} k_{drag}}}$. Since arctan is bounded, we conclude that *independently of initial velocity the motion terminates no later than the instant of time*

$$ T_{\max} = \frac{m\pi}{2\sqrt{k_{dry} k_{drag}}}. $$

The model of friction force $F$ is not homogeneous function of the velocity $v$, but locally (close to the origin or close to infinity) it is. Namely,

$$ F \approx -k_{dry} \text{sign}(v) \text{ as } v \to 0 \quad \text{and} \quad F \approx -k_{drag} v^2 \text{sign}(v) \text{ as } v \to \infty. $$

Therefore, the approximation at zero is a homogeneous function with degree $\nu = 0$, but the approximation at infinity is a homogeneous function with degree $\nu = 2$. The first one allows us to conclude finite-time convergence of the velocity to zero, while the second one guarantees fixed-time convergence to the neighborhood of zero. Such a combination implies fixed-time deceleration of the mechanical system independently of the initial velocity. This property is known as *fixed-time stability* (or *fixed-time convergence*) [54], [63].
## Weighted Dilation

Standard (uniform) homogeneity considered above has been introduced by means of uniform dilation $x \to \lambda x$, $\lambda > 0$. It is clear that changing of this dilation to some non-uniform one defines another type of homogeneity. The weighted dilation (introduced by V. Zubov in 1958, [79]) of the vector $x = (x_1, x_2, ..., x_n)^\top \in \mathbb{R}^n$ is the simplest case of the so-called generalized (non-uniform) dilation:

$$ (x_1, x_2, ..., x_n) \to (\lambda^{r_1} x_1, \lambda^{r_2} x_2, ..., \lambda^{r_n} x_n), $$

where $\lambda > 0$ as before is the scaling factor, but the positive numbers $r_1, r_2, ..., r_n$ are the weights specifying the dilation rates along different coordinates. If $r_1 = r_2 = ... = r_n$ then the weighted dilation becomes uniform. The introduced transformation of coordinates is a linear map $\mathbb{R}^n \to \mathbb{R}^n$ defined by the dilation matrix:

$$ x \to \Lambda x, \quad \text{where} \quad \Lambda = \begin{pmatrix} \lambda^{r_1} & 0 & \dots & 0 \\ 0 & \lambda^{r_2} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \lambda^{r_n} \end{pmatrix}. $$

The symmetry (homogeneity) of a function with respect to the weighted dilation can be identified analogously to the uniform case.

**Definition.** A function $f : \mathbb{R}^n \to \mathbb{R}$ is homogeneous with respect to the weighted dilation $\Lambda$ if

$$ f(\Lambda x) = \lambda^{\nu} f(x), \quad x \in \mathbb{R}^n, \quad \lambda > 0, $$

where $\nu \in \mathbb{R}$ is the homogeneity degree.

The weighted dilation extends the class of homogeneous functions under consideration. For example, the polynomial function

$$ x = (x_1, x_2)^\top \xrightarrow{f} x_1^2 + x_1 x_2^2 + x_2^4 $$

is homogeneous with respect to the weighted dilation $(x_1, x_2) \to (\lambda^2 x_1, \lambda x_2)$, but it is not homogeneous with respect to the uniform one $(x_1, x_2) \to (\lambda x_1, \lambda x_2)$.

Weighted homogeneous functions demonstrate similar properties to the standard homogeneous case. In particular, symmetry of solutions of homogeneous differential equations can be established easily (see, e.g. [79], [29]). Weighted homogeneous models frequently appear in control theory and applications. For example, the most of high order sliding mode control algorithms are weighted homogeneous [41], [48]. The classical minimum time control problem

$$ T \to \min_u $$
subject to

$$ \begin{cases} \dot{x}_1 = x_2, & u \in L^\infty((0, T), \mathbb{R}) \text{ such that } |u(t)| \le 1 \\ \dot{x}_2 = u, & x_1(T) = x_2(T) = 0, \end{cases} $$

has solution (see, e.g. [13]) in the form of the weighted homogeneous feedback

$$ u = -\operatorname{sign}(|x_2|x_2 + 2x_1). $$

Indeed, $u(\lambda^2 x_1, \lambda x_2) = u(x_1, x_2)$, i.e. $u$ is the weighted homogeneous function with the zero degree.

Below, for simplicity, the scaling factor $\lambda > 0$ is denoted by $e^s$ with $s \in \mathbb{R}$, where $e$ is the Euler number.

## Linear Dilation

From the previous section we conclude that once a homogeneity of a function with respect to a group of transformations (dilations) is established, many properties useful for analysis and design of dynamical control system modeled by the corresponding ODE can be discovered. In fact, the same conclusion can be done for infinite-dimensional dynamical models [60].

The different types of dilations have been studied in systems and control theory. The brief overview of different homogeneity concepts is given below

* Uniform dilation - L. Euler 17th century

$$ x \to e^s x, \quad s \in \mathbb{R}; $$

standard homogeneous ODEs (V. Zubov 1957 [78], W. Hahn [26]);

* Weighted dilation - V. Zubov 1958, [79]

$$ (x_1, x_2, \dots, x_n) \to (e^{r_1 s} x_1, e^{r_2 s} x_2, \dots, e^{r_n s} x_n), \quad s \in \mathbb{R}; $$

weighted homogeneous ODEs (V. Zubov 1958, [79]); weighted homogeneous PDEs (G. Folland 1975, [23]);

* Geometric dilation induced by a flow of a vector field - V. Khomenuk 1961, [36]; M. Kawski 1991 [33]; L. Rosier 1993, [67];

In this chapter we deal with the group of **linear dilations** in a Banach space $\mathbb{B}$:

$$ x \to \mathbf{d}(s)x, \quad x \in \mathbb{B}, \quad s \in \mathbb{R}, $$

where $\mathbf{d}$ is a strongly continuous group (i.e. $\mathbf{d}(\cdot)x : \mathbb{R} \to \mathbb{B}$ is continuous for any $x \in \mathbb{B}$) of **linear bounded operators** $\mathbb{B} \to \mathbb{B}$ and $s \in \mathbb{R}$ is the group parameter. For
$\mathbb{B} = \mathbb{R}^n$, the linear dilation includes uniform and weighted dilations as particular cases, but it is less general than the geometric dilation, which allows the group $d$ to be nonlinear. In other words, the geometric homogeneity (geometric dilation) can be split in two sub-classes: linear and non-linear. In this chapter we restrict ourself to **linear geometric homogeneity** only. The detailed study of linear dilations is presented in the next section.

Obviously that all linear operators are homogeneous with respect to the uni-
form dilation $d(s) = e^s I$, where $I$ is an identity operator in $\mathcal{L}(\mathbb{B}, \mathbb{B})$. Moreover,
the most of nonlinear models of mathematical physics are generalized homoge-
neous under a proper selection of the dilation group $d(s)$, e.g. Navier stocks, KdV
equation, Fast diffusion equation, Burgers equation, Saint-Venant equation are
homogeneous in the generalized sense [60].

The homogeneity allows many important results to be obtained, for example,
in the context of controllability, stabilizability and observability of nonlinear sys-
tems [32], [34], [70], [71], [25], controllers and observers design [3], [17], [52], [4], [61],
[44], [55], [58]. It simplifies robustness analysis for both delay-free [69], [4], [8] and
time-delay systems [2], [76]. The most of high order sliding mode algorithms are
homogeneous [41], [48], [7]. Below we introduce the linear geometric homogeneity
of vector fields in $\mathbb{R}^n$ and (unbounded) operators in the Banach space $\mathbb{B}$; discov-
ers some important properties of homogeneous systems and study some control/
estimation problems in order to demonstrate how homogeneity can be utilized for
analysis and design of both finite and infinite dimensional control systems.
## 2.2 Homogeneous Systems

### 2.2.1 Linear Dilations

Let $\mathcal{L}$ be a space of linear bounded operators $\mathbb{B} \to \mathbb{B}$, where $\mathbb{B}$ is a real Banach space with a norm $\| \cdot \|$.

**Definition 2.1.** A map $d: \mathbb{R} \to \mathcal{L}$ is said to be **dilation** in $\mathbb{B}$ if

* (Group property) $d(0) = I \in \mathcal{L}$ and $d(t+s) = d(t)d(s) = d(s)d(t)$ for $t, s \in \mathbb{R}$;

* (Limit property) $\lim_{s \to -\infty} \|d(s)z\| = 0$ and $\lim_{s \to +\infty} \|d(s)z\| = \infty$ uniformly on $z \in S$, where $S = \{z \in \mathbb{B} : \|z\| = 1\}$ is the unit sphere in $\mathbb{B}$.

Obviously, the dilation $d$ is a group of linear bounded and invertible operators

$$d(-s) = (d(s))^{-1}.$$

The limit property given above is just a version of the so-called Teresaka's condition (see, for example, [33] for more details) that is required for a group $d$ to be a dilation in an abstract space.

**Definition 2.2.** A dilation $d$ is

* strongly continuous if $d(\cdot)z: \mathbb{R} \to \mathbb{B}$ is continuous for any $z \in \mathbb{B}$;

* uniformly continuous if $d(\cdot): \mathbb{R} \to \mathcal{L}$ is continuous in the operator norm

$$\|d(\cdot)\| := \sup_{z \neq 0} \frac{\|d(\cdot)z\|}{\|z\|}.$$

The strong continuity plays an important role for dilations in infinite dimensional spaces, where the uniform continuity is sometimes a too restrictive property. In a finite dimensional space (e.g. $\mathbb{B} = \mathbb{R}^n$) any strongly continuous dilation is always uniformly continuous.

**Example 2.1.**

* Let $\mathbb{B}$ be the space of bounded uniformly continuous functions $\mathbb{R} \to \mathbb{R}$ with the supremum norm. The dilation $d$ in $\mathbb{B}$ can be defined as

$$(d(s)z)(x) = e^{\alpha s} z(x + \beta s),$$
where $s \in \mathbb{R}$ is the group parameter, $z \in \mathbb{B}$, $x \in \mathbb{R}$ and $\alpha > 0$ and $\beta \in \mathbb{R}$ are constant parameters.

* Let $\mathbb{B} = L^2(\mathbb{R}, \mathbb{R})$ and $\mathbf{d}: \mathbb{R} \to \mathcal{L}$ be defined as

$$ (\mathbf{d}(s)z)(x) = e^{\alpha s}z(e^{\beta s}x), $$

where $s \in \mathbb{R}$ is the group parameter, $z \in L^2(\mathbb{R}, \mathbb{R})$, $x \in \mathbb{R}$, $\alpha, \beta \in \mathbb{R}$ are constant parameters. If

$$ \beta < 2\alpha $$

then $\mathbf{d}$ is a strongly continuous dilation in $L^2(\mathbb{R}, \mathbb{R})$.

* Let $\mathbb{B} = \mathbb{R}^n$ and $Q$ is a Hurwitz matrix then $\mathbf{d}(s) = e^{-sQ}$ is a dilation in $\mathbb{R}^n$.

Let us recall the following notations

$$ \|\mathbf{d}(s)\| = \sup_{u \in S} \|\mathbf{d}(s)u\| = \sup_{u \neq 0} \frac{\|\mathbf{d}(s)u\|}{\|u\|} \quad \text{and} \quad \|\mathbf{d}(s)\| = \inf_{u \in S} \|\mathbf{d}(s)u\| = \inf_{u \neq 0} \frac{\|\mathbf{d}(s)u\|}{\|u\|}. $$

These quantities characterize a rate of dilation in $\mathbb{B}$.

**Proposition 2.1.** [60] If $\mathbf{d}$ is a strongly continuous dilation in $\mathbb{B}$ then

1) $0 < \|\mathbf{d}(s)\| \le \|\mathbf{d}(s)\| < +\infty$, $s \in \mathbb{R}$;

2) $\|\mathbf{d}(s)\| \to 0$ as $s \to -\infty$ and $\|\mathbf{d}(s)\| \to +\infty$ as $s \to +\infty$;

3) $\|\mathbf{d}(s)\| < 1$ for $s < 0$ and $\|\mathbf{d}(s)\| > 1$ for $s > 0$;

4) $\mathbf{d}(s) \neq I \in \mathcal{L}$ for $s \neq 0$.

If $\mathbf{d}$ is uniformly continuous then $\|\mathbf{d}(\cdot)\|: \mathbb{R} \to \mathbb{R}$ and $\|\mathbf{d}(\cdot)\|: \mathbb{R} \to \mathbb{R}$ are continuous.

It is well known that any nontrivial element from $\mathbb{B}$ can be projected to the unit sphere using the uniform dilation. The corresponding projector $P: \mathbb{B}\setminus\{0\} \to S$ is defined as follows

$$ P(u) = \frac{u}{\|u\|} = \mathbf{d}(-\ln \|u\|)u, $$

where $\mathbf{d}(s) = e^s I$ is the uniform dilation, $I$ is the identity operator on $\mathbb{B}$. In fact, the similar projection can be constructed using the linear dilation $\mathbf{d}$ given above.

**Proposition 2.2.** [60] If $\mathbf{d}$ is a strongly continuous dilation in $\mathbb{B}$ then for any $z \in \mathbb{B}\setminus\{0\}$ there exist $s_0 \in \mathbb{R}$ and $z_0 \in S$ such that

$$ z_0 = \mathbf{d}(s_0)z. $$
The point $z_0$ defined in the latter proposition is called homogeneous projection of $z$ to the unit sphere $S$. The homogeneous projection may be non-unique even in the Euclidean space $\mathbb{R}^2$.

**Example 2.2.** Let us consider the weighted dilation

$$d(s) = \begin{pmatrix} e^{\alpha s} & 0 \\ 0 & e^s \end{pmatrix}, \quad s \in \mathbb{R}, \quad \alpha > 1 + \frac{2}{\sqrt{5}-1},$$

in the Euclidean space $\mathbb{B} = \mathbb{R}^2$ with the norm

$$\|z\| = \sqrt{z^\top P z}, \quad z \in \mathbb{R}^2, \quad P = P^\top \in \mathbb{R}^{n \times n}, \quad P > 0,$$

where

$$P = P_0 + \varepsilon \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} > 0, \quad P_0 = \begin{pmatrix} \frac{3-\sqrt{5}}{2} & -1 \\ -1 & \frac{2}{3-\sqrt{5}} \end{pmatrix}, \quad \varepsilon > 0.$$

Let us show that for sufficiently small $\varepsilon > 0$, the function $\xi : \mathbb{R} \to \mathbb{R}$,

$$\xi(s) = \left\| d(s) \begin{pmatrix} 1 \\ 1 \end{pmatrix} \right\|^2 - 1, \quad s \in \mathbb{R}$$

has at least two zeros: $\xi(s_1) = \xi(s_2) = 0$ and $s_1 \neq s_2$. Let us denote

$$\xi_0(s) = \begin{pmatrix} 1 \\ 1 \end{pmatrix}^\top d^\top(s) P_0 d(s) \begin{pmatrix} 1 \\ 1 \end{pmatrix} - 1.$$

Notice that

$$\xi_0(0) = \begin{pmatrix} 1 \\ 1 \end{pmatrix}^\top P_0 \begin{pmatrix} 1 \\ 1 \end{pmatrix} - 1 = \frac{3-\sqrt{5}}{2} + \frac{2}{3-\sqrt{5}} - 3 = 0.$$

The function $\xi_0 : \mathbb{R} \to \mathbb{R}$ is continuous differentiable on $\mathbb{R}$ and decays in the neighborhood of the point $s = 0$, since

$$\xi'_{0}(0) = \begin{pmatrix} 1 \\ 1 \end{pmatrix}^{\top} \left( \begin{pmatrix} \alpha & 0 \\ 0 & 1 \end{pmatrix}^{\top} P_{0} + P_{0} \begin{pmatrix} \alpha & 0 \\ 0 & 1 \end{pmatrix} \right) \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}^{\top} \left( \begin{pmatrix} \alpha(3-\sqrt{5}) & -1-\alpha \\ -1-\alpha & \frac{4}{3-\sqrt{5}} \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \right) < 0.$$

This means that $\xi_0(s)$ is negative for small $s > 0$ and there exists $\delta > 0$ such that

$$\xi_0(s) < 0 \quad \text{for all} \quad 0 < s < \delta.$$

Since $\xi_0(s) \to +\infty$ as $s \to +\infty$ and $\xi_0(0) = 0$ then for sufficiently small $\varepsilon > 0$ the function $\xi$ given by

$$\xi(s) = \xi_0(s) + \varepsilon e^{2s}$$

has at least two zeros. Therefore, the point $z = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ admits at least two different homogeneous projections $d(s_1)z$ and $d(s_2)z$, $s_1 \neq s_2$ to the unit sphere $S$.

Uniqueness of homogeneous projection may depend on the unit sphere $S$, i.e. on the norm in $\mathbb{B}$. Notice that in the case of uniform dilation $d(s) = e^s I$, the homogeneous projection is always unique due to homogeneity of the norm: $\|\gamma u\| = |\gamma||u||$, $\gamma \in \mathbb{R}$, $u \in \mathbb{B}$. The problem of uniqueness of homogeneous projection in a more general case is studied below, where the so-called monotone dilation is introduced.
## Geometric Structures Induced By Dilation

Linear dilation allows a sort of norm-topology to be introduced in $\mathbb{B}$ by means of the so-called canonical homogeneous norm studied in the next section. Here we just consider some related geometric structures (balls, spheres and cones).

**Definition 2.3.** The set

$$S_d(r, z_0) = \{z \in \mathbb{B} : \|d(-\ln(r))(z - z_0)\| = 1\},$$

is called **d-homogeneous sphere** of the radius $r > 0$ with the center at $z_0 \in \mathbb{B}$ and the set

$$B_d(r, z_0) = \{z \in \mathbb{B} : \|d(-\ln(r))(z - z_0)\| < 1\},$$

is called **open d-homogeneous ball** of the radius $r > 0$ with the center at $z_0 \in \mathbb{B}$. If $z_0 = 0$ we use the notations $S_d(r)$ and $B_d(r)$.

Obviously, the unit homogeneous sphere $S_d(1)$ (resp. ball $B_d(1)$) coincides with the unit sphere $S$ (resp. ball $B$) in $\mathbb{B}$, so

$$S_d(r) = d(\ln r)S \quad \text{and} \quad B_d(r) = d(\ln r)B.$$

Note also that $0 \in B_d(r)$ for any $r > 0$.

**Corollary 2.1.** The following set-theoretic relations hold

$$1) B_d(r) \setminus \{0\} \subseteq \bigcup_{0<\delta<r} S_d(\delta);$$

$$2) \mathbb{B} \setminus \{0\} = \bigcup_{r>0} S_d(r).$$

Recall that the homogeneous projection is not unique in the general case. This means that the homogeneous sphere $S_d(r)$ may have intersections with the unit sphere for $r \neq 1$.

**Definition 2.4.** A nonempty set $\mathcal{D} \subseteq \mathbb{B}$ is said to be **d-homogeneous cone** if $\mathcal{D}$ is invariant with respect to dilation $d: \mathbb{R} \to \mathcal{L}(\mathbb{B}, \mathbb{B})$, i.e.

$$d(s)z \in \mathcal{D} \text{ for } z \in \mathcal{D} \text{ and } s \in \mathbb{R}.$$

Any point $z \in \mathcal{D}$ belongs to the homogeneous cone $\mathcal{D}$ together with the
continuous *curve* (homogeneous curve)

$$
\Gamma_d(z) = \{y \in \mathbb{B} : y = \mathbf{d}(s)z, s \in \mathbb{R}\}.
$$

In particular, if **d** is the uniform dilation (i.e. **d**(s) = e^s I, s ∈ R), then D becomes
the conventional positive cone in B.

**Example 2.3.** Let $\mathbb{B} = C([0, p], \mathbb{R})$ and the dilation in $\mathbb{B}$ be defined as $(\mathbf{d}(s)z)(x) = e^{s-0.5sx/p}z(x)$, where $s \in \mathbb{R}$, $x \in [0, p]$. The set

$$
\mathcal{D} = \{z \in C([0, p], \mathbb{R}) : z(0) = z^2(p)\}
$$

is **d**-homogeneous cone in C([0, p], R). Indeed, if z ∈ D then z(0) = z²(p) and

$$
(\mathbf{d}(s)z)(0) = e^s z(0) = e^s z^2(p) = (e^{0.5s} z(p))^2 = ((\mathbf{d}(s)z)(p))^2.
$$

Generators of Dilations

It is well known (see e.g. [50]) that any strongly continuous group of linear bounded
operators on $\mathbb{B}$ can be identified by means of an infinitesimal *generator*.

**Definition 2.5.** An operator $G_d : \mathcal{D}(G_d) \subset \mathbb{B} \to \mathbb{B}$ defined as

$$
G_d z = \lim_{s \to 0^+} \frac{d(s)z - z}{s}
$$

is called an infinitesimal generator of a strongly continuous dilation **d**. The
domain of the generator G<sub>d</sub> is given by

$$
\mathcal{D}(G_d) = \left\{ z \in \mathbb{B} : \lim_{s \to 0^+} \frac{\mathbf{d}(s)z - z}{s} \text{ exists} \right\}.
$$

The generator is a linear possibly unbounded operator [50]. The next result
follows, basically, from the fact that **d**(s) is a strongly continuous group of linear
bounded operators.

**Proposition 2.3.** Let **d** be a dilation in $\mathbb{B}$ then

1) there exists $M \ge 1$, $\omega^+ \in \mathbb{R}$ and $\omega^- \in \mathbb{R}$ such that

$$
\| \mathbf{d}(s) \|_{\mathcal{L}} \le M \begin{cases} e^{\omega^+ s} & \mathbf{if} \quad s \ge 0, \\ e^{\omega^- s} & \mathbf{if} \quad s \le 0; \end{cases} \qquad (2.1)
$$
2) if the operator $G_d^-$ is defined as follows

$$G_d^- u = \lim_{s \to 0^-} \frac{\mathbf{d}(s)u - u}{s}, \quad u \in \mathbb{B}$$

with the domain

$$\mathcal{D}(G_d^-) = \{ u \in \mathbb{B} : \lim_{s \to 0^-} \frac{\mathbf{d}(s)u - u}{s} \text{ exists} \}$$

then $G_d^- = G_d$, i.e. $G_d^-z = G_d z$ for any $z \in \mathcal{D}(G_d^-) = \mathcal{D}(G_d)$;

3) $\mathcal{D}(G_d)$ is a $\mathbf{d}$-homogeneous cone and it is a linear subspace dense in $\mathbb{B}$;

4) if $z \in B$ then

$$\lim_{h \to 0} \int_t^{t+h} \mathbf{d}(s)z \, ds = \mathbf{d}(t)z, \quad \int_0^t \mathbf{d}(s)z \, ds \in \mathcal{D}(G_d)$$

and

$$G_d \left( \int_0^t \mathbf{d}(s) z \, ds \right) = \mathbf{d}(t) z - z;$$

5) if $z \in \mathcal{D}(G_d)$ then

$$\frac{d}{ds} \mathbf{d}(s)z = G_d \mathbf{d}(s)z = \mathbf{d}(s)G_d z, \quad s \in \mathbb{R};$$

6) if the operator $G_d$ is bounded then the dilation $\mathbf{d}$ is uniformly continuous and

$$\frac{d}{ds} \mathbf{d}(s) = G_d \mathbf{d}(s) = \mathbf{d}(s) G_d, \quad s \in \mathbb{R};$$

$$\mathbf{d}(s) = e^{sG_d} := \sum_{n=0}^{+\infty} \frac{(sG_d)^n}{n!}, \quad s \in \mathbb{R};$$

$$\| \mathbf{d}(s) \|_{\mathcal{L}} \le e^{s \| G_d \|_{\mathcal{L}}}, \quad s \in \mathbb{R}.$$

In the above proposition, all integrals are understood in the sense of Brochner (see e.g. [20, page 187]). Most of the proofs of these properties can be found in [50, Theorem 2.4, page 4]. The others are straightforward consequences of the introduced definitions.

