infectiious disease MOdeling a hybrid system approach

Aside from modeling the spread of an infectiousdisease using a hybrid and switched system, a new approach to mathematicaldisease modeling, the unique features of this monograph can be su

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Series Editor: Albert C J Luo Nonlinear Systems and Complexity

Xinzhi Liu

Peter Stechlinski

Infectious Disease

Modeling

A Hybrid System Approach

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Nonlinear Systems and Complexity

Nonlinear Systems and Complexity provides a place to systematically summarizerecent developments, applications, and overall advance in all aspects of nonlinearity,chaos, and complexity as part of the established research literature, beyond thenovel and recent findings published in primary journals The aims of the bookseries are to publish theories and techniques in nonlinear systems and complexity;stimulate more research interest on nonlinearity, synchronization, and complexity

in nonlinear science; and fast-scatter the new knowledge to scientists, engineers,and students in the corresponding fields Books in this series will focus on therecent developments, findings and progress on theories, principles, methodology,computational techniques in nonlinear systems and mathematics with engineeringapplications The Series establishes highly relevant monographs on wide rangingtopics covering fundamental advances and new applications in the field Topicalareas include, but are not limited to: Nonlinear dynamics Complexity, nonlinearity,and chaos; Computational methods for nonlinear systems; Stability, bifurcation,chaos and fractals in engineering; Nonlinear chemical and biological phenomena;Fractional dynamics and applications; Discontinuity, synchronization and control

Xinzhi Liu • Peter Stechlinski

Infectious Disease Modeling

A Hybrid System Approach

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Human life expectancy has increased over the past three centuries, from mately 30 years in 1700 to approximately 70 years in 1970 [4]; one of the mainfactors of this improvement is a result of the decline in deaths caused by infectiousdiseases In contrast to this decline in mortality, both the magnitude and frequency ofepidemics increased during the eighteenth and nineteenth centuries, principally as aresult of an increase of large population centers in industrialized societies [4] Thistrend then reversed in the twentieth century, mainly due to the development andwidespread use of vaccines to immunize susceptible populations [4] The humaninvasion of new ecosystems, global warming, increased international travel, andchanges in economic patterns will continue to provide opportunities for the spread

approxi-of new and existing infectious diseases [65]

New infectious diseases have emerged in the twentieth century and some existingdiseases have reemerged [65]: Measles, a serious disease of childhood, still causesapproximately one million deaths each year worldwide Type A influenza led tothe 1918 pandemic (a worldwide epidemic) that killed over 20 million people.Examples of newly emerging infectious diseases include Lyme disease (1975),Legionnaire’s disease (1976), hepatitis C (1989), hepatitis E (1990), and hantavirus(1993) The appearance of the human immunodeficiency virus (HIV) in 1981, whichleads to acquired immunodeficiency syndrome (AIDS), has become a significantsexually transmitted disease throughout the world New antibiotic-resistant strains

of tuberculosis, pneumonia, and gonorrhea have emerged Malaria, dengue, andyellow fever have reemerged and, as a result of climate changes, are spreading intonew regions Plague, cholera, and hemorrhagic fevers (e.g., Ebola) continue to eruptoccasionally

In 1796, an English country doctor, Edward Jenner, observed that milkmaids whohad been infected with cowpox did not get smallpox, and so he began inoculatingpeople with cowpox to protect them from getting smallpox (this was the world’sfirst vaccine, taken from the Latin word vacca for cow) [65] Mathematical modelshave become important tools in analyzing both the spread and control of infectiousdiseases The first known mathematical epidemiology model was formulated andsolved by Daniel Bernoulli in 1760 [92] The pioneering work on infectious

v

vi Preface

disease modeling by Kermack and McKendrick has had a major influence in thedevelopment of mathematical models of infectious diseases [116] These authorswere the first to obtain a threshold result that showed the density of susceptiblesmust exceed a critical value for an outbreak to occur [65] In the early twentiethcentury, the foundations of modern mathematical epidemiology based on compart-ment models were laid, and mathematical epidemiology has grown exponentiallysince the middle of the previous century [92] An extensive number of modelshave been formulated, analyzed, and applied to a variety of infectious diseases,including measles, rubella, chickenpox, whooping cough, smallpox, malaria, rabies,gonorrhea, herpes, syphillis, and HIV/AIDS [64]