**Example 2.4.**
* Let $\mathbb{B}$ be the space of bounded uniformly continuous functions $\mathbb{R} \to \mathbb{R}$ with the supremum norm. In Example 2.1 the dilation in $\mathbb{B}$ is defined as $(\mathbf{d}(s)z)(x) = e^{\alpha s}z(x + \beta s)$, $s \in \mathbb{R}$, $z \in \mathbb{B}$, $x \in \mathbb{R}$ and $\alpha > 0$, $\beta \in \mathbb{R}$ are constant parameters. The generator of this dilation is defined as follows

$$ (G_d z)(x) = \alpha z(x) + \beta z'(x), \quad x \in \mathbb{R} $$

with the domain

$$ \mathcal{D}(G_d) = \{z \in \mathbb{B} : z' \in \mathbb{B}\}. $$

* In Example 2.1 the dilation in $\mathbb{B} = L^2(\mathbb{R}, \mathbb{R})$ is defined as $(\mathbf{d}(s)z)(x) = e^{\alpha s}z(e^{\beta s})$, $s \in \mathbb{R}$, $z \in \mathbb{B}$, $x \in \mathbb{R}$ and $\alpha \in \mathbb{R}$, $\beta < 2\alpha$ are some constant parameters. The generator of this dilation is defined as follows

$$ (G_d z)(x) = \alpha z(x) + \beta x z'(x), \quad x \in \mathbb{R} $$

with the domain

$$ \mathcal{D}(G_d) = \{z \in L^2(\mathbb{R}, \mathbb{R}) : x z' \in L^2(\mathbb{R}, \mathbb{R})\}. $$

where $z'$ denotes the weak derivative of $z \in L^2(\mathbb{R}, \mathbb{R})$ (see e.g. [20, page 623]).

* If $\mathbb{B} = \mathbb{R}^n$ then any continuous dilation in $\mathbb{R}^n$ is defined as follows

$$ \mathbf{d}(s) = e^{s G_d}, \quad G_d \in \mathbb{R}^{n \times n}. $$

Moreover, the limit property of the dilation implies that all eigenvalues of the matrix $G_d$ are placed in the right complex half-plane. Indeed, otherwise there exists an eigenvalue $\lambda_* : \mathbb{R}(\lambda_*) \le 0$ and there exists a vector $x_{\lambda_*}$ belonging to an eigensubspace of $\lambda_*$ such that $\|e^{sG_d}x_{\lambda_*}\|$ is bounded for any $s \ge 0$.

## Monotonicity of Dilations

The notion of monotonicity of dilation introduced below is very important for analysis and design of homogeneous control systems. In particular, it allows uniqueness of homogeneous projection and existence of the so-called canonical homogeneous norm (see below).

**Definition 2.6.** A dilation $\mathbf{d}$ is **monotone** in $\mathbb{B}$ if it is a strong contraction for any $s < 0$, i.e.

$$ \|\mathbf{d}(s)\| < 1, \quad \forall s < 0. \qquad (2.2) $$

Note that if $\mathbf{d}(s)$ is a strong contraction, then due to group property we conclude that $\mathbf{d}(-s) = \mathbf{d}(s)^{-1}$ is a strong expansion, i.e.

$$ |\mathbf{d}(s)| > 1 \quad \text{for any } s > 0. \qquad (2.3) $$
Indeed, if $z \in S$ then $1 = \|z\| = \|\mathbf{d}(-s)\mathbf{d}(s)z\| \le \|\mathbf{d}(-s)\| \cdot \|\mathbf{d}(s)z\|$ for any $s > 0$. Hence, $\|\mathbf{d}(s)z\| \ge \frac{1}{\|\mathbf{d}(-s)\|}$ for any fixed $s > 0$ and arbitrary $z \in S$. Due to strong contraction condition we derive

$$1 < \frac{1}{\|\mathbf{d}(-s)\|} \le \inf_{z \in S} \|\mathbf{d}(s)z\| = \lfloor \mathbf{d}(s) \rfloor$$

for any $s > 0$.

**Proposition 2.4.** [56] If $\mathbf{d}$ is a strongly continuous monotone dilation then

1) homogeneous projection is unique, i.e. for any $z \in \mathbb{B}$ there exists a unique pair $(s_0, z_0) \in \mathbb{R} \times S$ such that $z = \mathbf{d}(s_0)z_0$;

2) the function $\|\mathbf{d}(\cdot)z\| : \mathbb{R} \to \mathbb{R}_+$ (with $z \in \mathbb{B}\setminus\{0\}$) is continuous and strictly increasing on $\mathbb{R}$;

3) the function $\|\mathbf{d}(\cdot)\| : \mathbb{R} \to \mathbb{R}_+(\text{resp. } \lfloor \mathbf{d}(\cdot) \rfloor : \mathbb{R} \to \mathbb{R}_+)$ is lower(upper) semicontinuous and strictly increasing, moreover,

$$\lfloor \mathbf{d}(0^{-}) \rfloor \cdot \| \mathbf{d}(0^{+}) \| = \| \mathbf{d}(0^{-}) \| = \lfloor \mathbf{d}(0^{+}) \rfloor = 1.$$

According to the latter proposition, the monotonicity of the dilation implies the uniqueness of the homogeneous projection. This property is important for the later constructions, in particular, for the proof of an existence/uniqueness of solutions of non-linear homogeneous evolution systems, stability analysis and homogeneous control design.

To specify the rate of monotone dilation the next notion is utilized.

**Definition 2.7.** The dilation $\mathbf{d}$ is strictly monotone in $\mathbb{B}$ if $\exists \beta > 0$ such that

$$\|\mathbf{d}(s)\| \le e^{\beta s}, \quad \forall s \le 0. \qquad (2.4)$$

Let us note that if the set $\mathcal{D} \subset \mathbb{B}$ is dense in $\mathbb{B}$ and

$$\sup_{z \in \mathcal{D} \cap S} \|\mathbf{d}(s)z\| < 1 \quad (\text{resp. } \sup_{z \in \mathcal{D} \cap S} \|\mathbf{d}(s)z\| \le e^{\beta s} \text{ with } \beta > 0)$$

for $s < 0$ then $\mathbf{d}$ is monotone (resp. strictly monotone). Moreover, repeating the considerations given right after Definition 2.6 we derive

$$\lfloor \mathbf{d}(s) \rfloor \ge e^{\beta s}, \quad \forall s \ge 0.$$
**Proposition 2.5.** A strongly continuous dilation **d** in a real Hilbert space **H** is strictly monotone if and only if there exists **β** > 0 such that

$$ \langle G_d z, z \rangle \geq \beta \quad \text{for any} \quad z \in S \cap \mathcal{D}, \qquad (2.5) $$

where $G_d$ is the generator of **d** and $\mathcal{D} \subset \mathcal{D}(G_d)$ is an arbitrary **d**-homogeneous cone dense in $\mathcal{D}(G_d)$ and $S$ is the unit sphere in $\mathbb{B}$.

**Proof.** The sufficiency has been proven in [58].

*Necessity.* Let **d** be strictly monotone and $z \in S \cap \mathcal{D}(G_d)$. Using Cauchy-Swartz inequality for $s > 0$ we derive

$$ 1 = \langle z, z \rangle = \langle \mathbf{d}(-s)\mathbf{d}(s)z, z \rangle \le \|\mathbf{d}(-s)\| \cdot |\langle \mathbf{d}(s)z, z \rangle| \le e^{-\beta s} \|\mathbf{d}(s)z\|. $$

Hence we have

$$ e^{2\beta s} - 1 \le \langle \mathbf{d}(s)z, \mathbf{d}(s)z \rangle - \langle z, z \rangle = \langle \mathbf{d}(s)z - z, \mathbf{d}(s)z \rangle + \langle z, \mathbf{d}(s)z - z \rangle. $$

Dividing both sides of the last inequality by $s$ taking the limit $s \to 0^+$ we obtain

$$ 2\beta \le \langle G_d z, z \rangle + \langle z, G_d z \rangle = 2 \langle G_d z, z \rangle. $$

**Example 2.5.** Let $\mathbb{B} = L^2(\mathbb{R}, \mathbb{R})$ and $\mathbf{d}: \mathbb{R} \to \mathcal{L}$ be defined as

$$ (\mathbf{d}(s)z)(x) = e^{\alpha s} z(e^{\beta s} x), $$

where $s \in \mathbb{R}$ is the group parameter, $z \in L^2(\mathbb{R}, \mathbb{R})$, $x \in \mathbb{R}$, $\alpha, \beta \in \mathbb{R}$ are constant parameters. If

$$ \beta < 2\alpha $$

then this dilation is strongly continuous and strictly monotone since

$$ \| \mathbf{d}(s)z \| = e^{(\alpha - 0.5\beta)s} \|z\|. $$

It is worth stressing that monotonicity of dilation depends on the norm $\| \cdot \|$. The next corollary is the straightforward consequence of Proposition 2.5 in the case $\mathbb{B} = \mathbb{R}^n$.

**Corollary 2.2.** [55] Let **d** be a dilation in the Euclidean space $\mathbb{B} = \mathbb{R}^n$ with the scalar product

$$ \langle u, v \rangle = u^\top P v, \quad u, v \in \mathbb{R}^n, $$
where $0 < P = P^\top \in \mathbb{R}^{n \times n}$ is a positive definite symmetric matrix. The dilation $d$ is strictly monotone in $\mathbb{R}^n$ equipped with the norm $\|z\| = \sqrt{\langle z, z \rangle}$ if and only if the following linear matrix inequality holds

$$PG_d + G_d^\top P > 0, \qquad (2.6)$$

where $G_d \in \mathbb{R}^n$ is the generator of the dilation $d$. Moreover, one has

$$ \begin{aligned} e^{\alpha s} \le \|d(s)\| &\le \|d(s)\| \le e^{\beta s} & \text{if } s \le 0, \\ e^{\beta s} \le \|d(s)\| &\le \|d(s)\| \le e^{\alpha s} & \text{if } s \ge 0, \end{aligned} \qquad (2.7) $$

where

$$ \alpha = \frac{1}{2} \lambda_{\max} \left( P^{\frac{1}{2}} G_d P^{-\frac{1}{2}} + P^{-\frac{1}{2}} G_d^\top P^{\frac{1}{2}} \right) > 0 $$

and

$$ \beta = \frac{1}{2} \lambda_{\min} \left( P^{\frac{1}{2}} G_d P^{-\frac{1}{2}} + P^{-\frac{1}{2}} G_d^\top P^{\frac{1}{2}} \right) > 0. $$

Therefore, any continuous dilation $d(s) = e^{sG_d}$ in the Euclidean space $\mathbb{B} = \mathbb{R}^n$ is strictly monotone provided that the inner product in $\mathbb{R}^n$ and the weighted Euclidean norm, respectively, are selected according to the inequality (2.6).

Notice also that the parameter $\alpha$ given in (2.7) may not exist for infinite dimensional case, since the operator $G_d$ may be unbounded.

## Canonical Homogeneous Norm

The next definition introduces the so-called homogeneous norm in $\mathbb{B}$.

**Definition 2.8.** A continuous functional $p : \mathbb{B} \to \mathbb{R}_+$ is said to be $d$-homogeneous norm in $\mathbb{B}$ if

* $p(u) \to 0$ as $u \to 0$ and

* $p(d(s)u) = e^s p(u) > 0$ for $u \in \mathbb{B} \setminus \{0\}$, $s \in \mathbb{R}$.

The functional $p$ may not satisfy the triangle inequality $p(u+v) \le p(u)+p(v)$, so it is not even a semi-norm. However, many authors (see e.g. [32], [4], [21], [6]) call some functionals satisfying the above definition by the "homogeneous norm". We follow this tradition and consider the **canonical homogeneous norm** introduced by means of the following formula
$$ \|u\|_{d} = e^{s_{u}}, \text{ where } s_{u} \in \mathbb{R} \text{ is such that } \|\mathbf{d}(-s_{u})u\| = 1. \quad (2.8) $$

The next lemma proves that the canonical homogeneous norm is well defined for monotone dilations.

**Lemma 2.1.** [58]

* If the dilation $d$ is monotone on $\mathbb{B}$ then the functional $\|\cdot\|_d : \mathbb{B}\setminus\{0\} \to \mathbb{R}_+$ defined by (2.8) is single-valued and positive. It can be prolonged to zero by continuity since $\|u\|_d \to 0$ as $\|u\| \to 0$.

* If $d$ is strictly monotone on $\mathbb{B}$ then $\|\cdot\|_d$ is locally Lipschitz continuous on $\mathbb{B}\setminus\{0\}$.

In [56] such a homogeneous norm was called canonical since it is induced by the canonical norm $\|\cdot\|$ in $\mathbb{B}$. The canonical homogeneous norm has several obvious but rather important properties:

* $\|u\|_d = 1 \Leftrightarrow \|u\| = 1$;

* $\|\mathbf{d}(s)u\|_d = e^s \|u\|_d$;

* $\|u\|_d = r \Leftrightarrow u \in S_d(r)$, where $r \ge 0$

* for any $u \in \mathbb{B}$ one has

$$ \lfloor \mathbf{d}(\ln(\|u\|_d)) \rfloor \le \|u\| \le \|\mathbf{d}(\ln(\|u\|_d))\| \quad (2.9) $$

The canonical homogeneous norm $\|\cdot\|_d$ is not a norm in the usual sense, but it defines a sort of norm topology in $\mathbb{B}$ in terms of homogeneous spheres and homogeneous balls defined above.

**Lemma 2.2.** [58] Let $d$ be a strictly monotone dilation in a Hilbert space $\mathbb{H}$ then the homogeneous norm $\|\cdot\|_d$ is Fréchet differentiable on $\mathcal{D}(G_d)\setminus\{0\}$ and the Fréchet derivative of $\|\cdot\|_d$ at $u \in \mathcal{D}(G_d)\setminus\{0\}$ is given by

$$ (D\|u\|_d)(\cdot) = \frac{\langle \mathbf{d}(-\ln \|u\|_d) \cdot, \mathbf{d}(-\ln \|u\|_d)u \rangle}{\langle G_d \mathbf{d}(-\ln \|u\|_d)u, \mathbf{d}(-\ln \|u\|_d)u \rangle} \|u\|_d. \quad (2.10) $$

To compute a homogeneous norm in practice an appropriate numerical procedure is required [61], [55]. In some particular cases it can be calculated explicitly.
**Example 2.6.** Sometimes, for $\mathbb{B} = \mathbb{R}^2$ the canonical homogeneous norm can be calculated explicitly. Indeed, if

$$ \mathbf{d}(s) = e^{sG_d} \quad \text{with} \quad G_d = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} $$

and the norm of $u = (u_1, u_2)^\top \in \mathbb{R}^2$ is defined as

$$ \|u\| = \sqrt{u^\top P u}, \quad P = \begin{pmatrix} p_{11} & p_{12} \\ p_{12} & p_{22} \end{pmatrix} \in \mathbb{R}^{2 \times 2} : P \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix} P > 0 $$

then the equation (2.8) becomes

$$ u^\top \begin{pmatrix} e^{-2s_u} & 0 \\ 0 & e^{-s_u} \end{pmatrix} P \begin{pmatrix} e^{-2s_u} & 0 \\ 0 & e^{-s_u} \end{pmatrix} u = 1, $$

or, equivalently,

$$ V^4 = aV^2 + bV + c, $$

where $V = e^{s_u} = \|u\|_d$ and

$$ a = p_{22}u_2^2, \quad b = 2p_{12}u_1u_2, \quad c = p_{11}u_1^2. $$

Hence, using the Ferrari formulas we can solve the quartic equation to derive an explicit representation of the canonical homogeneous norm.

### 2.2.2 Homogeneous Mappings and Equations

#### Homogeneous operators

Operators to be studied in this chapter are introduced by the next definition, which extends the homogeneity concept proposed in [23].

**Definition 2.9 (Homogeneous operator).** An operator $f: \mathcal{D}(f) \subset \mathbb{B} \to \mathbb{B}$ (a functional $g: \mathcal{D}(g) \subset \mathbb{B} \to \mathbb{R}$) is said to be **d-homogeneous of degree $\nu$** if $\mathcal{D}(f)$ (resp. $\mathcal{D}(g)$) is a **d-homogeneous cone and**

$$ f(\mathbf{d}(s)u) = e^{\nu s} \mathbf{d}(s)f(u), \quad s \in \mathbb{R}, u \in \mathcal{D}(f), \qquad (2.11) $$

$$ (\text{resp. } g(\mathbf{d}(s)u) = e^{\nu s} g(u), \quad s \in \mathbb{R}, u \in \mathcal{D}(g)) \qquad (2.12) $$

where $\mathbf{d}$ is a dilation in $\mathbb{B}$ and $\nu \in \mathbb{R}$ is a homogeneity degree.

Any linear operator is always **d-homogeneous** of zero degree if $\mathbf{d}(s) = e^s I \in \mathcal{L}(\mathbb{B})$. However, in some cases the same linear operator may also have negative or positive homogeneity degree dependently of the dilation group as shown in the following example.
**Example 2.7.** [60] The operator $A = \frac{\partial^2}{\partial x^2} : H^2(\mathbb{R}, \mathbb{R}) \subset H^0(\mathbb{R}, \mathbb{R}) \to H^0(\mathbb{R}, \mathbb{R})$ is $\mathbf{d}$-homogeneous of degree $\nu \in \mathbb{R}$ provided that the dilation $\mathbf{d}$ is defined as follows

$$ (\mathbf{d}(s)u)(x) = e^{\alpha s} u(e^{0.5\nu s} x), \quad s \in \mathbb{R}, \quad u \in H^0(\mathbb{R}, \mathbb{R}), \quad x \in \mathbb{R} $$

with $\nu < 4\alpha$. Indeed,

$$ (Ad(s)u)(x) = \frac{\partial^2}{\partial x}(\mathbf{d}(s)u)(x) = \frac{\partial^2}{\partial x}e^{\alpha s}u(e^{0.5\nu s}x) = e^{\nu s}(\mathbf{d}(s)Au)(x). $$

**Example 2.8.** [55] Let $\mathbb{B} = \mathbb{R}^n$ and $f(x) = Ax$, where $x \in \mathbb{R}^n$ and $A \in \mathbb{R}^{n \times n}$. Then the vector field $f$ is $\mathbf{d}$-homogeneous of the degree $\nu \in \mathbb{R}$ if and only if

$$ AG_d - G_d A = \nu A, $$

where $G_d \in \mathbb{R}^{n \times n}$ is the generator of the dilation. Indeed, to fulfill the homogeneity condition $Ad(s) = e^{\nu s} \mathbf{d}(s)A$ with $s \in \mathbb{R}$, $\mathbf{d}(s) = e^{sG_d}$, the mentioned identity is, obviously, necessary and sufficient.

**Example 2.9.** [60] The Saint-Venant equation is an example of a system of conservation laws studied in [14]. In the field of hydraulics, it represents the flow in open-channels by the following model

$$ \begin{aligned} \frac{\partial H}{\partial t} &= -\frac{\partial}{\partial x}(HV), \\ \frac{\partial V}{\partial t} &= -\frac{\partial}{\partial x}\left(\frac{1}{2}V^2 + gH\right), \end{aligned} \qquad (2.13) $$

where $H$ and $V$ are scalar functions of time and space variables. The quantity $H(t, x)$ is the water level at the instant of time $t \in \mathbb{R}_+$ in the point $x \in \mathbb{R}$, and $V(t, x)$ is the water velocity in the same position. The parameter $g$ denotes the gravitation constant. Let us consider the case when the water channel is supported by two overflow spillways, which adjust an input and output flows in a pool (between spillways). The space argument is restricted on the segment $[0, 1]$, where $x=0$ and $x=1$ are positions of spillways, and the equation (2.13) is supported with the boundary conditions [14]:

$$ \begin{aligned} H(t, 0)V(t, 0) - (Z_0 - L_0)^{3/2} &= 0, \\ H(t, 1)V(t, 1) - (H(t, 1) - L_1)^{3/2} &= 0, \end{aligned} $$

where $Z_0$ is the water level above the pool and $L_0$, $L_1$ are spillways heights.

Let us show that for $L_0 = Z_0$ and $L_1 = 0$ the corresponding evolution equation is homogeneous. Let us consider the operator $f: \mathcal{D}(f) \subset \mathbb{B} \to \mathbb{B} := C([0, 1], \mathbb{R}) \times C([0, 1], \mathbb{R})$ defined on the domain

$$ \mathcal{D}(f) = \left\{ (u_1, u_2) \in C^1([0, 1], \mathbb{R}_+) \times C^1([0, 1], \mathbb{R}) : \begin{array}{l} u_1(0)u_2(0) = 0; \\ u_1(1)u_2(1) = u_1^{3/2}(1) \end{array} \right\} $$
as follows

$$f(u) = \begin{pmatrix} -\frac{\partial}{\partial x}(u_1 u_2) \\ -\frac{\partial}{\partial x}(g u_1 + \frac{1}{2} u_2^2) \end{pmatrix},$$

where $u = (u_1, u_2) \in \mathcal{D}(f)$. The operator $f$ is **d-homogeneous** of degree $\nu = 1$ with respect to the weighted dilation $\mathbf{d}(s)u = (e^{2s}u_1, e^s u_2)$, where $u = (u_1, u_2) \in C([0, 1], \mathbb{R}) \times C([0, 1], \mathbb{R})$ and $s \in \mathbb{R}$. Indeed,

$$f(\mathbf{d}(s)u) = \begin{pmatrix} -\frac{\partial}{\partial x}(e^{2s}u_1 e^s u_2) \\ -\frac{\partial}{\partial x}(g e^{2s}u_1 + \frac{1}{2}[e^s u_2]^2) \end{pmatrix} = \begin{pmatrix} -e^{3s} \frac{\partial}{\partial x}(u_1 u_2) \\ -e^{2s} \frac{\partial}{\partial x}(g u_1 + \frac{1}{2} u_2^2) \end{pmatrix} = e^s d(s) f(u).$$

Finally, the equality $(d(s)u)(x) = (e^{2s}u_1(x), e^s u_2(x))$, $x \in \mathbb{R}$ implies that for any $u \in \mathcal{D}(f)$ one has $d(s)u \in \mathcal{D}(f)$, i.e., $\mathcal{D}(f)$ is a **d-homogeneous cone**.