Studying these somewhat simple mathematical epidemiology models is crucial

in order to gain important knowledge of the underlying aspects of the spread ofinfectious diseases [64]; one such purpose of analyzing epidemiology models is

to get a clear understanding of the similarities and differences in the behavior ofsolutions of the models, as this allows us to make decisions in choosing models forcertain applications Mathematical models and computer simulations are extremelyuseful tools for building and testing theories, assessing quantitative conjectures,answering qualitative questions, and estimating key parameters from data; epidemicmodeling can help to identify trends, suggest crucial data that should be collected,make general forecasts, and estimate the uncertainty in forecasts [65]

The transmission of a disease, which depends on its intrinsic infectiousity aswell as population behavior, is a crucial part in the medical and statistical study of

an epidemic [38] In mathematical modeling, these two aspects are summarized inthe contact rate and the incidence rate of a disease, which are the average number

of contacts between individuals that would be sufficient for transmitting the diseaseand the number of new cases of a disease per unit time, respectively [65] Empiricalstudies have shown that there are seasonal variations in the transmission of manyinfections [69] Examples include differences in the abundance of vectors due toweather changes (e.g., dry season vs rainy season), changes in the survivability

of pathogens (outside hosts), differences in host immunity, and variations in hostbehavior (e.g., increased contacts between individuals in the winter season frombeing indoors) [39,53] For childhood infections such as measles, chickenpox, andrubella, it has been observed that the rates of transmission peak at the start of theschool year and decline significantly during the summer months [69] An analysis

of measles data in New York demonstrates that sufficiently large seasonal variations

in transmission can generate a biennial-looking cycle [134] Data from Englandand Wales displays a four-year cycle in poliomyelitis incidence, while measles hasbeen observed to have a biennial cycle for the same countries [134] Reports havefound that many diseases show periodicity in their transmission, such as measles,chickenpox, mumps, rubella, poliomyelitis, diphtheria, pertussis, and influenza[66] Depending on the particular disease of interest and population behavior, anappropriate model of the disease’s spread may require term-time forcing where themodel parameters change abruptly in time

The recent increase in seaborne trade and air travel has removed many geographicbarriers to insect disease vectors [26] For example, the vector responsible in part

Preface vii

for transmitting diseases such as chikungunya and, more recently, Zika virus, Aedes

albopictus, has developed capabilities to adapt to nontropical regions and is now

found in Southeast Asia, the Pacific and Indian Ocean islands, Europe, the USA, andAustralia [41,113,114] Consequently, studying mathematical models on the spread

of vector-borne diseases has become a large focus in the literature, for example, thedengue virus [165,166] and the chikungunya virus [7,40–43,113,114] Seasonalchanges are an important factor in how these vector-borne diseases spread in apopulation and relate to changes in the abundance of vectors and the host populationbehavior For example, Bacặr [7] noted that seasonality plays an important role inthe spread of the chikungunya virus The 2005 outbreak of chikungunya virus inRéunion occurred when the mosquito population was at its highest, the end of the hotseason and beginning of the winter season [42] The transmission of dengue fever ishigher during wet and humid periods with high temperatures ideal for mosquitoesand lower when the temperature is low [126,165]

One of the most important aspects of epidemic modeling is the application ofcontrol schemes to eradicate, or at least suppress, an impending epidemic Infectiousdisease models are a vital component of comparing, implementing, evaluating,and optimizing various detection, prevention, and control programs [65]; epidemicmodels are useful in approximating vaccination levels needed for the control of adisease [116] For example, in 1967, there were approximately 15 million cases

of smallpox per year which led the World Health Organization (WHO) to develop

an initiative against smallpox The WHO strategy involved extensive vaccinationprograms, surveillance for outbreaks, and containment of these outbreaks by localvaccination programs [65] This has been considered the most spectacular success

of a vaccination program [101]; smallpox was eventually eradicated worldwide by

1977, and the WHO estimates that the elimination of smallpox worldwide savesover two billion dollars per year [65] There are now vaccines that are effective inpreventing rabies, yellow fever, poliovirus, hepatitis B, parotitis, and encephalitis B,among others [83]