**Remark 2.1 (On a set-valued homogeneous operator).** Definition 2.9 is applicable to a set-valued operator $F: \mathcal{D}(F) \subset \mathbb{B} \Rightarrow \mathbb{B}$ provided that the identity (2.11) is understood in the set-theoretic sense (as equality of sets). It remains meaningful even for unbounded sets (values of $F$) since $\mathbf{d}(s)$ is a linear bounded invertible operator on $\mathbb{B}$. In [58] it was introduced the relaxed set-theoretic condition of homogeneity of the operator $F$ given by the inclusion

$$e^{\nu s} \mathbf{d}(s) F(u) \subset F(\mathbf{d}(s)u), \quad s \in \mathbb{R}, \ u \in \mathcal{D}(F). \qquad (2.14)$$

Evolution equations with **d-homogeneous** operators were studied in [60] and [56]. Below we basically use the next property: if a uniformly asymptotically stable evolution system is **d-homogeneous** with negative degree then it is finite-time stable.

## Homogeneous Extension

Homogeneous extension (as well as homogeneous approximation) allows some homogeneity-based methods to be applied for non-homogeneous systems.

**Definition 2.10. (Homogeneous extension)** A set-valued **d-homogeneous operator** $F: \mathcal{D}(F) \subset \mathbb{B} \Rightarrow \mathbb{B}$ is said to be **d-homogeneous extension** of an operator $f: \mathcal{D}(f) \subset \mathbb{B} \to \mathbb{B}$ if $f(u) \in F(u)$ for any $u \in \mathcal{D}(f) \subset \mathcal{D}(F)$.

Set-valued homogeneous extensions appear, for example, as a result of Filippov regularization procedure applied to discontinuous vector field [48], [41]. It also allows us to simplify robust stability analysis of uncertain non-linear affine system

$$s^{(n)} = a + b\xi,$$

$s \in \mathbb{R}, |a| < C, 0 < b_{\min} < b < b_{\max}$ provided that the extended differential inclusion

$$s^{(n)} \in [-C, C] + [b_{\min}, b_{\max}]\xi$$
is homogeneous, where $\xi = \xi(s, \dot{s}, ..., s^{(n-1)})$ is a homogeneous feedback law [48], [41].

Below we deal with the following homogeneous extension

$$F(u) = \bigcup_{s \in \mathbb{R}: \mathbf{d}(s) u \in \mathcal{D}(f)} \{e^{-\nu s} \mathbf{d}(-s) f(\mathbf{d}(s)u)\}, \quad (2.15)$$

having the domain $\mathcal{D}(F) = \bigcup_{s \in \mathbb{R}} \mathbf{d}(s) \mathcal{D}(f)$.

By construction the set-valued operator $F$ is $\mathbf{d}$-homogeneous of degree $\nu$. It satisfies (2.11) in the set-theoretic sense (see, Remark 2.1).

Formally, such a set-valued homogeneous extension is defined for any fixed $\nu \in \mathbb{R}$. For example, if $f(u) = -(1 + 0.5 \sin(u))u$ with $u \in \mathbb{R}$ then $F(u) = -u \cdot (0, +\infty)$ for $\nu \neq 0$ and $F(u) = -u \cdot (\frac{1}{2}, \frac{3}{2})$ for $\nu = 0$. The issue of an appropriate selection of the parameter $\nu$ can be solved for a concrete $f$.

It is worth stressing that a properly defined homogeneous extension remains informative and useful even when a homogeneous approximation [79], [4]) simply does not exist.

**Example 2.10.** Let $\mathbb{B} = \mathbb{R}$ and $f(x) = (\cos(1/x) - 2)x^{1/3}$, $x \in \mathbb{R}$, $\mathbf{d}(s) = e^s$. Obviously, a homogeneous approximation of the function $f$ at zero (see, [4]) does not exists, but the formula (2.15) gives

$$F(x) = [-3, -1]x^{1/3}.$$

and the stability of the original system can be easily analysed using the homogeneous extension.

**Remark 2.2.** Below we utilize the homogeneous extension for Lyapunov analysis of a well-posed non-homogeneous evolution system

$$\dot{u}(t) = f(u(t))$$

having classical (or strong) solutions $u(t) \in D(f) \subset \mathbb{H}$. Notice that in the latter case we do not need to prove existence of solution of the extended evolution inclusion

$$\dot{u}(t) \in F(u(t))$$

but we simply use the fact $f(u(t)) \in F(u(t))$ in order to estimate the time derivative of the Lyapunov-like functional $V: \mathcal{D}(f)\setminus\{\mathbf{0}\} \to \mathbb{R}_+$ defined as

$$V(u) = \|u\|_{\mathbf{d}}.$$
Indeed, if $\mathcal{D}(f) \subset \mathcal{D}(G_d)$ and $\mathbf{d}$ is a strictly monotone dilation then the formula (2.10) gives

$$ 
\begin{aligned}
\dot{V}(u) &= (D\|u\|_d)(f(u)) = \frac{\langle d(-\ln \|u\|_d)f(u), d(-\ln \|u\|_d)u \rangle}{\langle G_d d(-\ln \|u\|_d)u, d(-\ln \|u\|_d)u \rangle} \|u\|_d \\
&\leq \sup_{y \in F(u)} \frac{\langle d(-\ln \|u\|_d)y, d(-\ln \|u\|_d)u \rangle}{\langle G_d d(-\ln \|u\|_d)u, d(-\ln \|u\|_d)u \rangle} \|u\|_d.
\end{aligned}
$$

for $u \in \mathcal{D}(f)\setminus\{0\}$. Since $F$ is $\mathbf{d}$-homogeneous operator then $e^{\nu s}\mathbf{d}(s)F(u) \subset F(\mathbf{d}(s)u)$ for any $s \in \mathbb{R}$ and

$$ \dot{V}(u) \leq \|u\|_d^{1+\nu} \frac{\sup_{v \in F(d(-\ln \|u\|_d)u)} \langle v, d(-\ln \|u\|_d)u \rangle}{\langle G_d d(-\ln \|u\|_d)u, d(-\ln \|u\|_d)u \rangle}. $$

Using $\|d(-\ln \|u\|_d)u\| = 1$ we conclude that the inequality

$$ \alpha \langle G_d z, z \rangle + \sup_{v \in F(z)} \langle v, z \rangle \leq 0, \quad \forall z \in \mathcal{D}(f) : \|z\| = 1 \quad (2.16) $$

implies

$$ \dot{V}(u) \leq -\alpha V^{1+\nu}(u), \quad \alpha > 0, \quad u \in \mathcal{D}(f)\setminus\{0\}. \quad (2.17) $$

In the latter case the functional $V$ decays along any classical solution $u(t) \in \mathcal{D}(f)$ and vanishes in a finite time provided that $\nu < 0$.

**Example 2.11.** The above considerations may remain consistent for the case of unbounded sets $F(u), u \in \mathcal{D}(f)$. For example, if $f: \mathbb{R} \to \mathbb{R}$ is defined as

$$ f(u) = -(1 + |u|) \sqrt{|u|} \operatorname{sign}(|u|), \quad u \in \mathbb{R} $$

and $\mathbf{d}(s) = e^s, \nu = -1/2$, then the formula (2.15) gives the set-valued $\mathbf{d}$-homogeneous extension with

$$ F(u) = -\sqrt{|u|} \operatorname{sign}(u) \cdot (1, +\infty). $$

The differential equation $\dot{x}(t) = f(x(t))$, obviously, has continuous right-hand side so its solutions are well-defined (at least locally). For the uniform dilation $\mathbf{d}(s) = e^s$ one has $G_d = 1$ and $V(u) = \|u\|_d = |u|$. Hence, from (2.16) and (2.17) we derive $\dot{V} \leq -V^{1/2}$ since $z^2 + \sup_{y \in F(z)} yz \leq 0$ for all $z \in \mathbb{R}: |z| = 1$. We refer the reader to [61], [62] for more details about application of these ideas in the finite-dimensional case.

## Homogeneous Evolution Equations

Let us consider the following nonlinear evolution equation

$$ \dot{x} = f(x) := Ax + g(x), \quad (2.18) $$
with the initial condition

$$x(0) = x_0 \in \mathbb{B},$$

where $A: \mathcal{D}(A) \subset \mathbb{B} \to \mathbb{B}$ is a linear densely defined closed operator, which generates a strongly continuous semigroup $\Phi$ of linear bounded operators on $\mathbb{B}$; $g: \mathcal{D}(g) \subset \mathbb{B} \to \mathbb{B}$ is a nonlinear mapping, such that $\mathcal{D}(g) \subset \mathcal{D}(A)$. We refer the reader, for example, to [50] for more details about existence of solutions for the evolution equations like (2.18). Notice that for $\mathbb{B} = \mathbb{R}^n$ and $A = 0$ the latter equation defines a non-linear ODE.

In this chapter we deal with the classical solutions<fnref n="1" /> to (2.18) and the so-called mild solutions satisfying

$$x_{x_0}(t) = \Phi(t)x_0 + \int_0^t \Phi(t-s)g(x_{x_0}(s))ds, \quad (2.19)$$

where the integral is understood in the sense of Brochner (see e.g. [20, page 187]). The next results can be proven for both classical and mild solutions.

**Theorem 2.1.** [60] Let the nonlinear operator $f: \mathcal{D}(f) \subset \mathbb{B} \to \mathbb{B}$ be **d-homogeneous of degree $\nu \in \mathbb{R}$** and let $x_{x_0}: [0, T) \to \mathbb{B}$ be a solution to (2.18) with the initial condition $x(0) = \mathbf{d}(s)x_0$ then

$$x_{d(s)x_0}(t) = \mathbf{d}(s)x_{x_0}(e^{\nu s}t), \quad s \in \mathbb{R}, \quad t \in [0, e^{-\nu s}T),$$

is a solution to (2.18) with the initial condition $x(0) = \mathbf{d}(s)x_0$.

This simple result has a lot of corollaries useful for analysis and design of homogeneous control systems [78], [79], [66], [9]. Below we use the next one.

**Corollary 2.3.** [60] Let the origin of the system (2.18) be globally uniformly asymptotically stable and the operator $f: \mathcal{D}(f) \subset \mathbb{B} \to \mathbb{B}$ be **d-homogeneous of degree $\nu < 0$**, then the origin is **globally uniformly finite-time stable**, i.e. there exists $T(x_0): \mathbb{B} \to \mathbb{R}_+$ such that any solution $x_{x_0}$ of (2.18) with $x_{x_0}(0) = x_0$ satisfies $x_{x_0}(t) = 0$ for all $t \ge T(x_0)$.

This corollary introduces a general way for finite-time control and observation, namely, to design a stable homogeneous system with negative degree.

<footnote n="1">¹A function $x_{x_0}: [0, T) \to \mathbb{B}$ is a classical solution to (2.18) if $x_{x_0}(0) \in \mathcal{D}(A)$, $x_{x_0}(t) \in \mathcal{D}(A)$ and $\dot{x}_{x_0}(t) = g(x_{x_0}(t))$ for all $t \in (0, T)$.</footnote>
## 2.3 Control Design and State Estimation Using Linear Dilations

### 2.3.1 Homogeneous Stabilization in a Hilbert Space

Let us consider the following control system

$$ \dot{u}(t) = Au(t) + B\xi(u(t)), \quad t > 0 \qquad (2.20) $$

$$ u(0) = u_0 \in \mathcal{D}(A), \qquad (2.21) $$

where $A: \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H}$ is a (possibly unbounded) closed linear operator with the domain $\mathcal{D}(A)$ dense in $\mathbb{H}$, $B: X \to \mathbb{H}$ is a linear bounded operator, $X$ is a real Banach space, $\mathbb{H}$ is a real Hilbert space with the inner product $\langle \cdot, \cdot \rangle$, $u(t)$ is the system state, $\xi: \mathbb{H} \to X$ is a (locally or globally) bounded feedback control.

It is well known that algorithms for a linear feedback design $\xi = Ku$ with $K: \mathbb{H} \to X$ for the systems (2.20) are developed for both finite-time dimensional (see e.g. [75]) and infinite dimensional (see e.g. [73], [12]) cases. One of the most popular technique is based on linear matrix inequalities [11]. The closed-loop system is exponentially stable in the linear case.

Our aim is to design a control which steers any solution of the closed-loop system to zero in a finite time. The control design procedure is assumed to be based on the homogeneity with negative degree.

The problem of finite-time stabilization is studied since 1960 [24], where some finite-time control algorithms have been derived as solutions to certain optimization problems. In particular, the well-known time-optimal feedback control is a finite-time controller (see, e.g. [13]). Lyapunov function method for finite-time stability analysis is introduced in [68]. Finite-time controllers have been also studied in [27], [74], [64], [9], [41], [48].

The finite-time control of PDEs is also the topic of intensive research [38], [15, 16, 30, 51]. It is related with controllability analysis of evolution systems [73] as well as with sliding mode control method [47], [53].

In [61], [62] finite-time control is designed for linear finite-dimensional plants using implicit Lyapunov function method and homogeneity. First, a linear stabilizing feedback and a corresponding quadratic Lyapunov function are designed. Next, homogeneous dilation is applied in order to derive both a finite-time stabilizing feedback and a corresponding implicit Lyapunov function. Below we follow the same idea and use the canonical norm $\| \cdot \|_d$ defined by (2.8) as a Lyapunov function. Alternative approaches of finite-time stabilization of linear systems in $\mathbb{R}^n$ can be found, for example, in [64], [37], [9].
**Theorem 2.2.** [58] If $A: \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H}$ is a generator of a strongly continuous semigroup on $\mathbb{H}$, $B: X \to \mathbb{H}$ is a linear bounded operator and

A) $d_1$ is a uniformly continuous group of linear bounded operators in $\mathbb{H}$ such that $d_1(s)\mathcal{D}(A) \subset \mathcal{D}(A)$, $s \in \mathbb{R}$ and $\gamma\|z\|^2 \le \langle G_{d_1}z, z \rangle$, $\forall z \in \mathbb{H}$ with $\gamma \ge 0$,

B) $d$ be a dilation on $\mathbb{H}$ such that $\beta\|z\|^2 \le \langle G_d z, z \rangle$ for $z \in \mathcal{D}(G_d)$ and $\beta > \|G_{d_1}\|$,

C) $\exists K \in \mathcal{L}(\mathbb{H}, X)$ such that $BK(\mathcal{D}(A)) \subset \mathcal{D}(A)$ and there exists a set $\mathcal{D} \subset \mathcal{D}(A) \cap \mathcal{D}(G_d)$ dense in $\mathbb{H}$ such that

$$ \langle (A + BK + \alpha G_d)z, z \rangle \le 0, \forall z \in \mathcal{D} $$

where $\alpha > 0$ is a positive number,

D) the operator $A$ be $d$-homogeneous of degree $\mu < 0$ and

$$ d(s)BKd_1(s) = BKd(s), \quad s \in \mathbb{R}, $$

then the feedback control $\xi: \mathbb{H} \to \mathbb{H}$ defined by

$$ \xi(u) = \begin{cases} \|u\|_d^\mu K d_1(-\ln(\|u\|_d))u & \text{if } u \ne 0, \\ 0 & \text{if } u = 0 \end{cases} \quad (2.22) $$

is locally Lipschitz continuous on $\mathbb{H}\setminus\{0\}$, Fréchet differentiable on $\mathcal{D}(G_d)\setminus\{0\}$ and $B\xi(\mathcal{D}(A)) \subset \mathcal{D}(A)$.

Moreover, if $\exists M \ge 1$ such that $\|d(s)\| \le Me^{(\mu+\gamma)s}$ for $s > 0$ then $\|\xi(\cdot)\| \le M\|K\|_{\mathcal{L}}$.

For any $u_0 \in \mathbb{H}\setminus\{0\}$ the closed-loop system (2.20), (2.21), (2.22) has a mild solution $u: [0, T(u_0)) \to \mathbb{H}$ such that $\|u(t)\| \to 0$ as $t \to T(u_0)$ and

$$ \frac{d}{dt} \|u(t)\|_d^{a.e.} \le -\alpha \|u(t)\|_d^{1-\mu}, \quad t \in [0, T(u_0)), \quad (2.23) $$

and the settling time admits the estimate $T(u_0) \le \frac{\|u_0\|_d^{-\mu}}{-\alpha\mu}$.

**Sketch of the proof.** 1) The local Lipschitz continuity/differentiability of the feedback (2.22) follows from the local Lipschitz continuity/differentiability of $\|\cdot\|_d$. The inclusion $B\xi(\mathcal{D}(A)) \subset \mathcal{D}(A)$ follows from Conditions C) and D). Indeed, $d(s)\mathcal{D}(A) \subset \mathcal{D}(A)$ and $BK\mathcal{D}(A) \subset \mathcal{D}(A)$ imply $d(-s)BKd(s)\mathcal{D}(A) \subset \mathcal{D}(A)$. According to Condition D) we have $BKd_1(s) = d(-s)BKd(s)$.

2) Since $\xi$ is locally Lipschitz continuous then the closed-loop system has a mild solution any $x_0 \in \mathbb{H}\setminus\{0\}$ [50, Theorem 5.3.1, page 135]. The classical solutions can be obtained using [50, Theorem 5.4.6, page 146].
The finite-time extinction of classical solutions obviously follows from the inequality (2.23), which can be derived using conditions of Theorem 2.2. Indeed, Condition D) asks for **d**-homogeneity of the operator

$$f := A + B\xi$$

with negative degree $\mu < 0$, so one has

$$e^{\mu s} \mathbf{d}(s) f(u) = f(\mathbf{d}(s)u) = (A + BK)\mathbf{d}(s)u$$

for $u \in \mathcal{D}(A)$. Hence, for any classical solution, Condition C) and the formula (2.10) yield (2.23):

$$ \begin{aligned} \frac{d}{dt} \|u(t)\|_{\mathbf{d}} &= (D_u \|u(t)\|_{\mathbf{d}})(\dot{u}(t)) = \frac{\|u(t)\|_{\mathbf{d}} \langle \mathbf{d}(-\ln \|u(t)\|_{\mathbf{d}})\dot{u}(t), \mathbf{d}(-\ln \|u(t)\|_{\mathbf{d}})u(t) \rangle}{\langle G_{\mathbf{d}}\mathbf{d}(-\ln \|u(t)\|_{\mathbf{d}})u(t), \mathbf{d}(-\ln \|u(t)\|_{\mathbf{d}})u(t) \rangle} \\ &= \frac{\|u(t)\|_{\mathbf{d}}^{1+\mu} \langle (A+BK)\mathbf{d}(-\ln \|u(t)\|_{\mathbf{d}})u(t), \mathbf{d}(-\ln \|u(t)\|_{\mathbf{d}})u(t) \rangle}{\langle G_{\mathbf{d}}\mathbf{d}(-\ln \|u(t)\|_{\mathbf{d}})u(t), \mathbf{d}(-\ln \|u(t)\|_{\mathbf{d}})u(t) \rangle} \leq -\alpha \|u(t)\|_{\mathbf{d}}^{1+\mu}. \end{aligned} $$

One can be shown that the latter inequality holds for the mild solutions almost everywhere. The latter inequality immediately implies that $\|u(t)\|_{\mathbf{d}}$ vanishes to zero in a finite time $T(u_0) \le \frac{\|u_0\|_{\mathbf{d}}^{-\mu}}{-\alpha\mu}$. ■

**Example 2.12** (Distributed Finite-time Control for a Wave Equation on $\mathbb{R}$). Let

$$\mathbb{H} = H^1(\mathbb{R}, \mathbb{R}) \times L^2(\mathbb{R}, \mathbb{R})$$

with an inner product to be defined below.

Let us consider a wave equation represented in the form (2.20) using the notations:

$$A = \begin{pmatrix} O & I \\ \frac{\partial^2}{\partial z^2} & O \end{pmatrix} : \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H},$$

where

$$ \mathcal{D}(A) = H^2(\mathbb{R}, \mathbb{R}) \times H^1(\mathbb{R}, \mathbb{R}), \\ O \in \mathcal{L}(L^2(\mathbb{R}, \mathbb{R}), L^2(\mathbb{R}, \mathbb{R})) \quad \text{and} \quad I \in \mathcal{L}(L^2(\mathbb{R}, \mathbb{R}), L^2(\mathbb{R}, \mathbb{R})) $$

are zero and identity operators, respectively, and

$$B = \begin{pmatrix} O \\ I \end{pmatrix} : L^2(\mathbb{R}, \mathbb{R}) \to \mathbb{H}.$$

Let the inner product in $\mathbb{H}$ be defined as follows

$$\langle x, \tilde{x} \rangle = \int_{\mathbb{R}} x^{\top}(z) P \tilde{x}(z) + p_{22} \frac{\partial x_1}{\partial z} \frac{\partial \tilde{x}_1}{\partial z} dz,$$

where

$$P := \begin{pmatrix} p_{11} & p_{12} \\ p_{12} & p_{22} \end{pmatrix} = Q^{-1}$$
with

$$Q = \begin{pmatrix} q_{11} & q_{12} \\ q_{12} & q_{22} \end{pmatrix} \in \mathbb{R}^{2 \times 2}$$

being a positive definite solution of the system of linear matrix inequalities (LMIs) and equations:

$$\begin{pmatrix} \nu - 1.5\mu & 1 \\ 0 & \nu - 0.5\mu \end{pmatrix} Q + Q \begin{pmatrix} \nu - 1.5\mu & 0 \\ 1 & \nu - 0.5\mu \end{pmatrix} + y^\top b^\top + by = 0, \quad (2.24)$$

$$\begin{pmatrix} \nu - 1.5\mu & 0 \\ 0 & \nu - 0.5\mu \end{pmatrix} Q + Q \begin{pmatrix} \nu - 1.5\mu & 0 \\ 0 & \nu - 0.5\mu \end{pmatrix} > 0, \quad Q > 0, \quad (2.25)$$

where $\nu > \max\{0, 1.5\mu\}$, $b = (0 \ 1)^\top \in \mathbb{R}^2$, $y = (y_1 \ y_2) \in \mathbb{R}^{1 \times 2}$.

The operator A is a generator of a strongly continuous semigroup of linear bounded operators on $\mathbb{H}$ (see e.g. [50, Section 7.4] for more details).

This system of LMIs is always feasible with respect to Q and y. Since from (2.24) we conclude

$$(\nu - 1.5\mu)q_{11} + q_{12} = 0$$

then taking into account $P = Q^{-1} > 0$ it is easy to see that

$$0 < (\nu - 1.5\mu)p_{22} = p_{12}.$$

Let the operator $K: \mathbb{H} \to L^1(\mathbb{R}, \mathbb{R})$ be defined as

$$Kx := yQ^{-1}x, \quad x \in \mathbb{H},$$

where the pair $(Q, y)$ is a solution of the LMIs (2.24), (2.25).

A) Let the uniformly continuous semigroup $d_1$ of linear bounded operators on $\mathbb{H}$ be defined as follows

$$d_1(s)x = \begin{pmatrix} e^{-\mu s} & 0 \\ 0 & 1 \end{pmatrix} x.$$

Its generator

$$G_{d_1}x = \begin{pmatrix} -\mu & 0 \\ 0 & 0 \end{pmatrix} x,$$

obviously, satisfies the condition A) of Theorem 2.2 with $\gamma_{\min} = \min\{0, -\mu\}$ and $\gamma_{\max} = \max\{0, -\mu\} \ge 0$.

B) Let us introduce the dilation $d$ on $\mathbb{H}$ as follows

$$(d(s)x)(z) = \begin{pmatrix} e^{s(\nu-\mu)} & 0 \\ 0 & e^{s\nu} \end{pmatrix} x(e^{\mu s}z), \quad x \in \mathbb{H}, \ z \in \mathbb{R}.$$

The dilation $d$ has the generator $G_d: \mathcal{D}(G_d) \subset \mathbb{H} \to \mathbb{H}$ defined as

$$G_d x = \begin{pmatrix} \nu - \mu & 0 \\ 0 & \nu \end{pmatrix} x + \mu z \frac{\partial x}{\partial z},$$

where $\mathcal{D}(G_d) = \{x \in \mathbb{H} : z \frac{\partial x}{\partial z} \in \mathbb{H}\}$.
According to Proposition 2.5 the dilation **d** is strictly monotone on $\mathbb{H}$ if there exists $\beta > 0$ such that $\langle G_d x, x \rangle \ge \beta \|x\|^2$ for any $x \in C_c^\infty(\mathbb{R}, \mathbb{R}) \times C_c^\infty(\mathbb{R}, \mathbb{R})$, where $\|x\|^2 = \sqrt{\langle x, x \rangle}$. Using integration by parts we derive

$$ \langle G_d x, x \rangle = \int_{\mathbb{R}} x^{\top}(z) P \left( \begin{pmatrix} \nu-\mu & 0 \\ 0 & \nu \end{pmatrix} x(z) + \mu z \frac{\partial x(s)}{\partial z} \right) dz + $$

$$ p_{22} \int_{\mathbb{R}} (\nu - \mu) \left( \frac{\partial x_1(z)}{\partial z} \right)^2 + \mu \frac{\partial x_1(z)}{\partial z} \frac{\partial}{\partial z} \left( z \frac{\partial x_1(z)}{\partial z} \right) dz = $$

$$ \int_{\mathbb{R}} x^{\top}(z) P \begin{pmatrix} \nu-\mu & 0 \\ 0 & \nu \end{pmatrix} x(z) dz + \frac{\mu}{2} \int_{\mathbb{R}} z \frac{\partial}{\partial z} \left( x^{\top}(z) P x(z) \right) dz + $$

$$ p_{22} \nu \int_{\mathbb{R}} \left( \frac{\partial x_1(z)}{\partial z} \right)^2 dz + \frac{\mu}{2} \int_{\mathbb{R}} z \frac{\partial}{\partial z} \left( \frac{\partial x_1(z)}{\partial z} \right)^2 dz = $$

$$ \int_{\mathbb{R}} x^{\top}(z) P \begin{pmatrix} \nu-1.5\mu & 0 \\ 0 & \nu-0.5\mu \end{pmatrix} x(z) + p_{22}(\nu - 0.5\mu) \left( \frac{\partial x_1(z)}{\partial z} \right)^2 dz \geq \beta \|x\|^2, $$

where

$$ \beta \geq \min \left\{ \lambda_{\min} \left( P^{\frac{1}{2}} \begin{pmatrix} \nu - \frac{3\mu}{2} & 0 \\ 0 & \nu - \frac{\mu}{2} \end{pmatrix} P^{-\frac{1}{2}} + P^{-\frac{1}{2}} \begin{pmatrix} \nu - \frac{3\mu}{2} & 0 \\ 0 & \nu - \frac{\mu}{2} \end{pmatrix} P^{\frac{1}{2}} \right), \nu - \frac{\mu}{2} \right\} $$

The LMI (2.25) implies $\beta > 0$.

C) Given $x = (x_1, x_2)^{\top} \in C_c^\infty(\mathbb{R}, \mathbb{R}) \times C_c^\infty(\mathbb{R}, \mathbb{R})$ and $A_0 = \begin{pmatrix} 0 & I \\ 0 & 0 \end{pmatrix}$ we have

$$ \langle (A+BK+G_d)x, x \rangle = \langle (A_0+BK+G_d)x, x \rangle + \left\langle \begin{pmatrix} 0 \\ \frac{\partial^2 x_1}{\partial z^2} \end{pmatrix}, x \right\rangle. $$

Using (2.24) we derive

$$ \langle x, (A_0 + BK + G_d)x \rangle = p_{22} \int_{\mathbb{R}} \frac{\partial x_1(z)}{\partial z} \frac{\partial x_2(z)}{\partial z} dz + p_{22}(\nu - 0.5\mu) \int_{\mathbb{R}} \left( \frac{\partial x_1(z)}{\partial z} \right)^2 dz. $$

Using integration by parts we derive

$$ \langle x, \begin{pmatrix} 0 \\ \frac{\partial^2 x_1}{\partial z^2} \end{pmatrix} \rangle = \int_{\mathbb{R}} -p_{12} \left( \frac{\partial x_1(z)}{\partial z} \right)^2 - p_{22} \frac{\partial x_1(z)}{\partial z} \frac{\partial x_2(z)}{\partial z} dz, $$

Therefore, taking into account the identity $0 < (\nu - 1.5\mu)p_{22} = p_{12}$ we conclude

$$ \langle x, (A+BK+G_d)x \rangle = \mu p_{22} \int_{\mathbb{R}} \left( \frac{\partial x_1(z)}{\partial z} \right) dz. $$

For $\mu \le 0$ we, obviously, have $\langle x, (A+BK+G_d)x \rangle \le 0$. On the other hand, if $\nu \ge 2.5\mu > 0$ then using the representation for $\langle x, G_d x \rangle$ we derive

$$ \langle x, (A+BK+0.5G_d)x \rangle \le -\frac{p_{22}(\nu-2.5\mu)}{2} \int_{\mathbb{R}} \left( \frac{\partial x_1(z)}{\partial z} \right) dz \le 0. $$
Therefore, the condition C) of Theorem 2.2 holds for any $\mu \in \mathbb{R}$ and any $\nu \ge \max\{0, 2.5\mu\}$.

D) Finally, $\mathbf{d}(s)BK\mathbf{d}_1(s) = BK\mathbf{d}(s)$ for any $s \in \mathbb{R}$ and the operator A is $\mathbf{d}$-homogeneous of the degree $\mu$. Indeed,

$$ (A\mathbf{d}(s)x)(z) = A\left(\frac{e^{(\nu-\mu)s}x_1(e^{\mu s}z)}{e^{\nu s}x_2(e^{\mu s}z)}\right) = \left(\frac{e^{\nu s}x_2(e^{\mu s}z)}{e^{(\nu+\mu)s}\left.\frac{\partial^2 x_1(q)}{\partial q^2}\right|_{q=e^{\mu s}z}}\right) = e^{\mu s}(\mathbf{d}(s)Ax)(z) $$

and the condition D) of Theorem 2.2 is fulfilled.

Therefore, the homogeneous feedback control of the form (2.22) steers all trajectories of the wave system to the origin in a finite time if $\mu < 0$.

In the general case, the homogeneous norm $\|\cdot\|_d$ is defined implicitly, see (2.8). For $\mu = -1$, $\nu = \frac{1}{2}$ the canonical homogeneous norm can be found as a unique positive solution of the quartic equation

$$ V^4 = aV^2 + bV + c $$

with $V = e^{s_x} = \|x\|_d$,

$$ a = p_{22} \int_{\mathbb{R}} x_2^2(z) + (x_1'(z))^2 dz, \quad b = 2p_{12} \int_{\mathbb{R}} x_1(z)x_2(z) dz, \quad c = p_{11} \int_{\mathbb{R}} x_1^2(z) dz. $$

In this case $\|x\|_d$ can be computed using Ferrari formulas, i.e., the homogeneous finite-time stabilizing feedback (2.22) for the wave equation admits an explicit representation.

## Non-homogeneous operators allowing homogeneous extensions

The above examples introduce dilations for systems on unbounded domains. If the domain is bounded then the mentioned dilations are not applicable and the previous kind of operators on bounded domains may become non-homogeneous. If the condition D) in Theorem 2.2 does not hold (non-homogeneous case) then the finite-time control can be still be designed for an operator A allowing a homogeneous extension.

**Corollary 2.4.** [58] Theorem 2.2 remains true if D) is replaced with
D*) there exists a set-valued operator $F: D(F) \subset \mathbb{H} \Rightarrow \mathbb{H}$ such that

$$ Au + B\xi(u) \in F(u) \text{ for } u \in D(A) \subset D(F), $$

$F$ is $\mathbf{d}$-homogeneous of degree $-\mu < 0$ and there exists $D \subset D(F) \cap D(G_d)$ dense in $D(A)$,

$$ \alpha \langle G_d z, z \rangle + \sup_{y \in F(z)} \langle y, z \rangle \le 0, \quad z \in D : \|z\| = 1 $$

for some $\alpha > 0$.
The operator $F$ is the homogeneous extension of the operator $A + B\xi$ (see Definition 2.10). Recall that we do not need to study an evolution system $\dot{u} \in F(u)$. The existence of classical solutions of the closed-loop system (2.20) comes from Conditions A)-C) of Theorem 2.2. The inclusion $Au + B\xi(u) \in F(u)$ is utilized only for analysis of finite-time extinction of these solutions (see Remark 2.2 for more details).

This corollary allows any mild solution of the closed-loop system to be prolonged to zero by continuity $u(t) = 0$ for $t \ge T(u_0)$ since the canonical homogeneous norm $\| \cdot \|_d$ is the Lyapunov function to the closed-loop system implying uniqueness and finite-time stability of the zero solution.

### Finite-time stabilization by means of switched linear feedback

Realization of the control law (2.22) needs calculation of the homogeneous norm $\|u\|_d$, which is defined implicitly by the functional equation (2.8). Sometimes the homogeneous norm can be calculated explicitly (see, [61], [56] and the above example). The control law (2.22) can also be applied in a sampled way by means of replacing $\|u(t)\|_d$ in (2.22) with $\|u(t_j)\|_d$ for $t \in [t_j, t_{j+1})$. The value $\|u(t_j)\|_d$ can be found by means of an appropriate numerical procedure (see, e.g. [61]). The control (2.31) becomes linear on each interval of time $[t_j, t_{j+1})$.

Let us consider the following switched linear feedback

$$ \xi_{\sigma(t)}(u(t)) = r_{\sigma(t)}^{-\mu} K d_1(-\ln(r_{\sigma(t)}))u(t), \quad (2.26) $$

with state dependent switching function $\sigma$ such that $\sigma(0) \in \mathbb{Z}$,

$$ \sigma(t) = S(\sigma(t^{-}), u(t^{-})), \quad S: \mathbb{Z} \times \mathbb{H} \to \mathbb{Z}, \\ S(\sigma, u) := \begin{cases} i-1 & \text{if } \sigma=i \text{ and } \|u\|_d \ge r_{i-1}, \\ i & \text{if } \sigma=i \text{ and } r_{i+1} \le \|u\|_d \le r_{i-1}, \\ i+1 & \text{if } \sigma=i \text{ and } r_{i+1} \le \|u\|_d, \end{cases} \quad (2.27) $$

where $r_i \in \mathbb{R}_+ : r_{i+1} < r_i$, $i \in \mathbb{Z}$ and $\Omega = \bigcup_{i \in \mathbb{Z}} \{r_i\}$. The switching variable $\sigma$ has its own evolution equation given by (2.27) so the pair $(u(t), \sigma(t))$ can be treated as an augmented state of the closed-loop switched system having a strong solution<fnref n="2" />. We refer the reader to [43] for more details about switched systems.

**Corollary 2.5 (On switched linear finite-time feedback law).** [58] If all conditions of Theorem 2.2 hold and the control is given by (2.26), (2.27) with

$$ r_0 = 1, \quad r_{i+1} = e^{-q_i} r_i, \quad i \in \mathbb{Z} $$

<footnote n="2">²A continuous function $u: [0, T) \to \mathbb{H}$ which is differentiable almost everywhere on $[0, T]$ such that $\dot{u} \in L^1((0, T), \mathbb{H})$ is called strong solution to the initial value problem (2.20), (2.21), (2.26), (2.27) if $u(t) \in D(A)$ for $t \in (0, T)$, $u(0) = u_0$ and there exists a (locally) piecewise constant function $\sigma: [0, T) \to \mathbb{Z}$ such that $\sigma$ satisfies (2.27) on $(0, T)$ and $\dot{u}(t) = Au(t) + B\xi_{\sigma(t)}(u(t))$ almost everywhere on $(0, T)$.</footnote>
and $q_i > 0$ such that

$$ \sum_{i=0}^{+\infty} q_i = +\infty \quad \text{and} \quad \lim_{i \to +\infty} q_{i+1} e^{-\mu q_i} / q_i < 1, $$

then

* for $u_0 \in D(A) \setminus \{0\}$ the closed-loop system (2.20), (2.21), (2.26), (2.27) has a unique strong solution and

$$ \|u(t)\| \to 0 \text{ as } t \to T(u_0), $$

where the settling time functional $T: \mathbb{H} \to \mathbb{R}$ admits the estimate

$$ T(u_0) \le \alpha^{-1} \sum_{j=i_0}^{\infty} r_j^{\mu} q_j < +\infty $$

provided that

$$ \|u_0\|_d \in (r_{i_0+1}, r_{i_0}]; $$

* the time instants $t_j : \|u(t_j)\|_d = r_j$ are isolated and the set $\{t_j\}_{j=i_0+1}^{+\infty}$ has the unique accumulation point $T(u_0)$;

* the control is a linear continuous feedback on each $(t_j, t_{j+1})$ and $\|\xi_{\sigma(t)}(t)\| \le M \|K\|_{\mathcal{L}}$ if $\|d(s)\|_{\mathcal{H}} \le Me^{(\mu+\gamma)s}$ for $s > 0$.

Existence, uniqueness and finite-time extinction of mild solutions to the closed-loop system (2.20), (2.21), (2.26), (2.27) with $u_0 \in \mathbb{H} \setminus \{0\}$ can be shown repeating the proof of Corollary ??.

Corollary 2.5 proposes a scheme for homogeneity-based finite-time control design in the form of switched linear feedback. Its applicability to boundary control case looks like a promising research direction, especially, in the context of the papers [16], [57], where a finite-time linear switched boundary control has been designed for heat system using the back-stepping approach [39].

**Example 2.13** (Distributed Finite-time Control for Heat System on a Segment). To demonstrate a possible way of application of Corollary 2.4, we design a distributed finite-time control for the following heat system on $[0, 1]$:

$$ \frac{\partial x}{\partial t} = \frac{\partial^2 x}{\partial z^2} + \phi(z)u(t, z), \quad x(t, 0) = x(t, 1) = 0, \quad x(0, z) = x_0(z), $$

where $x$ is the system state, $u$ is the distributed control, $\phi: [0, 1] \to \mathbb{R}_+$ is a continuous function such that
$$cz^2 \leq \phi(z) \quad \text{for} \quad z \in [0, 1], \qquad (2.28)$$

for some $c > 0$. The function $\phi$ represents possibly non-uniform feedback gain on $[0, 1]$. The design procedure is realized in several steps

* First, we extend the considered system virtually to $\mathbb{R}_+$ (with the zero operator on $(1, +\infty)$);

* Next, we design a feedback control using conditions A)-C) of Theorem 2.2) and Condition D*) of Corollary 2.4.

* Finally, we reduce the extended closed-loop system to the original one assuming that the initial condition of the extended system has a support only in $(0, 1)$.

Homogeneous finite-time control design on $\mathbb{R}_+$. Let $\mathbb{H} = H^0(\mathbb{R}_+, \mathbb{R}) = L^2(\mathbb{R}_+, \mathbb{R})$ and $x_{(a,b)}$ denotes a restriction of a function $x: \mathbb{R}_+ \to \mathbb{R}$ to the interval $(a, b)$. For any $x \in \mathbb{H}$ we have $x_{(0,1)} \in H^0((0,1), \mathbb{R})$ and $x_{(1,+\infty)} \in H^0((1,+\infty), \mathbb{R})$.

Let the operator $A: \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H}$ be defined as the connection of the Laplace operator with Dirichlet boundary conditions on $[0, 1]$ and the zero operator on $[1, +\infty)$, i.e. $(Ax)(z) = \frac{\partial^2 x(z)}{\partial z^2}$ if $z \in (0, 1)$ and $(Ax)(z) = 0$ if $z \ge 1$ for

$$x \in \mathcal{D}(A) := \{x \in \mathbb{H} : x_{(0,1)} \in H_0^1((0,1), \mathbb{R}) \cap H^2((0,1), \mathbb{R})\}.$$

The operator $A$ is closed and densely-defined. It generates the strongly continuous semigroup $\Phi$ of linear bounded operators on the Hilbert space $\mathbb{H}$:

$$(\Phi(t)u)(x) = \begin{cases} 2 \sum_{n=1}^{\infty} e^{-n^2 \pi^2 t} \sin(\pi nx) \int_{0}^{1} \sin(\pi ny)u(y)dy & \text{if } x \in (0,1), \\ u(x) & \text{if } x \in [1, +\infty). \end{cases}$$

Let us consider the equation (2.20) with $A$ given above and $B: \mathbb{H} \to \mathbb{H}$ defined as follows $(Bx)(z) = \phi(z)x(z)$ if $z \in (0, 1)$ and $(Bx)(z) = \phi(1)x(z)$ if $z \ge 1$ for $x \in \mathbb{H}$.

Let us show that conditions A)-C) of Theorem 2.2 and the condition D*) of Corollary 2.4 hold for $K = -\tilde{k}I \in \mathcal{L}(\mathbb{H})$ with a sufficiently large scalar $\tilde{k} > 0$.

A) Let a uniformly continuous group $d_1$ be defined as follows

$$(d_1(s)x)(z) = e^{-\mu s}x(z) \quad \text{if } z \in (0, 1) \quad \text{and} \quad (d_1(s)x)(z) = x(z) \quad \text{if } z \ge 1,$$

where $\mu < 0$. Obviously, $\gamma_{\min} = 0$ and $\gamma_{\max} = -\mu$, where $\gamma_{\min}$ and $\gamma_{\max}$ are defined in Theorem 2.2.

B) Let the dilation $d(s)$ in $\mathbb{H}$ be defined as

$$(d(s)x)(z) = e^s x(e^{0.5\mu s}z).$$

It has the generator $G_d x = x + 0.5\mu z \frac{\partial x}{\partial z}$ with the domain $\mathcal{D}(G_d) = \{x \in \mathbb{H}: x + 0.5\mu z \frac{\partial x}{\partial z} \in \mathbb{H}\}$. Obviously,

$$\|d(s)x\| = \sqrt{\int_{\mathbb{R}_+} e^{2s} x^2 (e^{0.5\mu s} z) dz} = e^{(1-0.25\mu)s} \|x\|$$
and $(1 - 0.25\mu)\|x\|^2 = \langle G_d x, x \rangle$ if $x \in \mathcal{D}(G_d)$, where, as before, $\| \cdot \|$ denotes the norm in $\mathbb{H}$. Hence, we derive

$$ \|x\|_{d} = \|x\|^{\frac{1}{1-0.25\mu}} $$

and the formula (2.22) gives

$$ u(x) = -\tilde{k} \|x\|_{d}^{\mu} d_{1}(-\ln \|x\|_{d})x = -\tilde{k} \|x\|^{\frac{2\mu}{1-0.25\mu}} x. $$

Theorem 2.2, guarantees that this feedback control is locally Lipschitz continuous on $\mathbb{H}\setminus\{0\}$. Moreover, it is continuous at $0$ if $-4/7 < \mu < 0$ and globally bounded if $\mu = -4/7$.

C)-D*) Let us consider the homogeneous extension (2.15):

$$ F(x) = \bigcup_{s \in \mathbb{R}: d(s)x \in \mathcal{D}(A)} \{A_s x + q_s(x)\}, $$

where $A_s = e^{-\mu s} d(-s) A d(s)$ and $g_s(x) = e^{-\mu s} d(-s) B u(d(s)x)$. It is easy to see that $(A_s x)(z) = \frac{\partial^2 x(z)}{\partial z^2}$ if $z \in (0, e^{0.5\mu s})$ and $(A_s x)(z) = 0$ if $z \ge e^{0.5\mu s}$,

$$ (g_s(x))(z) = -\tilde{k} \begin{cases} \phi(e^{-0.5\mu s}z)e^{\mu s} \frac{x(z)}{\|x\|_d^{-2\mu}}, & z \in (0, e^{0.5\mu s}), \\ \phi(1) \frac{x(z)}{\|x\|_d^{\mu}}, & z \ge e^{0.5\mu s}. \end{cases} $$

The set-valued operator $F: \mathcal{D}(F) \subset \mathbb{H} \Rightarrow \mathbb{H}$ is $d$-homogeneous of degree $\mu < 0$ and $Ax + Bu(x) \in F(x)$ for any $x \in \mathcal{D}(A)$, where $\mathcal{D}(F) = \bigcup_{s \in \mathbb{R}} d(s)\mathcal{D}(A)$.

Let us denote $q_s = e^{0.5\mu s}$ and consider the set

$$ \tilde{D}_s = \left\{ x \in \mathbb{H} : \begin{array}{l} x_{(0,q_s)} \in C_c^\infty((0,q_s), \mathbb{R}), \\ x_{(q_s, +\infty)} \in C_c^\infty((q_s, +\infty), \mathbb{R}). \end{array} \right\} $$

Obviously, $\tilde{D}_0 \subset \mathcal{D}(A)$ is dense in $\mathcal{D}(A)$ and $d(s)x \in \tilde{D}_0 \Rightarrow x \in \tilde{D}_s$.