Aside from seasonal changes in population behavior, the conduct of the ulation can shift due to, for example, psychological effects (widespread panic of

pop-an impending outbreak) or from public health campaigns to prevent a diseasespread The aim of this study is to mathematically model infectious diseases,which take these important factors into account, using a switched and hybridsystems framework The scope of coverage includes background on mathematicalepidemiology, including classical formulations and results; a motivation for seasonaleffects and changes in population behavior; an investigation into term-time forcedepidemic models with switching parameters; and a detailed account of severaldifferent control strategies The main goal is to study these models theoreticallyand to establish conditions under which eradication or persistence of the disease isguaranteed In doing so, the long-term behavior of the models is determined throughmathematical techniques from switched systems theory Numerical simulations arealso given to augment and illustrate the theoretical results and to help study theefficacy of the control schemes

viii Preface

The objective of this monograph is to formulate new epidemiology models withtime-varying contact rates or time-varying incidence rate structures, and to study thelong-time behavior of diseases More specifically, we look to extend epidemiologymodels in the literature by the addition of switching, which is the abrupt change

of the dynamics governing the systems at certain switching times This switchingframework allows the contact rate to be approximated by a piecewise constantfunction Since relatively modest variations in the contact rate can result in largeamplitude fluctuations in the transmission of a disease [69], this is an importantphenomenon that requires attention Switching is a new approach to this problemthat has not been studied before as an application to epidemiology models Aspecific incidence rate must be chosen appropriately based on the scenario anddisease being modeled for any given infectious disease model There are numerousincidence rates which have been used in models in the literature, for example, thestandard incidence, psychological-effect incidences, saturation incidences, mediacoverage incidences, and more general nonlinear forms (see [38, 64,73,122]).With regard to different forms of the incidence rate, one of the possible causes

of unexpected failures of a vaccination campaign may be the nonlinearity of theincidence rate not being properly modeled [38], which gives extra motivation instudying switching incidence rate structures The focus of this monograph is topresent new methods for formulating and analyzing epidemic models with time-varying model parameters and function forms, which are easily extendable to manydifferent models, as will be shown

The area of hybrid dynamical systems (HDS) is a new discipline which bridgesapplied mathematics, control engineering, and theoretical computer science [45].HDS frameworks provide a natural fit for many problems scientists face as theyseek to control complex physical systems using computers [45] Indeed, there

is a growing demand in industry for methods to model, analyze, and stand systems that combine continuous components with logic-based switching[136] Practical examples of switched systems, a type of HDS, include areas asdiverse as mechanical systems, the automotive industry, air traffic control, robotics,intelligent vehicle/highway systems, chaos generators, integrated circuit design,multimedia, manufacturing, high-level flexible manufacturing systems, power elec-tronics, interconnected power systems, switched-capacitor networks, computer diskdrives, automotive engine management, chemical processes, and job scheduling[31,45,54,85] Examples of systems which can be described by switching systemswith abrupt changes at the switching instances include biological neural networks,optimal control modes in economics, flying object motions, bursting rhythm models

under-in pathology, and frequency-modulated signal processunder-ing systems [54] Impulsivesystems will be important when we look to add pulse control to the switched models

As mentioned, switched systems are described using a mixture of continuousdynamics and logic-based switching, in that they evolve according to mode-dependent continuous dynamics and experience transitions between modes that aretriggered by certain events [136] There are typically two cases in which a switchedsystem arises [31]: One is when there are abrupt changes in the structure or the

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parameters of a dynamical system, which can be due to, for example, environmentalfactors (i.e., outside forces) The second is when a continuous system is controlledusing a switched controller

This monograph is not meant to be a comprehensive analysis of every modelingchoice possible for mathematical models of infectious diseases Rather, its aim is

to provide theoretical tools which are applicable to a wide variety of problems

in epidemic modeling The mathematical methods are revealed one at a time

as this monograph progresses Aside from modeling the spread of an infectiousdisease using a hybrid and switched system, a new approach to mathematicaldisease modeling, the unique features of this monograph can be summarized asfollows: (1) using techniques from switched systems theory to study the stability

of epidemic models, (2) focusing on the role seasonality plays in the spread ofinfectious diseases, and (3) investigating how abrupt changes in model parameters orfunction forms affect control schemes Accessible to individuals with a background

in dynamical systems theory or mathematical modeling of epidemics, this work isintended as a graduate-level book for individuals with an interest in mathematicalbiology, epidemic models, and, more generally, physical problems exhibiting amixture of continuous and discrete dynamics (i.e., hybrid behavior)