Let us denote $\langle x, y \rangle_{(a,b)} = \int_a^b x(z)y(z)dz$. Using integration by parts for any $s \in \mathbb{R}$ and for any $x \in \tilde{D}_s$ we derive

$$ \langle z \frac{\partial x}{\partial z}, x \rangle_{(0,q_s)} = \frac{-\langle x, x \rangle_{(0,q_s)}}{2} $$

and

$$ \langle z \frac{\partial x}{\partial z}, x \rangle_{(q_s, +\infty)} = \frac{-\langle x, x \rangle_{(q_s, +\infty)}}{2}. $$

Due to (2.28) one has

$$ \phi(e^{-0.5\mu s}z)e^{\mu s} \ge cz^2 \quad \text{if } z \in [0, q_s] $$

and

$$ \langle \phi(e^{-0.5\mu s}z)e^{\mu s}x, x \rangle_{(0, e^{0.5\mu s})} \ge c \langle zx, zx \rangle_{(0, q_s)}. $$
Therefore, for any $x \in \tilde{D}_s$, $\|x\| = 1$ we obtain

$$ \begin{aligned} \langle (A_s x + g_s(x) + \rho G_d)x, x \rangle &\le \\ &- \langle \frac{\partial x}{\partial z}, \frac{\partial x}{\partial z} \rangle_{(0,q_s)} - 2\rho(1 - 0.25\mu) \langle z \frac{\partial x}{\partial z}, x \rangle_{(0,q_s)} - \tilde{k}c \langle zx, zx \rangle_{(0,q_s)} \\ &- (\tilde{k}\phi(1) - \rho(1 - 0.25\mu)) \langle x, x \rangle_{(q_s, +\infty)} \le 0 \end{aligned} $$

provided that $\tilde{k} \ge \max \left\{ \frac{\rho(1-0.25\mu)}{\phi(1)}, \frac{\rho^2(1-0.25\mu)^2}{c} \right\}$.

The condition $D^*$) of Corollary 2.4 holds for

$$ \mathcal{D} = \bigcup_{s \in \mathbb{R}} \tilde{D}_s $$

since

* $\mathcal{D}$ is a $d$-homogeneous cone by construction;

* $\mathcal{D}$ contains the set $\tilde{D}_0$ dense in $\mathcal{D}(A)$;

* $x \in \mathcal{D} : d(s)x \in \mathcal{D}(A) \Rightarrow x \in \tilde{D}_s$.

Therefore, the feedback law (2.22) steers any solution of the extended evolution system (2.20) to zero in a finite time $T \le (-\rho\mu)^{-1}\|x_0\|_d^{-\mu}$, provided that $\mu \in [-4/7, 0)$.

Restriction of the finite-time control to the segment $[0, 1]$. If $\text{supp}(x) \subset [0, 1]$ and $x \in \mathbb{H}\{0\}$ then $\text{supp}(\Phi(t)x) \subset [0, 1]$, $\text{supp}(u(x)) \subset [0, 1]$ and $\|x\| = \sqrt{\int_0^1 x^2(z)dz}$, where $\text{supp}(x)$ denotes the support of a function $x \in \mathcal{D}(A)$. Therefore, any solution of the closed-loop system (2.20), (2.22) with initial condition

$$ x_0 \in \{x \in \mathcal{D}(A) : \text{supp}(x) \subset [0, 1]\} $$

coincides on $[0, 1]$ with the solution to the equation

$$ \frac{\partial x}{\partial t} = \frac{\partial^2 x}{\partial z^2} + \phi(z) \frac{-\tilde{k}}{\|x\|^{\frac{-2\mu}{1-0.25\mu}}} x, \quad \begin{array}{l} x(t,0)=x(t,1)=0, \\ x(0,z)=u_0(x), \quad z \in [0,1]. \end{array} $$

Consequently, any solution of the latter heat system vanishes in a finite time

$$ T(x_0) \le (-\rho\mu)^{-1} \left( \int_0^1 x_0^2(z) dz \right)^{-0.5\mu/(1-0.25\mu)} . $$

Piecewise linear finite-time control on $[0, 1]$. Since $\|x\|_d = \|x\|^{\frac{1}{1-0.25\mu}}$ then the parameters of Lemma 2.1 are $M = 1, \alpha = \beta = 1 - 0.25\mu$. Applying Corollary 2.5 for $q_i = \ln 2$ we derive $u(t) = -\lambda(t)x(t)$, where the switched gain $\lambda(t) =$
<figure><img src="image_6.png" /><figcaption>Figure 2.1: Evolution of L²-norm of the heat equation with a linear (red dotted line), switched linear (green dashed line) and a non-linear (blue solid line) homogeneous control (plotted in the logarithmic scale)</figcaption></figure>

$\tilde{k}/r_{\sigma(t)}^{-2\mu/1-0.25\mu}$ with $r_i = 2^{-i}$ and $\sigma(t)$ defined by (2.27). Therefore, for any $x_0 \in H_0^1((0,1), \mathbb{R}) \cap H^2((0,1), \mathbb{R})$ the equation

$$ \frac{\partial x}{\partial t} = \frac{\partial^2 x}{\partial z^2} - \lambda(t)\phi(z)x, \quad \begin{array}{l} x(t,0)=x(t,1)=0, \\ x(0,z)=u_0(z), \quad z \in [0,1]. \end{array} \qquad (2.29) $$

has a unique strong solution which vanishes in a finite-time

$$ T(x_0) \le \frac{\ln 2}{\rho} \sum_{j=i_0}^{+\infty} 2^{\mu j} $$

for $\|x_0\|_d \in (r_{i_0+1}, r_{i_0}]$. Notice that in contrast to the linear switched feedback presented in [16], the settling time of our algorithm depends on the initial condition.

Numerical simulations. The simulation results of the system (2.29) (for $\phi(z) = z^2$) with homogeneous and switched linear finite-time controls ($\mu = -0.5, \tilde{k} = 1$) as well as with the static linear feedback ($\lambda = \tilde{k}$) are depicted at Fig. 2.1. The initial condition is given by $x_0(z) = 5.25z(1-z)$. For numerical simulations, the system has been discretized by means of divided differences on a uniform grid with the step $h = 0.005$ for the space variable. The discretization with respect to time has been done using implicit scheme with the step size $\tau = h^2$. The norm $\|x\|$ is calculated by Simpson rule. Numerical simulations show that the heat system with homogeneous control has a fast decay (typical for finite-time stable systems) with the settling time $T(x_0) \le 0.3$.
**Example 2.14.** Let us consider the linear control system in $\mathbb{R}^n$

$$ \dot{x}(t) = Ax(t) + Bu(t), \quad t > 0, $$

where $x(t) \in \mathbb{R}^n$, $u(t) \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$. Let $x \to Ax$ be **d**-homogeneous of degree $-\mu < 0$, i.e.

$$ AG_d = (G_d - \mu I_n)A, $$

where $G_d \in \mathbb{R}^{n \times m}$ is a generator of the dilation $\mathbf{d}(s) = e^{sG_d}$. Let the Euclidean space $\mathbb{R}^n$ be equipped with the inner product

$$ \langle x, y \rangle = x^\top P y, \quad 0 < P \in \mathbb{R}^{n \times n}, \quad x, y \in \mathbb{R}^n. $$

and the dilation $\mathbf{d}$ be monotone with respect to the norm $\|x\| = \sqrt{x^\top P x}$, i.e.

$$ G_d^\top P + PG_d > 0. $$

If $G_d B = \mu B$ and there exists a matrix $K \in \mathbb{R}^{m \times n}$ such that

$$ (A + BK)^\top P + P(A + BK) + \alpha(G_d^\top P + PG_d) \le 0, \quad \alpha > 0, $$

then all conditions of Theorem 2.2 hold for $G_{d_1} = G_d - \mu I_n$, $\mathbf{d}_1(s) = e^{sG_{d_1}}$ and the homogeneous globally bounded feedback

$$ u(t) = K \mathbf{d}(-\ln \|x(t)\|_d)x(t), $$

stabilizes the origin of the closed-loop system in a finite time. If the pair $\{A, B\}$ is controllable then the obtained system of matrix inequalities is always feasible and the homogeneous feedback can always be designed (see, [62] for more details).

Other examples of an application of Theorem 2.2 for $\mathbb{H} = \mathbb{R}^n$ can be found in [61], [62] and in the next section.

### 2.3.2 Quadratic Stabilizability of Homogeneous Systems in $\mathbb{R}^n$

#### Problem Statement

Let $\mathbb{B} = \mathbb{R}^n$ and the model of a control system is given by

$$ \dot{\sigma}(t) = g(\sigma(t), u(t)), \qquad (2.30) $$

where $t \in \mathbb{R}_+$ is time variable, $\sigma(t) \in \mathbb{R}^n$ is the vector of state variables at the time instant $t$, $u(t) \in \mathbb{R}^m$ is the control input, $g \in C(\mathbb{R}^{n+m}, \mathbb{R}^n)$, $g(\mathbf{0}, \mathbf{0}) = \mathbf{0}$.

We assume that the whole state vector $\sigma$ can be measured on-line and utilized for control purposes. In this section we deal (mainly) with the problem of a dynamical feedback control design

$$ \dot{u}(t) = k(\sigma(t), u(t)), \quad k \in C(\mathbb{R}^{n+m} \setminus \{\mathbf{0}\}, \mathbb{R}^m). \qquad (2.31) $$
and we restrict ourself with the class of the homogeneous nonlinear systems. The main goal is to derive a necessary and sufficient condition of (finite-time or fixed-time) stabilizability of the system (2.30) by means of a homogeneous dynamical feedback (2.31). In particular, we are interested in a (more or less) constructive stabilizability criteria allowing a unified scheme for control design. Obviously, under assumption

$$f = \begin{pmatrix} g \\ k \end{pmatrix} \in C(\mathbb{R}^{n+m} \setminus \{0\}, \mathbb{R}^{n+m}), \qquad (2.32)$$

the closed-loop system (2.30), (2.31) may have discontinuity only at the origin similarly to the unit control [74] or "quasi-continuous" control [41]. It is also worth stressing that sliding mode control method frequently uses dynamical feedbacks as a part of the control law (see, e.g. super-twisting algorithm [40]).

Let $\mathbb{F}_d(\mathbb{R}^n)$ (resp. $\mathbb{H}_d(\mathbb{R}^n)$) be the set of vector fields $\mathbb{R}^n \to \mathbb{R}^n$ (resp. functions $\mathbb{R}^n \to \mathbb{R}$) satisfying the identity (2.11) (resp. (2.12)), which are continuous on $\mathbb{R}^n \setminus \{0\}$. Let $\deg_{\mathbb{F}_d}(f)$ (resp. $\deg_{\mathbb{H}_d}(h)$) denote the homogeneity degree of $f \in \mathbb{F}_d(\mathbb{R}^n)$ (resp. $h \in \mathbb{H}_d(\mathbb{R}^n)$).

## Quadratic Stability

The further constructions are essentially based on the theorem given below for the system

$$\dot{\xi} = f(\xi), \quad f \in \mathbb{F}_d(\mathbb{R}^n). \qquad (2.33)$$

**Theorem 2.3.** [55] The next five claims are equivalent

1) The origin of the system (2.33) is asymptotically stable.

2) There exists a Lyapunov function $V \in \mathbb{H}_d(\mathbb{R}^n) \cap C^\infty(\mathbb{R}^n)$;

3) The origin of the system

$$\dot{z} = \|z\|^{1+\deg_{\mathbb{F}_d}(f)} \left( \frac{(I_n - G_d)z^\top z P}{z^\top P G_d z} + I_n \right) f \left( \frac{z}{\|z\|} \right) \qquad (2.34)$$

is asymptotically stable, where $\|z\| = \sqrt{z^\top P z}$ with $P$ satisfying (2.6).

4) For any matrix $P \in \mathbb{R}^{n \times n}$ satisfying (2.6) there exists $\Psi \in \mathbb{F}_d(\mathbb{R}^n) \cap C^\infty(\mathbb{R}^n \setminus \{0\})$, $\deg_{\mathbb{F}_d}(\Psi) = 0$ such that $\Psi$ is diffeomorphism on $\mathbb{R}^n \setminus \{0\}$, homeomorphism on $\mathbb{R}^n$, $\Psi(0) = 0$ and

$$\frac{\partial(\Psi^\top(\xi)P\Psi(\xi))}{\partial\xi} f(\xi) < 0 \quad \text{if } \Psi^\top(\xi)P\Psi(\xi) = 1. \qquad (2.35)$$
Moreover, $\|\Psi\|_d \in \mathbb{H}_d(\mathbb{R}^n) \cap C^\infty(\mathbb{R}^n\setminus\{0\})$ is Lyapunov function to the system (2.33), where $\|\cdot\|_d$ is the canonical homogeneous norm induced by $\|\xi\| = \sqrt{\xi^\top P \xi}$.

5) For any matrix $P \in \mathbb{R}^{n \times n}$ satisfying (2.6) there exists a map

$$ \Xi \in C^\infty(\mathbb{R}^n \setminus \{0\}, \mathbb{R}^{n \times n}) $$

such that

$$ \det(\Xi(z)) \neq 0, \quad \frac{\partial \Xi(z)}{\partial z_i} z = 0, \quad \Xi(e^z z) = \Xi(z) $$

for $z = (z_1, ..., z_n)^\top \in \mathbb{R}^n \setminus \{0\}$, $s \in \mathbb{R}$, $i = 1, ..., n$ and

$$ z^\top \Xi^\top(z) P \Xi(z) \left( \frac{(I_n - G_d) z z^\top P}{z^\top P G_d z} + I_n \right) f\left(\frac{z}{\sqrt{z^\top P z}}\right) < 0. \quad (2.36) $$

This theorem proves three important facts:

* Any homogeneous (with respect to linear dilations) system is homeomorphic on $\mathbb{R}^n$ and diffeomorphic on $\mathbb{R}^n \setminus \{0\}$ to a standard homogeneous one.

* Any asymptotically stable homogeneous system is homeomorphic on $\mathbb{R}^n$ and diffeomorphic on $\mathbb{R}^n \setminus \{0\}$ to a **quadratically stable** one. Indeed, making the change of variables $z = \Psi(\xi)$ we derive

$$ \dot{z} = \tilde{f}(z), $$

where

$$ \tilde{f}(z) = \left. \frac{\partial \Psi(\xi)}{\partial \xi} f(\xi) \right|_{\xi = \Psi^{-1}(z)}, $$

but the criterion (2.35) implies that

$$ z^\top P \dot{z} < 0 \quad \text{if} \quad z^\top P z = 1, $$

so the homogeneous norm $\|\cdot\|_d$ is the Lyapunov function to the latter system. Finally, the change of variable

$$ x = \|z\|_d d(-\ln \|z\|_d)z $$

gives $\|z\|_d = \|x\|$, so the transformed system

$$ \dot{x} = \hat{f}(x) $$

is quadratically stable. The fact that the Lyapunov function for the homogeneous system has a level set diffeomorphic to the unit sphere, has been
noticed in [33]. The canonical homogeneous norm has allowed us to construct
the required transformation in the simple explicit form

$$
\Psi(\xi) = \mathbf{d} \left( \ln \frac{V^{\frac{1}{\deg_{H_d}(V)}}(\xi)}{\|\xi\|_{\mathbf{d}}} \right) \xi.
$$

* The formula (2.36) gives a more or less constructive stability criterion. Since $\Xi(z) = \Xi(e^z z)$ for $z \in \mathbb{R}^n \setminus \{\mathbf{0}\}$, $s \in \mathbb{R}$ then the map $\Xi$ is constant along any straight line $\{e^z z : s \in \mathbb{R}\}$ if $z \neq \mathbf{0}$. In addition, the property $\frac{\partial \Xi(z)}{\partial z_i} z = 0$ motivates the conjecture that a quadratic Lyapunov function ($\Xi \equiv \text{const}$) always exists for asymptotically stable standard homogeneous system. Such assumption was also studied in [3]. However, in the view of [6, Proposition 5.2] this conjecture seems to be wrong. Therefore, the condition

$$
\begin{equation} \tag{2.37}
\begin{split}
z^\top Q_0 \left( \frac{(I_n - G_d) z z^\top P}{z^\top P G_d z} + I_n \right) f \left( \frac{z}{\sqrt{z^\top P z}} \right) < 0 \quad \text{if } z \neq \mathbf{0}, \\
PG_d + G_d^\top P > 0, \quad P > 0, \quad Q_0 > 0
\end{split}
\end{equation}
$$

derived from (2.36) is just sufficient stability condition. For $Q_0 = P$ (or,
equivalently, $\Xi = I_n$) we derive the more simplified stability condition

$$
z^T P f \left( \frac{z}{\sqrt{z^\top P z}} \right) < 0.
$$

**Finite-time Stabilization of a Linear System in R<sup>n</sup>**

As an example to both Theorem 2.2 and Theorem 2.3 let us consider the system

$$
\dot{\sigma} = A\sigma + Bu(\sigma), \quad A = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ 0 & 0 & 0 & \cdots & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}, \quad (2.38)
$$

$$
u(\sigma) = \sum_{i=1}^{n} \frac{k_i \sigma_i}{N(\sigma)^{n-i+1}}, \quad \sigma = (\sigma_1, \sigma_2, \dots, \sigma_n)^{\top} \in \mathbb{R}^n, \quad (2.39)
$$

and the weighted dilation

$$
\mathbf{d}(s) = e^{G_d s} \quad \text{with} \quad G_d = \begin{pmatrix} n & 0 & \dots & 0 & 0 \\ 0 & n-1 & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & 2 & 0 \\ 0 & 0 & \dots & 0 & 1 \end{pmatrix},
$$
where $N: \mathbb{R}^n \to \mathbb{R}$ is a continuous positive definite function that is $\mathbf{d}$-homogeneous of degree 1, i.e. $N(\sigma) > 0$ for $\sigma \neq 0$ and

$$N(\mathbf{d}(s)\sigma) = e^s N(\sigma), \quad \sigma \in \mathbb{R}^n, \quad s \in \mathbb{R}.$$

In the view of Definition 2.8 the function $N$ is simply an arbitrary homogeneous norm and the control law can be rewritten in the form

$$u(\sigma) = K \mathbf{d}(-\ln N(\sigma)) \sigma,$$

where

$$K = (k_1 \ k_2 \ \dots \ k_{n-1} \ k_n)^{\top} \in \mathbb{R}^n$$

is a vector of control parameters. Obviously, if $N$ is a canonical homogeneous norm then the considered control law becomes the form (2.31).

The control $u: \mathbb{R}^n \to \mathbb{R}$ is $\mathbf{d}$-homogeneous of degree zero:

$$\begin{aligned} u(\mathbf{d}(s)\sigma) &= K \mathbf{d}(-\ln N(\mathbf{d}(s)\sigma))\mathbf{d}(s)\sigma = K \mathbf{d}(-\ln(e^s N(\sigma)))\mathbf{d}(s)\sigma \\ &= K \mathbf{d}(-s - \ln N(\sigma))\mathbf{d}(s)\sigma = K \mathbf{d}(-\ln N(\sigma))\mathbf{d}(-s)\mathbf{d}(s)\sigma = u(\sigma), \end{aligned}$$

$\sigma \in \mathbb{R}^n \setminus \{\mathbf{0}\}, s \in \mathbb{R}$. Therefore, $u$ is a version of the so-called **quasi-continuous homogeneous sliding mode control** [42]. Indeed, it is homogeneous of degree 0, continuous outside the origin and globally bounded:

$$\begin{aligned} \|Bu(\sigma)\| &\le \|BK\|_{\mathcal{L}} \|\mathbf{d}(-\ln N(\sigma))\sigma\| \le \|BK\|_{\mathcal{L}} \left\| \mathbf{d} \left(-\ln \frac{N(\sigma)}{\|\sigma\|_d}\right) \right\|_{\mathcal{L}} \|\mathbf{d}(-\ln \|\sigma\|_d)\sigma\| \\ &= \|BK\|_{\mathcal{L}} \left\| \mathbf{d} \left(-\ln \frac{N(\sigma)}{\|\sigma\|_d}\right) \right\|_{\mathcal{L}} \le C < +\infty, \quad \sigma \in \mathbb{R}^n, \end{aligned}$$

since the function $\chi: \mathbb{R}^n \to \mathbb{R}$, $\chi(\sigma) = \frac{N(\sigma)}{\|\sigma\|_d}$ is homogeneous of degree 0, and, consequently, globally bounded. Similarly, for $f(\sigma) = A\sigma + Bu(\sigma)$ we derive

$$\begin{aligned} f(\mathbf{d}(s)\sigma) &= A\mathbf{d}(s)\sigma + Bu(\mathbf{d}(s)) = e^{-s}\mathbf{d}(s)A\sigma + Bu(\sigma) = e^{-s}\mathbf{d}(s)(A\sigma + Bu(\sigma)) \\ &= e^{-s}\mathbf{d}(s)f(\sigma), \quad \sigma \in \mathbb{R}^n \setminus \{\mathbf{0}\}, \quad s \in \mathbb{R}, \end{aligned}$$

i.e. the closed loop system has negative homogeneity degree: $\deg_{\mathbb{F}_d}(f) = -1$.

Let us find some sufficient conditions (to the vector $K$ and to the function $N$) allowing the finite-time stability of the closed-loop system. Selecting $Q_0 = P$ in (2.37) we derive the following sufficient stability condition

$$z^{\top} P \tilde{f}(z) < 0, \quad \tilde{f}(z) = \frac{1}{\|z\|} (Az + BK \mathbf{d}(-\ln \tilde{N}(z))z), \quad z \in \mathbb{R}^n \setminus \{\mathbf{0}\}, \quad (2.40)$$

where $\tilde{N}(z) = N(\frac{z}{\|z\|})$, $\|z\| = \sqrt{z^{\top} P z}$ with $P$ satisfying (2.6). Since $N$ is a continuous and positive definite function then

$$0 < \min_{\xi \in \mathbb{R}^n : \xi^{\top} P \xi = 1} N(\xi) \le \tilde{N}(z) \le \max_{\xi \in \mathbb{R}^n : \xi^{\top} P \xi = 1} N(\xi) < +\infty$$
and there exist numbers $\delta_{\min}$ and $\delta_{\max}$ such that

$$0 < \delta_{\min} \le \tilde{N}(z) \le \delta_{\max}. \qquad (2.41)$$

Obviously, if $N(\sigma) = \|\sigma\|_{\mathbf{d}}$ is the canonical homogeneous norm then $\tilde{N}(z) = 1, z \in \mathbb{R}^n \setminus \{\mathbf{0}\}$ and the stability condition becomes

$$z^T P(A + BK)z < 0, \quad z \in \mathbb{R}^n \setminus \{\mathbf{0}\},$$

i.e., K can be selected as a gain matrix of a linear stabilizing feedback. Since the pair $\{A, B\}$ is controllable then appropriate K and P $\succ 0$ can always be found (see, e.g. [11]) to guarantee the required inequality, while the parameter $\gamma$ can be selected sufficiently large to fulfill (2.6). Moreover, Theorem 2.3 implies that $\|\cdot\|_{\mathbf{d}}$ is the Lyapunov function to the closed-loop system.

Therefore, the usage of the canonical homogeneous norm allows the very simple rule to be applied for selection of control parameters. However, the canonical homogeneous $\|\cdot\|_{\mathbf{d}}$ is defined implicitly, and for simplicity of practical realization it is worth to know if $\|\cdot\|_{\mathbf{d}}$ can be replaced with some explicitly defined homogeneous norm N (see also [61], [77]).

**Proposition 2.6.** [55] Let for some $0 < \delta_{\min} \le \delta_{\max}$ the system of LMIs

$$\left(
\begin{array}{ccc}
AX+XA^{\top}+BY+Y^{\top}B^{\top}-X & \sqrt{\ln\left(\frac{\delta_{\max}}{\delta_{\min}}\right)} BY & Xd\left(\ln\frac{\delta_{\max}}{\delta_{\min}}\right) \\
\sqrt{\ln\left(\frac{\delta_{\max}}{\delta_{\min}}\right)} Y^T B^T & -(G_d^{-1}X+XG_d^{-1}) & \mathbf{0} \\
d\left(\ln\frac{\delta_{\max}}{\delta_{\min}}\right) X & \mathbf{0} & -X
\end{array}
\right) < 0, \quad X \succ 0, \qquad (2.42)$$

be feasible with respect to $X \in \mathbb{R}^{n \times n}$, $Y \in \mathbb{R}^{1 \times n}$.

If a function $N: \mathbb{R}^n \setminus \{\mathbf{0}\} \to \mathbb{R}$ is continuous, positive definite, $N \in \mathbb{H}_d(\mathbb{R}^n)$, $\deg_{H_d}(N) = 1$ and it satisfies the inequalities (2.41) with $P = X^{-1}$ and $0 < \delta_{\min} \le \delta_{\max}$ then the closed-loop system (2.38), (2.39) with

$$K = YX^{-1}d(\ln \delta_{\max})$$

is globally uniformly finite-time stable and the canonical homogeneous norm $\|\cdot\|_{\mathbf{d}}$ is its Lyapunov function.

The system of LMIs (2.42) is always feasible provided that $\delta_{\min}$ and $\delta_{\max}$ are sufficiently close to each other. Indeed,

$$\ln(\delta_{\max}/\delta_{\min}) \to 0$$

and

$$d(\ln(\delta_{\max}/\delta_{\min})) \to I_n$$

as $\delta_{\max}/\delta_{\min} \to 1$, so the LMI (2.42) becomes feasible.
An explicitly-defined homogeneous norm N can be selected according to [25], [4] or [77]. Also it can be defined as follows

$$N(\sigma) = (\phi^{\top}(\sigma)Q\phi(\sigma))^{\frac{1}{2n}}, \qquad (2.43)$$

where $Q \in \mathbb{R}^{n \times n}$ is a positive definite matrix and

$$\phi(\sigma) = \begin{pmatrix} |\sigma_1|^{r_1} \operatorname{sign}(\sigma_1) \\ \vdots \\ |\sigma_n|^{r_n} \operatorname{sign}(\sigma_n) \end{pmatrix}, \quad r_i = \frac{n}{n-i+1}, \quad i=1, 2, \dots, n.$$

The only problem is to select N such that (2.41) is fulfilled.

To illustrate the proposition 2.6 we consider the system (2.38), (2.39), (2.43) for $n = 2$. Selecting $\delta_{\min} = 1$, $\delta_{\max} = 1.4$ we solve the LMI (2.42):

$$P = \begin{pmatrix} 1.1214 & 0.4247 \\ 0.4247 & 0.2327 \end{pmatrix} \quad \text{and} \quad K = \begin{pmatrix} -10.8695 & -6.2376 \end{pmatrix}.$$

One can be checked numerically (e.g. using a sufficiently dense grid on the unit sphere $\{z \in \mathbb{R}^n : z^\top P z = 1\}$) that the homogeneous norm (2.43) with

$$Q = \begin{pmatrix} 1.9403 & 0.4410 \\ 0.4410 & 0.1330 \end{pmatrix}$$

satisfies $1 \le N(z) \le 1.4$ for $z^\top P z = 1$, i.e. the condition (2.41) holds.

The similar control design scheme can be developed for an arbitrary matrix A (see, e.g. [62]) provided that the pair $\{A, B\}$ is controllable.

## Dynamic Feedback for Non-Linear Plants

Let us consider now the problem of homogeneous stabilization of the (2.30) in the disturbance free case ($\omega = 0$), i.e.

$$\dot{\sigma}(t) = g(\sigma(t), u(t)),$$

by means of the dynamical feedback (2.31):

$$\dot{u}(t) = k(\sigma(t), u(t)).$$

Based on the scheme of the universal stabilizing control design given in [72] we derive the following corollary of Theorem 2.3.

**Theorem 2.4 (On Homogeneous Dynamical Feedback).** [55] Let $d_x$ and $d_u$ be dilations in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively,

$$d := \begin{pmatrix} d_x(s) & 0 \\ 0 & d_u(s) \end{pmatrix} \quad \text{and} \quad \tilde{f} = \begin{pmatrix} g \\ 0 \end{pmatrix} \in \mathbb{F}_d(\mathbb{R}^{n+m}).$$
The origin of the system (2.30) is globally asymptotically stabilizable by means of the homogeneous dynamical feedback (2.31) with

$$ \begin{pmatrix} g \\ k \end{pmatrix} \in \mathbb{F}_d(\mathbb{R}^{n+m}) $$

if and only if there exist a number $\gamma \ge 0$, a symmetric matrix $P \in \mathbb{R}^{(n+m) \times (n+m)}$ satisfying (2.6) and a map $\Xi \in C^\infty(\mathbb{R}^{n+m} \setminus \{0\}, \mathbb{R}^{(n+m) \times (n+m)})$:

$$ \det(\Xi(z)) \neq 0, \quad \frac{\partial \Xi(z)}{\partial z_i} z = 0, \quad \Xi(e^s z) = \Xi(z) $$

$$ \text{for } z = (z_1, \dots, z_{n+m}) \in \mathbb{R}^{n+m} \setminus \{0\}, s \in \mathbb{R}, i = 1, \dots, n+m $$

such that

$$ a(z) < \gamma \sqrt{b^\top(z)b(z)} \quad \text{for } z \in S, \qquad (2.44) $$

where

$$ \begin{aligned} a(z) &= z^\top W(z) \tilde{f}(z), & b^\top(z) &= z^\top W(z) \begin{pmatrix} 0 \\ I_m \end{pmatrix}, \\ W(z) &= \Xi^\top(z) P \Xi(z) \begin{pmatrix} \frac{(I_n - G_d)z^\top z P}{z^\top P G_d z} + I_n \\ 0 \end{pmatrix} \end{aligned} $$

and $S$ is the unit sphere in $\mathbb{R}^{n+m}$ with $\|z\| = \sqrt{z^\top P z}$. Moreover, the corresponding stabilizing homogeneous feedback law can be designed as follows

$$ k(\xi) = \|\xi\|_{d}^{\text{deg}_d(\tilde{f})} \mathbf{d}_u (\ln \|\xi\|_d) k_0(\mathbf{d}(-\ln \|\xi\|_d)\xi) \quad \text{with} \quad \xi = \begin{pmatrix} \sigma \\ u \end{pmatrix}, \qquad (2.45) $$

where

$$ k_0(\cdot) = \begin{cases} -\frac{a(\cdot) + \sqrt{a^2(\cdot) + (b^\top(\cdot)b(\cdot))^2}}{b^\top(\cdot)b(\cdot)} b(\cdot) & \text{if } b(\cdot) \neq 0 \\ 0 & \text{if } b(\cdot) = 0 \end{cases} \qquad (2.46) $$

and $\|\cdot\|_d$ is the canonical homogeneous norm induced by $\|\xi\| = \sqrt{\xi^\top P \xi}$.

The control design scheme consists of two steps: first, we design a stabilizing control using Sontag's formula (2.46), next, we apply the homogeneous dilation in order to derive the homogeneous stabilizing control (2.45) allowing the closed-loop system to be homogeneous with a required degree.

For $\Xi(z) = I_n$ the inequality (2.44) gives the following sufficient condition of quadratic stabilizability:

$$ \begin{aligned} z^\top P \tilde{f} \left( \frac{z}{\sqrt{z^\top P z}} \right) &\le \gamma \sqrt{z^\top P \begin{pmatrix} 0_{n \times n} & 0_{n \times m} \\ 0_{m \times n} & I_m \end{pmatrix} P z}, & z \ne 0 & (2.47) \\ PG_d + G_d^\top P > 0, & P > 0. \end{aligned} $$

Taking into account Corollary 2.3 we derive criterion of finite-time stabilizability of the system (2.30) by means of the feedback (2.31).
**Example 2.15** (Quadratic Dynamical Stabilization in a Finite Time). Let us consider the stabilization problem for the system

$$ \dot{\sigma}_1 = |\sigma_1|^{\frac{1}{3}} u + \sigma_2, \quad \dot{\sigma}_2 = u, \qquad (2.48) $$

where $\sigma_1, \sigma_2, u \in \mathbb{R}$. The vector field $\tilde{f}: \mathbb{R}^3 \to \mathbb{R}^3$ defined by

$$ \tilde{f}(\xi) = \begin{pmatrix} |\xi_1|^{\frac{1}{3}} \xi_3 + \xi_2 \\ \xi_3 \\ 0 \end{pmatrix}, $$

$\xi = (\xi_1, \xi_2, \xi_3)^\top = (\sigma_1, \sigma_2, u)^\top$ is **d**-homogeneous of degree $-1$ with respect to the dilation **d** in $\mathbb{R}^3$ generated by $G_d = \text{diag}\{3, 2, 1\}$. Hence, the sufficient stabilizability condition (2.47) becomes

$$ z^\top P A(z) z < \gamma \sqrt{z^\top P z} \sqrt{z^\top P e_3 e_3^\top P z}, $$

$P > 0, PG_d + G_d^\top P > 0$, where $z = (z_1, z_2, z_3)^\top \in \mathbb{R}^3$ and

$$ A(z) = \begin{pmatrix} 0 & 1 & \left( \frac{|z_1|}{\sqrt{z^\top P z}} \right)^{\frac{1}{3}} \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}. $$

Let $X \in \mathbb{R}^{3 \times 3}$ and $\gamma_0 > 0$ satisfy the linear matrix inequalities

$$ X A_i + A_i^\top X - 2\gamma e_3 e_3^\top < 0, \quad X G_d + G_d X > 0, \quad X > 0, $$

$A_i = \begin{pmatrix} 0 & 1 & i \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, i = 0, 1, e_3 = (0, 0, 1)^\top \in \mathbb{R}^3$. Since $A(z) = \alpha(z)A_1 + (1 - \alpha(z))A_0$, where $\alpha(z) = \left( \frac{|z_1|}{\sqrt{z^\top P z}} \right)^{\frac{1}{3}}$ and $\alpha(z) \in [0, 1]$ provided that $p_{11} = 1$, then for $P = \tilde{P}/(\tilde{p}_{11})$, $\tilde{P} = X^{-1}$ the stabilizability condition (2.47) holds. So, the considered system can be stabilized in a finite time (due to $\text{deg}_d(f) = -1$) to zero by means of the **d**-homogeneous dynamical feedback

$$ \dot{u} = k_0 (d(-\ln \|\xi\|_d)\xi), \qquad (2.49) $$

where $k_0$ is given by (2.46) with $a(z) = z^\top P \tilde{f}(z)$ and $b(z) = p_{13}z_1 + p_{23}z_2 + p_{33}z_3$. For example, an appropriate matrix $P$ obtained using the above LMIs is

$$ P = \begin{pmatrix} 1.0000 & 0.8930 & 0.8846 \\ 0.8930 & 1.8679 & 1.3659 \\ 0.8846 & 1.3659 & 1.7523 \end{pmatrix}. $$

**Remark 2.3.** The canonical homogeneous norm is defined implicitly by the formula (2.8). Therefore, to implement the designed control law either the canonical homogeneous norm must be calculated numerically on-line (like in [61], [62]) or it must
be approximated with some explicitly defined homogeneous norm N (for example, using the formula (2.43) or the results of [77]). The construction of an appropriate approximation N of the canonical homogeneous norm ‖·‖, in the general case, is a non-trivial theoretical problem needed to be studied in future.

The problems of robustness analysis and sliding mode control design using the presented methodology are studied in [55].

### 2.3.3 Homogeneous Observers in $\mathbb{R}^n$

Let us consider the perturbed linear control system:

$$ \begin{cases} \dot{x}(t) = Ax(t) + Bu(t), \\ y(t) = Cx(t), \end{cases} \qquad (2.50) $$

where $x \in \mathbb{R}^n$ is the state variable, $y \in \mathbb{R}^k$ is the measured output, $u: \mathbb{R} \to \mathbb{R}^s$ is the control input, $A \in \mathbb{R}^{n \times n}$ is the system matrix, $B \in \mathbb{R}^{n \times s}$ is the matrix of input gains and the matrix $C \in \mathbb{R}^{k \times n}$ is the output matrix which links the measured outputs to the state variables. The pair $(A, C)$ is assumed to be observable and $rank(C) = k$.

The goal is to design two dynamic observers that estimate the state of the non-perturbed system (2.50) in a finite time (or in a fixed, defined a priori, time under the assumption that the domain of initial conditions is unknown).

The observers' design starts with a decomposition of the considered system into an appropriate block canonical form.

**Lemma 2.3.** Consider the system (2.30) with the pair $(A, C)$ being observable and rank$(C) = k$. Then there exists a nonsingular transformation $\Phi$ such that

$$ \Phi A \Phi^{-1} = F \tilde{C} + \tilde{A}, \quad C \Phi^{-1} = (c_0 \ 0 \ \dots \ 0), $$

$$ \tilde{C} = [I_k \ 0] \in \mathbb{R}^{k \times n}, \quad \tilde{A} = \begin{pmatrix} 0 & A_{12} & 0 & \cdots & 0 \\ 0 & 0 & A_{23} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & A_{m-1,m} \\ 0 & 0 & 0 & \cdots & 0 \end{pmatrix}, \quad C_0 \in \mathbb{R}^{k \times k}, \quad F \in \mathbb{R}^{n \times k}, $$

where $m$ is an integer, $A_{j-1,j} \in \mathbb{R}^{n_{j-1} \times n_j}$, $n_j = \text{rank}(A_{j-1,j})$, $j = 2, \dots, m$, so that $n_1 = \text{rank}(C) = k$ and $\sum_{i=1}^m n_i = n$.

This lemma is a trivial consequence of well known results on block observability and controllability forms (see [75], [18], [19], [45]).

If $k = 1$ then $m = n$, $n_i = 1$ and $\Phi$ transforms the matrix $A$ into the canonical Brunovsky form. It is also worth stressing that canonical forms and related transformations also exist for nonlinear systems (see for instance [31], [35]). Therefore, the observer design algorithms given below can be adapted to the nonlinear case.

Let us consider the following nonlinear observer

$$ \frac{d}{dt} \hat{x}(t) = A\hat{x}(t) + Bu(t) - g_{FT}(y(t) - C\hat{x}(t)), \qquad (2.51) $$
where $\hat{x}(t) \in \mathbb{R}^n$ is the observer state vector and the function $g_{FT}: \mathbb{R}^k \to \mathbb{R}^n$ is defined as

$$g_{FT}(\sigma) = \Phi^{-1} \left[ \mathbf{d}_{\tilde{r}} \left( \ln \| \tilde{P} C_0^{-1} \sigma \|^{-1} \right) L_{FT} - F \right] C_0^{-1} \sigma, \quad (2.52)$$

where $\sigma \in \mathbb{R}^k$, the matrices $\Phi \in \mathbb{R}^{n \times n}$, $C_0 \in \mathbb{R}^{k \times k}$ and $F \in \mathbb{R}^{n \times k}$ are defined in Lemma 2.3, $\mathbf{d}_{\tilde{r}}(\cdot)$ is the weighted dilation

$$\mathbf{d}(s) = e^{\text{diag}(r)s} \in \mathbb{R}^{n \times n}$$

with

$$\tilde{r} = \left( \frac{\mu}{1+(m-1)\mu}, \frac{2\mu}{1+(m-1)\mu}, \dots, \frac{m\mu}{1+(m-1)\mu} \right)^T, \mu \in (0, 1], \quad (2.53)$$

and $L_{FT} \in \mathbb{R}^{n \times k}$ and $\tilde{P} \in \mathbb{R}^{k \times k}$ are matrices of observer gains, to be determined. The error equation has the form

$$\dot{e} = (\tilde{A} + \mathbf{d}_{\tilde{r}} (\ln \| \tilde{P} \tilde{C} e \|^{-1}) L_{FT} \tilde{C}) e, \quad (2.54)$$

where $e = \Phi(x - \hat{x})$, $\tilde{A} \in \mathbb{R}^{n \times n}$ and $\tilde{C} \in \mathbb{R}^{k \times n}$ are defined in Lemma 2.3. Obviously, if $\mu = 0$ then $\mathbf{d}_{\tilde{r}}(s) = I_n$ for all $\sin \mathbb{R}$ and the presented observer becomes the classical Luenberger one.

**Remark 2.4.** If the term $\| \tilde{P} C_0^{-1} \sigma \|$ in (2.52) is replaced with $\epsilon^{(1+(m-1)\mu)/\mu}$ where $\epsilon > 0$ is a small constant, then the system (2.51)-(2.52) becomes a high-gain observer [65], [35] with the error dynamics given by

$$\dot{e} = (\tilde{A} + \text{diag}(\epsilon^{-1}I_{n_1}, \epsilon^{-2}I_{n_2}, \dots, \epsilon^{-m}I_{n_m})L_{FT}\tilde{C})e.$$

In our algorithms the gain factor $\epsilon$ depends on the available part of the observation error, namely, on $\sigma = y - C\hat{x} = Ce$. This allows the finite-time and fixed-time observers to be less sensitive with respect to noises.

Let us denote

$$H_r = \text{diag}(r_1 I_{n_1}, r_2 I_{n_2}, \dots, r_m I_{n_m}) \in \mathbb{R}^{n \times n},$$

$$r = \left[ 1 + \frac{\mu}{1 + (m-1)\mu} \right] (1, 1, \dots, 1)^{\top} - \tilde{r}$$

and

$$\Xi(\lambda) = \lambda(D_{\tilde{r}}(\lambda^{-1}) - I_n).$$

**Theorem 2.5.** [44] Let for some $\mu \in (0, 1]$, $\alpha > 0$, $\xi > 0$ and $\tau \ge 1$ the system of matrix inequalities

$$\left( \begin{array}{cc} P\tilde{A} + \tilde{A}^T P + \tilde{C}^T Y^T + Y\tilde{C} + \xi P + \alpha(PH_r + H_r P) & P \\ P & -\xi Z \end{array} \right) \preceq 0, \quad (2.55a)$$

$$P \succ 0, \quad Z \succ 0, \quad X \succ 0, \quad (2.55b)$$
$$ \begin{pmatrix} \tau X & Y^T \\ Y & P \end{pmatrix} \succeq 0, \qquad (2.55c) $$

$$ PH_r + H_r P \succ 0, \qquad (2.55e) $$

$$ P \succeq \tilde{C}^T X \tilde{C}, \qquad (2.55d) $$

$$ \Xi(\lambda)Z\Xi(\lambda) \preceq \frac{1}{\tau}P, \quad \forall \lambda \in [0, 1], \qquad (2.55f) $$

be feasible for some $P, Z \in \mathbb{R}^{n \times n}, Y \in \mathbb{R}^{k \times n}$ and $X \in \mathbb{R}^{k \times k}$. Then the error equation (2.54) with $L_{FT} = P^{-1}Y$ and $\dot{P} = X^{1/2}$ is globally finite-time stable with settling time

$$ T \le \frac{\|e(0)\|_d^\rho}{\alpha \rho}, \qquad \rho = \frac{\mu}{1+(m-1)\mu}. $$

In other words, this theorem claims that any solution of the observer system (2.51) converges to a solution of the real system (2.30) in a finite time $T$, which is dependent on the initial estimation error $e(0) \in \mathbb{R}^n$. As before the main idea of the proof is to show that the canonical homogeneous norm $\| \cdot \|_d$ is a Lyapunov function of the error equation.

**Corollary 2.6.** The system of matrix inequalities (2.55) is feasible for sufficiently small $\mu > 0$.

Indeed, observability of the pair $(A, C)$ implies that the pair $(\tilde{A} + 0.5(\alpha H_r + \xi I_n), \tilde{C})$ is also observable. Hence, it can be easily shown that the inequality (2.55a) is feasible with some positive definite matrices $P, Z \in \mathbb{R}^{n \times n}$ and $Y \in \mathbb{R}^{k \times n}$. The matrix inequalities (2.55d) and (2.55c) are also feasible for some $X \in \mathbb{R}^{k \times k}$ and sufficiently large $\tau \ge 1$. Since $\|\Xi(\lambda)\| \to 0$ uniformly on $\lambda \in [0, 1]$ as $\mu \to 0$, then the inequalities (2.55e), (2.55f) will hold for sufficiently small $\mu > 0$.

**Proposition 2.7.** The parametric inequality (2.55f) holds if

$$ \Xi(q_i)Z\Xi(q_i) + \frac{1}{2}(q_i^2 - q_{i-1}^2)M \prec \frac{1}{\tau}P, \quad i = 1, \dots, N, \qquad (2.56a) $$

$$ P \succ 0, \quad Z \succ 0, \quad M \succ 0 \qquad (2.56b) $$

$$ \left( \begin{array}{cc} (I_n - H_{\tilde{r}})Z + Z(I_n - H_{\tilde{r}}) & -ZH_{\tilde{r}} \\ -H_{\tilde{r}}Z & M \end{array} \right) > 0, \qquad (2.56c) $$

where $0 = q_0 < q_1 < \dots < q_N = 1$, $H_{\tilde{r}} = \text{diag}(\tilde{r}_1 I_{n_1}, \tilde{r}_2 I_{n_2}, \dots, \tilde{r}_m I_{n_m})$, $P, M, Z \in \mathbb{R}^{n \times n}$.

The provided result allows the implementation of a simple algorithm to solve the parametrized system of matrix inequalities (2.55) with fixed $\alpha, \xi, \tau, \mu$.

**Algorithm 2.1.** [44]

**Initialization:** $N = 1, q_0 = 0, q_N = 1, \Sigma = \{q_0, q_N\}$.

**Loop:** While the system of LMI (2.55a)-(2.55d), (2.56a)-(2.56c) is not feasible, do
$\Sigma \leftarrow \Sigma \cup \left\{ \frac{q_{i-1}+q_i}{2} \right\}_{i=1}^N$ and $N \leftarrow 2N$.
Since the matrix inequality $(I_n-H_{\tilde{r}})Z+Z(I_n-H_{\tilde{r}})>0$ is obviously feasible for sufficiently small $\mu > 0$, then, in the view of Corollary 2.6, the presented algorithm always finds the required solution if $\mu$ is sufficiently small.

The fixed-time observer for linear plants is designed similarly in [44]. It is based on combination of systems with positive and negative degrees.

**Example 2.16.** The numerical simulations aim to show the main properties of the finite-time observers. In the example, an inverted-cart pendulum model will be used. The robustness of the finite-time observer will be studied by applying it to the nonlinear plant and a comparison with a high-gain observer will be included.

The state vector is given by $\mathbf{x} = [x, \dot{x}, \theta, \dot{\theta}]^T$, where as usual $(x, \dot{x})$ represents the position and the velocity of the cart and $(\theta, \dot{\theta})$ the angle (from the vertical down position) and the angular velocity of the pendulum. The model parameters are $M = 0.5$ Kg - mass of the cart, $m_c = 0.2$ Kg - mass of the pendulum, $b = 0.1$ N/m/s - cart friction coefficient, $l = 0.3$ m - length to pendulum center of mass, $I = 0.006$ Kg - moment of inertia of the pendulum.

It is assumed that only position and angle can be measured directly. The nonlinear equations describing the system motion are given by

$$ (M + m_c)\ddot{x} + b\dot{x} + m_c - l\ddot{\theta} \cos \theta - m_c l \dot{\theta}^2 \sin \theta = F_{in} \\ (I + m_c l^2)\ddot{\theta} + m_c g l \sin \theta + m_c l \ddot{x} \cos \theta = 0, $$

where $F_{in}$ represents the input force. A simple proportional control law $F_{in} = K_P \frac{d}{dt} \dot{\theta}(t)$ is used to stabilize the pendulum around the downward position, $\frac{d}{dt} \dot{\theta}(t)$ is an estimate of the angular velocity to be obtained using the observers. The linearization of the model around the downward equilibrium point $\mathbf{x}_0 = 0 \in \mathbb{R}^4$ gives the following parameters for (2.30):

$$ A = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & -0.1818 & 2.672 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0.4545 & -31.181 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 \\ 1.818 \\ 0 \\ -4.545 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}. $$

The first step of the observer design is to transform the linearized model into the observable canonical form, given in Lemma 2.3, by obtaining the matrix

$$ \Phi = \begin{pmatrix} 0 & 0 & 1 & 0 \\ -1 & 0 & 0 & 0 \\ a & -1 & 0 & 0 \\ b & 0 & 0 & 1 \end{pmatrix}, \text{ where } a = -0.1818 \text{ and } b = -0.4545. $$

**Comparison between finite-time and high-gain observers.** In order to make a fair comparison we adjusted the parameters of both observers to have similar time response for the initial condition $x_0 = (0, -2, \pi/4, 1)$; that is to say, the norm of the estimation errors is admitted to be less than 0.15 for $t \ge 0.5$ s. Using Algorithm 2.1 with $(\mu, \alpha, \tau, \xi) = (0.25, 2.5, 100, 4)$ we design the finite-time observer (2.51), (2.52) with

$$ L_{FT} = \begin{pmatrix} -9.9080 & 0 & 0 & -46.8452 \\ 0 & -9.9080 & -46.8452 & 0 \end{pmatrix}^\top, \quad \tilde{P} = 0.0969 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, $$

and we compare it with the high-gain observer, designed according to Remark 2.4. Namely, the term $||\tilde{P}C_0^{-1}\sigma||$ in (2.52) is replaced with $\epsilon^{(1+(m-1)\mu)/\mu}$.
Note that, the gain factor $\epsilon = 0.3$ (and correspondingly the admissible estimation error $\|e(0.5)\| \le 0.15$) is selected sufficiently large since the high-gain observer becomes more sensitive with respect to measurement noises as $\epsilon$ decreases. The estimation error of the finite-time observer turned out to be 10 times less<fnref n="3" /> ($\|e(t)\| \le 0.01$) for $t \ge 0.5$ s.

Figure 2.2 depicts the evolution of the observation errors of finite-time and high-gain observers for the noise-free case. Although the observers were designed using a linearized model, the system remains stable for the complete nonlinear model. It is also worth noting that the finite-time observer demonstrates a smaller peaking during transients.

To compare the observers for the case of noisy measurements, a band-limited white noise of power $10^{-5}$ has been added to the output signal during the simulation. The corresponding results are presented at Figure 2.3. They show almost twice better precision (in both $L^2(0.5,1.5)$ and $L^\infty(0.5,1.5)$ norms of the error) of the finite-time observer with respect to the high-gain one. This fact has a simple explanation in the context of high-gain observer theory: since the gain factor $\epsilon$ of the finite-time observer depends on the available part of the observation error (see Remark 2.4), namely,

$$ \epsilon = \epsilon(\sigma) = \|\tilde{P}C_0^{-1}\sigma\|^{\mu/(1+(m-1)\mu)}, \quad \sigma = y - C\hat{x} = Ce, $$

then its value is automatically adapted to the noises of different magnitude (the larger the noise magnitude, the smaller the gain). In the noisy case the convergence time of the finite-time observer slightly increases, allowing a better estimation precision.

## 2.4 Conclusions

In this chapter it is shown that the method of linear dilation and the canonical homogeneous norm simplify the analysis and design of homogeneous control systems. In particular, they allow simple extensions of linear control and state estimation algorithms to the non-linear homogeneous cases. Notice that the canonical homogeneous norm implicitly appears in the theory of controllability functions introduced in [37]. It also corresponds the radius vector of "homogeneous" spheric coordinates studied in [64]. The canonical homogeneous norm also defines the implicit Lyapunov functions considered in [1], [61], [62]. It allows to unify the finite-time control design methodology for linear models in both finite and infinite dimensional spaces. A further elaboration of this method looks promising for other control problems (like observer design and robustness analysis for evolution equations in Hilbert spaces, consistent discretization of homogeneous models, etc).

<footnote n="3">³The numerical simulations have been done using the explicit Euler method with sampling period $10^{-4}$.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 2.2: Simulation plots of $e_x$ and $e_\theta$ for the HG and finite-time observers applied to the nonlinear plant with $\text{IC } x(0) = (0, -2, \frac{\pi}{4}, 1)$ and $K_p = 10$.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 2.3: Simulation plots of $e_x$ and $e_\theta$ for the high-gain and finite-time observers with a band-limited white noise in the measurements.</figcaption></figure>
# Chapter 3

# Future Directions of Research

## 3.1 Fundamental research

### 3.1.1 Short term perspective

In a short term perspective I will concentrate on the problems announced in the actual research projects, in which I am involved (e.g. AND DIGITSLID, ANR Finite4SoS, CPER DATA, CPET ELSAT2020).

**Consistent Discretization of Homogeneous Systems in $\mathbb{R}^n$**

The discretization issues are very important for practical implementation of homogeneous algorithms in a digital control device. The development of discretization schemes for homogeneous systems, which would preserve the convergence properties of the continuous system, is a difficult task. Some of my recent results [59] gives us a possible way for successful solving of this problem in a short term perspective. The proposed scheme is based on a transformation of the original homogeneous model to an equivalent one, which admits consistent implicit or semi-implicit discretization. It has been tested that this approach allows to reject completely the numerical chattering of homogeneous sliding mode controllers and preserve finite-time/fixed-time stability in some systems with a sampled implementation of control laws. Extension of this results to the general case are expected to be provided in a short term perspective. This research direction is proposed by ANR DIGITSLID.

**Homogeneous observers for linear systems in $\mathbb{B}$**

Development of a unified approach to the design of finite-time control and estimation algorithms for systems modeled by ODEs, PDEs, time-delay equations, differential inclusions is planned in the context of ANR Finite4SoS. The simplest
way to cover all this models is to deal with abstract evolution equations in a Banach space. The finite-time control design has been presented in the previous chapter. Using the Kalman duality principle for linear system and homogeneity ideas, a universal finite-time observer for evolution systems is planned to be developed in a short term perspective.

## Quasi-Homogeneity and Homogeneous Extensions

The homogeneous systems cover rather large class of the nonlinear models. However, the conventional assumptions on the dilation symmetry and the conventional symmetry (homogeneity) relations are still very restrictive. It is rather difficult to use homogeneous arguments for finite-time boundary control of PDEs. For example, an infinite chain of integrators may appear as an approximated model of a boundary controlled heat system. It does not admit a well defined weighted dilation with respect to which the homogeneity property can be established. However, some quasi-homogeneity relations like

$$f(\mathbf{d}_1(s)u) = \mathbf{d}_2(s)f(u), \quad f: \mathcal{D}(f) \subset \mathbb{B} \to \mathbb{B}, \quad s \in \mathbb{R}, \quad u \in \mathcal{D}(f)$$

where $\mathbf{d}_1$ and $\mathbf{d}_2$ are some dilations in $\mathbb{B}$, are fulfilled. Under additional conditions on the dilations $\mathbf{d}_1$ and $\mathbf{d}_2$, some results about finite-time convergence can be proven similarly to the homogeneous case.

To treat this problem the method of homogeneous extensions introduced in the previous chapter can also be utilized in some cases. In the short term perspective the concepts of quasi-homogeneity and homogeneous extensions is expected developed in the context of ANR Finite4SoS.

### 3.1.2 Long term perspective

In a long term perspective my research activities in the context of homogeneous systems can be continued in the following directions.

**Homogeneous controllers and observers for nonlinear systems in $\mathbb{B}$**. Extension of the homogeneous control and estimation algorithms developed for linear plants in $\mathbb{B}$ to some classes of nonlinear evolution systems are planned for a long term perspective.

**Consistent Discretization of Homogeneous Systems in $\mathbb{B}$**. Homogeneity simplifies numerical simulations of the non-linear systems. Indeed, knowing the solutions of some homogeneous systems for initial conditions on a sphere we can easily construct solutions with initial conditions in a whole domain. Due to this symmetry, numerical simulations for some nonlinear infinite dimensional systems can be boosted. The latter is very important, in particular, for control/observation systems based on Navier-Stokes models of the flow, which on-line simulation is currently impossible. The problem of consistent discretization in $\mathbb{B}$ can be also studied in this context (similarly to consistent discretization in $\mathbb{R}^n$).

**Homogeneity in optimal control.** The ideas of the dilation symmetry can be useful
in the optimal control theory. For instance, homogeneity of the system will imply a homogeneity of the Hamiltonian and a homogeneity of the Bellman function. This may lead to a simplification of an optimal control design.

**Homogeneity in learning control.** If it is known (e.g. from physics) that a control plant is homogeneous in a generalized sense, then the learning control design can be essentially using this knowledge. This may imply a homogeneity of a learning function, of the corresponding data utilized for the learning process, etc. This is also an interesting problem to be considered in a long term perspective.

## 3.2 Applications

**Control of flow separation.** Active flow control is a very promising research direction in the transportation area. It is aimed at a drag (and, consequently, consumption) reduction of the vehicles (both flying and terrestrial) by means of some actuators (e.g. air-blowers). This task is also important for increase (or on-line control) of the lift force of an airplane under turbulence (e.g. during take off and landing). I develop flow control algorithms in cooperation with colleagues from ONERA, Centrale Lille, The University of Valenciennes, University of Poitiers. The efficiency of the developed control laws is confirmed by successful experimental applications (see, https://www.youtube.com/watch?v=b5NnAV2qeno). In a short term perspective, an efficiency of the drag reduction by means of the proposed control scheme is planned to be analyzed at University of Valenciennes during real-life experiments with the car Reanult Twingo.

**Fast control of flying vehicles.** The platform for control experimentation with autonomous vehicles is under construction in Inria Lille in the context of the project ControlHub (see below). In particular, it includes flying vehicles such as Q-Drone of Quanser. Application of the control algorithms proposed in the previous chapter is planned to be done using this platform.

**Interconnected systems.** The Internet of Things (IoT) is the union of physical devices embedded with electronics, software, sensors, actuators and network interconnectivity allowing these objects to gather and exchange the information. The interconnected systems are expected to be the main model of a control plant in a recent future. The issues of possible application of homogeneity approach to some real interconnected systems are planned to be considered in long term perspective.

## 3.3 ControlHub Platform

ControlHub is the platform for a rapid prototyping of control algorithms on real physical systems. ControlHub is under construction Inria Lille. I lead this project (http://controlhub.gforge.inria.fr/).

The control theory can be considered as a crossing point between engineering (and other sciences such as biology, economy ...) and mathematics. The control
practice deals with a real plant (e.g. distillation column) and a control device (e.g. digital controller), which sends appropriate signals to the plant in order to govern its behavior, while the mathematical control theory considers its mathematical model (usually a dynamical system) with some input (control signals) and output (sensors measurements) parameters. Control design in the mathematical sense is just a synthesis of an appropriate mathematical law. The practical validation of this law cannot be done without a direct interaction with a real system. Apart of a mathematical study, a lot of engineering activities are required for prototyping of control algorithms. ControlHub automatizes the most of them using the modern achievements of computer science and control engineering. It allows the end-user to have a remote access (i.e. using Internet) to real experimental setups in order to validate mathematical control algorithms realized, for example, in MATLAB.

The project ControlHub illustrates my activities related to an innovative research infrastructure for the control systems.
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# Appendix

## A. Publications

### International journals

IJ1. A. Polyakov, J.-M. Coron, L. Rosier, On Homogeneous Finite-Time Control for Linear Evolution Equation in Hilbert Space, **IEEE Transactions on Automatic Control**, 2018, .

IJ2. A. Polyakov, Sliding Mode Control Design Using Canonical Homogeneous Norm, **International Journal of Robust and Nonlinear Control**, 2018, .

IJ3. F. Lopez Ramirez, D. Efimov, A. Polyakov, W. Perruquetti, Fixed-Time Output Stabilization and Fixed-Time Estimation of a Chain of Integrators, **International Journal of Robust and Nonlinear Control**, 2018, .

IJ4. K. Zimenko, A. Polyakov, D. Efimov, W. Perruquetti, On simple scheme of finite/fixed-time control design, **International Journal of Control**, 2018, .

IJ5. B.H. Du, A. Polyakov, G. Zheng, Q. Quan, Quadrotor trajectory tracking by using fixed-time differentiator, **International Journal of Control**, 2018, .

IJ6. D. Efimov, A. Levant, A. Polyakov, W. Perruquetti, Convergence acceleration for observers by gain commutation, **International Journal of Control**, 2018, .

IJ7. T. Kharkovskaya, D. Efimov, A. Polyakov, J.-P. Richard, Design of interval observers and controls for PDEs using finite-element approximations, **Automatica**, 93(7), 2018, p. 302-310, .
IJ8. Y. Dvir, A. Levant, D. Efimov, A. Levant, A. Polyakov, W. Perruquetti, Acceleration of finite-time stable homogeneous systems, **International Journal of Robust and Nonlinear Control**, 2018, 28(5), p. 1757-1777.

IJ9. G. Zheng, A. Polyakov, A. Levant, Delay estimation via sliding mode for nonlinear time-delay systems, **Automatica**, 2018, 89 (3), p. 266-273, .

IJ10. F. Lopez Ramirez, A. Polyakov, D. Efimov, W. Perruquetti, Finite-time and fixed-time observer design: Implicit Lyapunov function approach, **Automatica**, 2018, 87, p. 52-60, .

IJ11. M. Feingesicht, A. Polyakov, F. Kerherve, J.-P. Richard, SISO model-based control of separated flows: Sliding mode and optimal control approaches, **International Journal of Robust and Nonlinear Control**, 2017, .

IJ12. D. Efimov, A. Levant, A. Polyakov, W. Perruquetti, Supervisory Acceleration of Convergence for Homogeneous Systems, **International Journal of Control**, 2017,

IJ13. K. Zimenko, D. Efimov, A. Polyakov, W Perruquetti, A note on delay robustness for homogeneous systems with negative degree, **Automatica**, 2017, 79(5), p. 178-184, .

IJ14. D. Efimov, A. Polyakov, A. Levant, W. Perruquetti, Realization and Discretization of Asymptotically Stable Homogeneous Systems, **IEEE Transactions on Automatic Control**, 2017, 62(11), p. 5962 - 5969, .

IJ15. D. Efimov, A. Levant, A. Polyakov, W. Perruquetti, Supervisory Acceleration of Convergence for Homogeneous Systems, **International Journal of Control**, 2017, 27(18), p. 5008-5027.

IJ16. A. Polyakov, L. Hetel, Relay Control Design for Robust Stabilization in a Finite-Time, **IEEE Transactions on Automatic Control**, 2017, 62(4), p. 1985-1991

IJ17. S. Zhuk, A. Polyakov, O. Nakonechnyi, Note on Minimax Sliding Mode Control Design for Linear Systems, **IEEE Transactions on Automatic Control**, 2017, 62(7), p. 3395 - 3400

IJ18. H. Rios, M. Mera, D Efimov, A. Polyakov, Robust output control for uncertain linear systems: Homogeneous differentiator-based observer approach, **Int. J. of Rob. and Nonlin. Contr.**, 2017, 27(11), p.1895-1914
IJ19. K. Zimenko, A. Polyakov, D. Efimov, A. Kremlev, Feedback sensitivity functions analysis of finite-time stabilizing control system, **International Journal of Robust and Nonlinear Control**, 2017, 27(15), p. 2475-2491

IJ20. M. Defoort, G. Demesure, Z. Zuo, A. Polyakov, M. Djemai, Fixed-time stabilisation and consensus of non-holonomic systems, **IET Control Theory & Applications**, 2016, 10(18), p. 2497 - 2505,

IJ21. D. Efimov, E. Fridman, A. Polyakov, W. Perruquetti, J.-P. Richard, On design of interval observers with sampled measurement, **Systems & Control Letters**, 2016, 96(10), p. 158-164 (IF: 2.656)

IJ22. A. Polyakov, D. Efimov, E. Fridman, W. Perruquetti, Homogenous Distributed Parameters Systems, **IEEE Transactions on Automatic Control**, 2016, 61(11), p. 3657 - 3662

IJ23. H. Rios, D. Efimov, A. Polyakov, W. Perruquetti, Homogeneous Time-Varying Systems: Robustness Analysis, **IEEE Transactions on Automatic Control**, 2016, 61(12), p. 4075 - 4080,

IJ24. D. Efimov, E. Fridman, A. Polyakov, W. Perruquetti, J.-P. Richard, Linear interval observers under delayed measurements and delay-dependent positivity, **Automatica**, 2016, 72(10), p. 123-130,

IJ25. M. Mera, A. Polyakov, W. Perruquetti, Finite-time attractive ellipsoid method: implicit Lyapunov function approach, **International Journal of Control**, 2016, 89(6), p. 1079-1090,

IJ26. D. Efimov, A. Polyakov, L. Fridman, W. Perruquetti, J.-P. Richard, Delayed sliding mode control, **Automatica**, 2016, 64(2), p. 37-43,

IJ27. A. Polyakov, D. Efimov, W. Perruquetti, Robust stabilization of MIMO systems in finite/fixed time, **International Journal of Robust and Nonlinear Control**, 2016, 26(1), p. 69-90.

IJ28. D. Efimov, A. Polyakov, W. Perruquetti, J.-P. Richard, Weighted Homogeneity for Time-Delay Systems: Finite-Time and Independent of Delay Stability, **IEEE Transactions on Automatic Control**, 2016, 61(1), p. 210-215,

IJ29. A. Polyakov, D. Efimov, W. Perruquetti, J.-P. Richard, Implicit Lyapunov-Krasovski Functionals For Stability Analysis and Control Design of Time-Delay Systems, **IEEE Transactions on Automatic Control**, 2015,
IJ30. M. Defoort, A. Polyakov, G. Demesure, M. Djemai, K. Veluvolu, Leader-follower fixed-time consensus for multi-agent systems with unknown non-linear inherent dynamics, **IET Control Theory & Applications**, 2015,

IJ31. A. Polyakov, D. Efimov, W. Perruquetti, Finite-time and Fixed-time Stabilization: Implicit Lyapunov Function Method, **Automatica**, 2015, 51(1), p.332-340,

IJ32. A. Polyakov, Time-Suboptimal Feedback Design via Linear Matrix Inequalities, *Automation and Remote Control*, 2015, 76 (5), *Special Issue Dedicated to 80th Anniversary of Prof. Boris T. Polyak*,

IJ33. D. Efimov, A. Polyakov, J.-P. Richard, Interval Observer Design for Estimation and Control of Time-Delay Descriptor Systems, *European Journal of Control*, 2015, 23 (5), pp.26-35,

IJ34. A. Polyakov, L. Fridman, Stability Notions and Lyapunov Functions of Sliding Mode Systems, **Journal of the Franklin Institute**, 2014, 351(4), p. 1831-1865, .

IJ35. D. Efimov, A. Polyakov, E. Fridman, W. Perruquetti, J.-P. Richard, Comments on Finite-Time Stability of Time Delay Systems, **Automatica**, 2014, 50(7), p. 1944-1947, .

IJ36. V. Azhmyakov, A. Polyakov, A. Poznyak, Consistent approximations and variational description of some classes of sliding mode control processes, **Journal of The Franklin Institute**, 2014, 351(4), P. 1964-1981,

IJ37. E. Bernuau, D. Efimov, W. Perruquetti and A. Polyakov, On Homogeneity and Its Application in Sliding Mode Control, **Journal of the Franklin Institute**, 2014, 351(4), p. 1866-1901,

IJ38. A. Polyakov, D. Efimov, W. Perruquetti, J.-P. Richard, Output Stabilization of Time-Varying Input Delay Systems Using Interval Observation Technique, **Automatica**, 2013, 49(11), p. 3402-3410,

IJ39. E. Bernuau, A. Polyakov, D. Efimov and W. Perruquetti, Verification of ISS, iISS and IOSS properties applying weighted homogeneity, **System & Control Letters**, 2013, 62(12), p. 1159-1167, (IF: 2.656)

IJ40. A. Polyakov, Nonlinear feedback design for fixed-time stabilization of Linear Control Systems, **IEEE Transactions on Automatic Control**, 2012, 57(8), p. 2106-2110,
IJ41. A. Polyakov, A. Poznyak, Unified Lyapunov function for a finite-time stability analysis of relay second-order sliding mode control systems, *IMA J. of Mathematical Control and Information*, 2012, 29(4), p. 529-550, (IF: 1.034).

IJ42. A. Polyakov, Minimization of disturbances effects in time delay predictor-based sliding mode control systems, *Journal of The Franklin Institute*, 2012, 349(4), p. 1380-1396,

IJ43. A. Polyakov, A. Poznyak, Invariant Ellipsoid Method for Minimization of Unmatched Disturbances Effects in Sliding Mode Control, 2011, Automatica, 47(7), p. 1450-1454,

IJ44. A. Poznyak, A. Polyakov, V. Strygin, Analysis of finite-time convergence by the method of Lyapunov functions in systems with second-order sliding modes, *J. of Applied Mathematics and Mechanics*, 2011, 75(3), p. 289-303.

IJ45. S. Gonzalez-Garcia, A. Polyakov, A. Poznyak, Using the Method of Invariant Ellipsoids for Linear Robust Output Stabilization of Spacecraft, *Automation and Remote Control*, 2011, 72(3), p.540-555,

IJ46. A. Polyakov, A. Poznyak, Method of Lyapunov Function for Systems with Higher-order Sliding Modes, *Automation and Remote Control*, 2011, 72(5), p.944-963,

IJ47. A. Polyakov, On Practical Stabilization of the Systems with Relay Delay Control, *Automation and Remote Control*, 2010, 71(11), p. 2331-2344,

IJ48. A. Polyakov, A. Poznyak, Lyapunov Function Design for Finite-Time Convergence Analysis: "Twisting" Controller for Second Order Sliding Mode Realization, *Automatica*, 2009. 45(2), p. 444-448,

IJ49. V. Azmyakov, R. Galvan-Guerra, A. Polyakov, On the Method of Dynamic Programming for Linear-Quadratic Problems of Optimal Control in Hybrid Systems, *Automation and Remote Control*, 2009, 70(5), p. 787-799,

IJ50. A. Polyakov, A. Poznyak, Reaching Time Estimation for "Super-Twisting" Second Order Sliding Mode Controller via Lyapunov Function Designing, *IEEE Transaction on Automatic Control*, 2009, 54(8), p. 1951-1955,

IJ51. M. Yefremov, A. Polyakov, V. Strygin, An active stabilization algorithm for a spacecraft with viscoelastic elements under uncertainties, *Journal of Applied Mathematics and Mechanics*, 2006, 70(5), p. 800-811.
IJ52. M. Yefremov, A. Polyakov, V. Strygin, A new tracking algorithm for certain mechanical systems, *Journal of Applied Mathematics and Mechanics*, 2005, 69(1), p. 27-37.

IJ53. L. Fridman, V. Strygin, A. Polyakov, Nonlocal Stabilization via Delayed Relay Control Rejecting Uncertainty in a Time Delay, **International Journal of Robust and Nonlinear Control**, 2004, 14(1), p.15-37,

IJ54. P. Acosta, A. Polyakov, L. Fridman, V. Strygin. Estimation of Amplitude of Oscillations in Sliding Mode Systems Caused by Time Delay, **Asian Journal of Control**, 2004, 6(4), p.513-519 (IF: 2.779).

IJ55. L. Fridman, V. Strygin, A. Polyakov, Stabilization of Oscillations Amplitudes via Relay Delay Control, **International Journal of Control**, 2003, 76(8), p. 770-780,

## Reviewed international conferences

IC1. N. Espitia, A. Polyakov, D. Efimov, W. Perruquetti, On continuous boundary time-varying feedbacks for finite-time stabilization of coupled reaction-diffusion systems, **IEEE Conference on Decision and Control**, 2018.

IC2. K. Zimenko, A. Polyakov, D. Efimov, On dynamical feedback control design for generalized homogeneous differential inclusions, **IEEE Conference on Decision and Control**, 2018.

IC3. F. Lopez-Ramirez, D. Efimov, A. Polyakov, W. Perruquetti, On Implicit Finite-Time and Fixed-Time ISS Lyapunov Functions, **IEEE Conference on Decision and Control**, 2018.

IC4. T. Kharkovskaya, D. Efimov, A. Polyakov, E. Fridman, J.-P. Richard, Interval estimation for second-order delay differential equations with delayed measurements and uncertainties, **IEEE Conf. on Decision and Contr.**, 2018.

IC5. J. P. Epperlein, O. V. Iftime, S. Zhuk, A. Polyakov, Asymptotic Output-Feedback Stabilization of Linear Evolution Equations with Uncertain Inputs via Equivalent Control Method, **IEEE Conf. on Decision and Control**, 2018.

IC6. D. Efimov, A. Polyakov, A. Aleksandrov, On State-dependent Discretization of Stable Homogeneous Systems, **IEEE Conference on Decision and Control**, 2018.
IC7. T. Sanchez, A. Polyakov, L. Hetel, E. Fridman, A switching controller for a class of MIMO bilinear time-delay systems, **IEEE International Conference on the Science of Electrical Engineering**, 2018.

IC8. A. Polyakov, D. Efimov, B. Brogliato, Consistent Discretization of Finite-time Stable Homogeneous Systems, **Workshop on Variable Structure Systems**, 2018.

IC9. B. Brogliato; A. Polyakov, D. Efimov, The implicit discretization of the super-twisting sliding-mode control algorithm, **Workshop on Variable Structure Systems**, 2018.

IC10. T. Sanchez, A. Polyakov, J.-P. Richard, D. Efimov, A robust Sliding Mode Controller for a class of SISO bilinear delayed systems, **Workshop on Variable Structure Systems**, 2018.

IC11. K. Zimenko, A. Polyakov, D. Efimov, Finite-Time and Fixed-Time Stabilization for Integrator Chain of Arbitrary Order, **European Control Conference**, 2018.

IC12. K. Zimenko, A. Polyakov, D. Efimov, Generalized Feedback Homogenization and Stabilization of Linear MIMO Systems, **European Control Conference**, 2018.

IC13. F. Lopez-Ramirez, D. Efimov, A. Polyakov, W. Perruquetti, On Necessary and Sufficient Conditions for Fixed-Time Stability of Continuous Autonomous Systems, **European Control Conference**, 2018.

IC14. K. Zimenko, D. Efimov, A. Polyakov, A. Kremlev, Control of Systems with Arbitrary Bounded Input Delay Using Implicit Lyapunov Function Technique, **European Control Conference**, 2018.

IC15. A. Polyakov, Quadratic-like Stability of Nonlinear Homogeneous Functions, **IEEE Conference on Decision and Control**, 12-15 December 2017.

IC16. A. Polyakov, L. Hetel, C. Fiter, Relay Control Design using Attractive Ellipsoids Method, **IEEE Conference on Decision and Control**, 12-15 December 2017.

IC17. T. Sanchez, D. Efimov, A. Polyakov, J. Moreno, W. Perruquetti, A homogeneity property of a class of discrete-time systems, **IEEE Conference on Decision and Control**, 12-15 December 2017.
IC18. A. Polyakov, D. Efimov, E. Fridman, W. Perruquetti, J.-P. Richard, On Hyper Exponential Stabilization of Linear State-Delay Systems, **IEEE Conference on Decision and Control**, 12-15 December 2017.

IC19. A. Polyakov, J.-M. Coron, L. Rosier, On Boundary Finite-Time Feedback Control for Heat Equation, **IFAC World Congress**, 9-14 July 2017.

IC20. M. Feingesicht, A. Polyakov, F. Kerherve, J.-P. Richard, Sliding Mode Control for Turbulent Flows, **IFAC World Congress**, 9-14 July 2017.

IC21. M. Feingesicht, A. Polyakov, F. Kerherve, J.-P. Richard, Model-Based Feedforward Optimal Control applied to a Turbulent Separated Flow, **IFAC World Congress**, 9-14 July 2017.

IC22. T. Kharkovskaya, D. Efimov, E. Fridman, A. Polyakov, J.-P. Richard, On design of interval observers for parabolic PDEs, **IFAC World Congress**, 9-14 July 2017.

IC23. D. Efimov, A. Polyakov, A. Levant, W. Perruquetti, Switched gain differentiator with fixed-time convergence, **IFAC World Congress**, 9-14 July 2017.

IC24. S. Zhuk, A. Polyakov, O. Nakonechnyi, Sliding Mode Control Design for Linear Evolution Equations with Uncertain Measurements and Exogenous Perturbations, **IFAC World Congress**, 9-14 July 2017.

IC25. A. Polyakov, J.-M. Coron, L. Rosier, On Finite-Time Stabilization of Evolution Equations: A Homogeneous Approach, **IEEE Conference on Decision and Control**, Las Vegas, USA, 12-14 December 2016.

IC26. D. Efimov, A. Polyakov, A. Levant, W. Perruquetti, Discretization of asymptotically stable homogeneous systems by explicit and implicit Euler methods, **IEEE Conf. on Decision and Control**, Las Vegas, USA, 12-14 December 2016.

IC27. A. Levant, D. Efimov, A. Polyakov, W. Perruquetti, Stability and Robustness of Homogeneous Differential Inclusions, **IEEE Conference on Decision and Control**, Las Vegas, USA, 12-14 December 2016.

IC28. D. Efimov, A. Levant, A. Polyakov, W. Perruquetti, On Acceleration of a Class of Asymptotically Stable Homogeneous Systems, **IEEE Conference on Decision and Control**, Las Vegas, USA, 12-14 December 2016.
IC29. F. Lopez-Ramirez, D. Efimov, A. Polyakov, W. Perruquetti, Fixed-Time Output Stabilization of a Chain of Integrators, **IEEE Conference on Decision and Control**, Las Vegas, USA, 12-14 December 2016.

IC30. T. Kharkovskaya, D. Efimov, A. Polyakov, J.-P. Richard, Interval observers for PDEs: approximation approach, **NOLCOS**, Monterey, USA, 22-25 August 2016

IC31. K. Zimenko, D. Efimov, A. Polyakov, W. Perruquetti, Time-delay Robustness Analysis for Systems with Negative Degree of Homogeneity, **NOLCOS**, Monterey, USA, 22-25 August 2016

IC32. K. Zimenko, A. Polyakov, D. Efimov, A. Kremlev, Frequency Domain Analysis of Control System Based on Implicit Lyapunov Function, **European Control Conference**, Aalborg, Denmark, 29 June-1 July 2016

IC33. M. Feingesicht, C. Raibudo, A. Polyakov, F. Kerherve, J.-P. Richard, A bilinear input-output model with state-dependent delay for separated flow control, **European Control Conf.**, Aalborg, Denmark, 29 June-1 July 2016

IC34. F. Lopez-Ramirez, A. Polyakov, D. Efimov, W. Perruquetti, Finite-Time and Fixed-Time Observers Design via Implicit Lyapunov Function, **European Control Conference**, Aalborg, Denmark, 29 June-1 July 2016

IC35. D. Efimov, E. Fridman, A. Polyakov, W. Perruquetti, J.P. Richard, On Design of Sampled-Data Interval Observers, **European Control Conference**, Aalborg, Denmark, 29 June-1 July 2016

IC36. A. Polyakov, L. Hetel, On Finite-Time Stabilization Via Relay Feedback Control, **IEEE Conference on Decision and Control**, 2015, Osaka, Japan, p. 4098 - 4102

IC37. A. Polyakov, Y. Orlov, H. Oza, S. Spurgeon, Robust Finite-time Stabilization and Observation of a Planar System Revisited, **IEEE Conference on Decision and Control**, 2015, Osaka, Japan, p. 5689-5694

IC38. K. Zimenko, A. Polyakov, D. Efimov, Stabilization of Chain of Integrators with Arbitrary Order in Finite-Time, **IEEE Conference on Decision and Control**, 2015, Osaka, Japan, p. 4637 - 4641

IC39. B. Brogliato, A. Polyakov, Globally Stable Implicit Euler Time-Discretization of a Nonlinear Single-Input Sliding-Mode Control System, **IEEE Conference on Decision and Control**, 2015, Osaka, Japan, p. 5426 - 5431
IC40. M. Mera, A. Polyakov, W. Perruquetti, G. Zheng, Finite-Time Attractive Ellipsoid Method Using Implicit Lyapunov Functions, **IEEE Conference on Decision and Control**, 2015, Osaka, Japan, p. 6892-6896

IC41. A. Polyakov, D. Efimov, E. Fridman, W. Perruquetti, Homogeneous Evolution Equation in a Banach Space, **European Control Conference**, 2015, Linz, Austria, p.2469-2474

IC42. S. Zhuk, A. Polyakov, Output-Based Sliding Mode Control Design for Linear Plants with Multiplicative Disturbances: The Minimax Approach, **European Control Conference**, 2015, Linz, Austria, p. 227-282

IC43. D. Efimov, A. Polyakov, E. Fridman, W. Perruquetti, J.-P. Richard, Delay-Dependent Positivity: Application to Interval Observers, **European Control Conference**, 2015, Linz, Austria, p. 2079-2083

IC44. D. Efimov, A. Polyakov, L. Fridman, W. Perruquetti, J.-P. Richard, A Note on Continuous Delayed Sliding Mode Control, **European Control Conference**, 2015, Linz, Austria, p. 2401-2406

IC45. A. Polyakov, D. Efimov, W. Perruquetti, J.-P. Richard, Implicit Lyapunov-Krasovski Functions for Time-Delay Systems, **IEEE Conference on Decision and Control**, 15-17 December 2014, Los Angeles, USA.

IC46. A. Polyakov, D. Efimov, W. Perruquetti, J.-P. Richard, Implicit Lyapunov-Krasovski Functionals for time delay systems, **IEEE Conference on Decision and Control**, 15-17 December 2014, Los Angeles, USA, 2014, p. 1191-1196.

IC47. D. Efimov, A. Polyakov, W. Perruquetti, J.-P. Richard, Stability Analysis for Nonlinear Time-Delay Systems Applying Homogeneity, **IEEE Conference on Decision and Control**, 15-17 December 2014, Los Angeles, USA, p. 37-42.

IC48. A. Polyakov, I. Chairez, A New Homogeneous Quasi-Continuous Second Order Sliding Mode Control, **Latinamerican Congress on Automatic Control**, 14-17 October 2014, Cancun, Mexico.

IC49. A. Polyakov, Discontinuous Lyapunov Functions for Nonasymptotic Stability Analysis, **IFAC World Congress**, 24-29 August 2014, Cape Town, South Africa, p. 5455-5460.

IC50. A. Polyakov, J.-P. Richard, Suboptimal LMI-based Solution of Minimum Time Control Problem, **IFAC World Congress**, 24-29 August 2014, Cape Town, South Africa, p. 2475-2480.
IC51. S. Zhuk, A. Polyakov, On Output-based Sliding Mode Control Design Using Minimax Observer, **International Workshop on Variable Structure Systems**, 29 June-2 July 2014, Nantes, France.

IC52. M. Kryachkov, A. Polyakov, Graph-based field automata for modeling of sliding mode systems, **1 International Workshop on Variable Structure Systems**, 29 June-2 July 2014, Nantes, France.

IC53. A. Polyakov, D. Efimov, W. Perruquetti, Sliding Mode Control Design for MIMO Systems: Implicit Lyapunov Function Approach, **European Control Conference**, 24-27 June 2014, Strasburg, France, p. 2612-2617.

IC54. A. Polyakov, D. Efimov, W. Perruquetti, Homogeneous Differentiator Design using Implicit Lyapunov Function Method, **European Control Conference**, 24-27 June 2014, Strasburg, France, p. 288-293.

IC55. D. Efimov, A. Polyakov, J.-P. Richard, Interval estimation for systems with time delays and algebraic constraints, **European Control Conference**, 24-27 June 2014, Strasburg, France, p.678-683.

IC56. S. Govindaswamy, L. Hetel, A. Polyakov, T. Floquet, On relay control for discrete time systems using linear matrix inequalities, **European Control Conference**, 24-27 June 2014, Strasburg, France, p. 2225-2230.

IC57. A. Polyakov, A. Poznyak, J.-P. Richard, Robust Output Stabilization of Time-Varying Input Delay Systems using Attractive Ellipsoid Method, **IEEE Conference on Decision and Control**, 10-13 December 2013, Florence, Italy.

IC58. S. Parsegov, A. Polyakov, P.S. Scherbakov, Fixed-Time Consensus Algorithm for Multi-Agent Systems with Integrator Dynamics, **IFAC Workshop on Distributed Estimation and Control in Networked Systems**, 2013, p. 110-115.

IC59. A. Polyakov, D. Efimov, W. Perruquetti, Finite-time Stabilization Using Implicit Lyapunov Function Technique, **Symposium on Nonlinear Control Systems**, Toulouse, France, 4-6 September 2013, p. 140-145.

IC60. A. Polyakov, D. Efimov, W. Perruquetti, J.-P. Richard, Interval Observer Approach to Output Stabilization of Time-Varying Input Delay System, **European Control Conference**, Zurich, 17-19 July 2013, p. 4412-4417.

IC61. E. Bernuau, D. Efimov, W. Perruquetti, A. Polyakov, On an extension of homogeneity notion for differential inclusions, **European Control Conference**, Zurich, 17-19 July 2013, p. 2204-2209.
IC62. E. Bernuau, A. Polyakov, D. Efimov, W. Perruquetti, On ISS and iISS properties of homogeneous systems, **European Control Conference**, 17-19 July 2013, Zurich, Switzerland, p. 1705-1710.

IC63. E. Bernuau, A. Polyakov, D. Efimov, W. Perruquetti, Robustness of finite-time stability property for sliding modes, **Symposium on System Structure and Control**, 4-6 February 2013, Grenoble, France, p. 391-396.

IC64. S. Parsegov, A. Polyakov, P. Shcherbakov, Nonlinear Fixed-Time Control Protocol for Uniform Allocation of Agents on a Segment, **IEEE Conference on Decision and Control**, 10-13 December 2012, Maui, Hawaii, USA, p. 7732-7737.

IC65. M. Mera, A. Poznyak, V. Azhmyakov, A. Polyakov, A Robust Dynamic Controller for a Class of Nonlinear Systems with Sample-Data Outputs, **Int. Conf. on Electrical Engineering, Comp. Science and Autom. Control**, 2012

IC66. A. Polyakov, Fixed-Time Stabilization via Second Order Sliding Mode Control, **Analysis & Design of Hybrid Systems**, 6-8 June 2012, Eindhoven, Netherlands, P. 254 - 258.

IC67. A. Polyakov, Fixed-Time Stabilization of Linear Systems via Sliding Mode Control, **International Workshop on Variable Structure Systems**, 12-14 January 2012, Mumbai, India, P. 1-6.

IC68. A. Polyakov, On Settling Time Function and Stability of Vector Relay Systems, **International Workshop on Variable Structure Systems**, 12-14 January 2012, Mumbai, India, P. 149-154.

IC69. A. Polyakov, A. Poznyak, The Lyapunov Function Design for the Stability Analysis of the "Italian Version" of the Second Order Sliding Mode Controllers, **IFAC World Congress**, 28 August - 2 September 2011, Milan, Italy.

IC70. A. Polyakov, Invariant Ellipsoid Method for Time-Delayed Predictor-Based Sliding Mode Control System, **International Workshop on Variable Structure Systems**, 26-28 June 2010, Mexico D.F., P. 95-99.

IC71. M. Kryachkov, A. Polyakov, V. Strygin, Finite-time stabilization of an integrator chain using only signs of the state variables, **International Workshop on Variable Structure Systems**, 26-28 June 2010, Mexico D.F.

IC72. S. Gonzalez-Garcia, A. Polyakov, A. Poznyak, Linear feedback spacecraft stabilization using the method of invariant ellipsoids, **Southeastern Symposium on System Theory**, Tullahoma, TN, USA, March 15-17, 2009.
IC73. S. Gonzalez-Garcia, A. Polyakov, A. Poznyak, Output Linear Controller for a Class of Nonlinear Systems Using the Invariant Ellipsoid Method, **American Control Conference**, St. Louis, Missouri, USA, 10-12 June 2009.

IC74. A. Polyakov, A. Poznyak, Minimization of the Unmatched Disturbances in the Sliding Mode Control Systems via Invariant Ellipsoid Method, **IEEE Multi Conference on Systems and Control**, 2009

IC75. A. Polyakov, A. Poznyak, Lyapunov Function Design for Finite-Time Convergence Analysis of "Twisting" and "Super-Twisting" Second Order Sliding Mode Controlers, **IFAC Workshop on Variable Structure Systems**, 2008

IC76. S. Gonzalez-Garcia, A. Polyakov, A. Poznyak, Output linear feedback for a class of nonlinear systems based on the invariant ellipsoid method, **Int. Conf. on Electr. Engineering, Comp. Science and Autom. Control**, 2008

IC77. A. Polyakov, Practical Stabilization via Relay Delayed Control, **IEEE Conference on Decision and Control**, 8-11 December 2008, Cancun, Mexico, P. 5306-5311.

IC78. S. Gonzalez-Garcia, A. Polyakov, A. Poznyak, Robust Stabilization of a spacecraft with flexible elements using invariant ellipsoid technique, **Int. Conf. on Electr. Engineering, Comp. Science and Autom. Control**, 2008

IC79. L. Fridman, V. Strygin, A. Polyakov, Semiglobal Stabilization via Relay Controller with Uncertain Time Delay, **IEEE Conference on Decision and Control**, Maui, Hawaii, 9-12 December 2003, p.4074-4079.

IC80. L. Fridman, P. Acosta, A. Polyakov, V. Strygin. Amplitude Estimation of Oscillations in Delayed Relay Systems, **American Control Conference**, Denver, Colorado, USA, 4-6 June 2003, pp. 2676-2681.

IC81. L. Fridman, V. Strygin, A. Polyakov. Delayed relay nonlocal stabilization suppressing uncertainty in a time delay. **IFAC Symposium on Robust Control Design**, Milan, 25-27 June 2003, pp.389-394.

IC82. L. Fridman, V. Strygin, A. Polyakov, Nonlocal Stabilization via Delay Relay Control Gain Adaptation, **American Control Conference**, Anchorage, Alaska, 8-10 May 2002, pp. 2221-2226.

IC83. L. Fridman, V. Strygin, A. Polyakov, Algorithm of Nonlocal Stabilization via Delayed Control Gain Adaptation, **IEEE Workshop on Variable Structure Systems**, Sarajevo, 17-19 July 2002, pp. 47-56.
IC84. L. Fridman, V. Strygin, A. Polyakov, Stabilization of Oscillations Amplitudes via Relay Delay Control, **IEEE Conference on Decision and Control**, Orlando, Florida, 4-7 December 2001, pp.3704-3709.

IC85. L. Fridman, P. Acosta, A. Polyakov, Robust eigenvalue assignment for uncertain delay control systems, **IFAC Workshop on Time Delay Systems**, Santa Fe, New Mexico, USA, 8-10 December 2001, pp.239-244.

IC86. L. Fridman, V. Strygin, A. Polyakov, Relay Control of Oscillations Amplitudes for Systems with Delay, **IFAC Workshop on Time Delay Systems**, Santa Fe, New Mexico, USA, 8-10 December 2001, pp.245-248.

## Books and book chapters

B1. A.Poznyak, A. Polyakov, V. Azhmyakov, **Attractive Ellipsoids in Robust Control**, Birkhauser, 2014.

B2. A. Polyakov, Fast Control Systems: Nonlinear Approach, in **New Perspectives and Applications of Modern Control Theory** (Editors: J. B. Clempner and W. Yu). - Springer, 2017. - pp. 287-316.

B3. S. Zhuk, A. Polyakov, Minimax Observer for Sliding Mode Control Design, in **Recent Trends in Sliding Mode Control** (Editors L. Fridman, J.-P. Barbot, F. Plestan). - London, IET, 2016. - pp. 119-130.

B4. E. Bernuau, D. Efimov, W. Perruquetti, A. Polyakov, Homogeneity of Differential Inclusions, in **Recent Trends in Sliding Mode Control** (Editors L. Fridman, J.-P. Barbot, F. Plestan). - London, IET, 2016. - pp. 103-118.

B5. L. Fridman, V. Strygin, A. Polyakov, Semiglobal Stabilization of Linear Uncertain System via Delayed Relay Control, in **Variable Structure Systems : from Principles to Implementation**. - London, 2004. - pp. 377-400.

## National journals

NJ1. K. A. Zimenko, A. S. Kremlev, A. E. Polyakov, D. V. Efimov, Robust stabilization of two-rotor MIMO plant, Scientific and Technical Journal of Information Technologies, Mechanics and Optics, Vol.18, N. 3, 2018, p. 392-398