The reader gains the fundamentals of compartmental infectious disease eling, as well as the necessary mathematical background (e.g., stability theory ofordinary and functional differential equations) The reader learns techniques fromswitched and hybrid systems, which are applicable to a variety of applications

mod-in engmod-ineermod-ing and computer science Knowledge is gamod-ined regardmod-ing the rolesseasonality and population behavior play in the spread of a disease, including theformulation and theoretical tools for analysis of epidemic models and infectiousdisease control strategies In doing so, the reader learns about the concept ofthreshold conditions in epidemic modeling, such as the basic reproduction number,used to prove eradication or persistence of the disease based on model parameters.Numerical simulations are also given, to help illustrate the results to the reader.The structure of the monograph is outlined as follows: In PartI, the theoreticalframework is established for the remainder of the monograph Chapter 2 detailsthe necessary foundational material Switching epidemic models are formalized andstudied in PartII: The classic SIR model is investigated in Chap.3while extensionsare studied in Chap.4 Control methods to achieve eradication of the diseaseare presented and thoroughly analyzed in Part III Switching control schemesare investigated in Chap.5 while impulsive strategies are studied in Chap.6 Acase study is given in Chap.7 detailing an outbreak of chikungunya virus andpossible control strategies for its containment and eradication Conclusions andfuture directions are given in PartIV

The authors were supported in part by the Natural Sciences and EngineeringResearch Council of Canada (NSERC) and the Ontario Graduate Scholarship (OGS)program, which are gratefully acknowledged

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Part I Mathematical Background

1 Basic Theory 3

1.1 Preliminaries 3

1.2 Ordinary Differential Equations 5

1.2.1 Fundamental Theory 5

1.2.2 Stability Theory 7

1.2.3 Partial Stability 13

1.3 Impulsive Systems 14

1.4 Delay Differential Equations 16

1.5 Stochastic Differential Equations 18

2 Hybrid and Switched Systems 21

2.1 Stability Under Arbitrary Switching 25

2.2 Stability Under Constrained Switching 27

2.3 Switching Control 30

Part II Hybrid Infectious Disease Models 3 The Switched SIR Model 43

3.1 Model Formulation 43

3.2 Threshold Criteria: The Basic Reproduction Number 49

3.3 Seasonal Variations in Disease Transmission: Term-Time Forcing 52

3.4 Adding Population Dynamics: The Classical Endemic Model 55

3.5 Generalizing the Incidence Rate of New Infections 69

3.6 Uncertainty in the Model: Stochastic Transmission 74

3.7 Discussions 81

4 Epidemic Models with Switching 83

4.1 Absence of Conferred Natural Immunity: The SIS Model 83

4.2 Multi-City Epidemics: Modeling Traveling Infections 97

4.3 Vector-Borne Diseases with Seasonality 108

xi

xii Contents

4.4 Other Epidemiological Considerations 112

4.4.1 Vertical Transmission 112

4.4.2 Disease-Induced Mortality: Varying Population Size 115

4.4.3 Waning Immunity: The Switched SIRS Model 120

4.4.4 Passive Immunity: The Switched MSIR Model 122

4.4.5 Infectious Disease Model with General Compartments 124

4.4.6 Summary of Mode Basic Reproduction Numbers and Eradication Results 127

4.5 Discussions 129

Part III Control Strategies 5 Switching Control Strategies 135

5.1 Vaccination of the Susceptible Group 135

5.2 Treatment Schedules for Classes of Infected 146

5.3 Introduction of the Exposed: A Controlled SEIR Model 151

5.4 Screening of Traveling Individuals 160

5.5 Switching Control for Vector-borne Diseases 165

5.6 Discussions 176

6 Pulse Control Strategies 179

6.1 Public Immunization Campaigns: Control by Pulse Vaccination and Treatment 179

6.1.1 Impulsive Control Applied to the Classical Endemic Model 180

6.1.2 Incorporating Impulsive Treatment into the Public Campaigns 186

6.1.3 The SIR Model with General Switched Incidence Rates 190

6.1.4 Vaccine Failures 194

6.1.5 Pulse Control Applied to an Epidemic Model with Media Coverage 197

6.1.6 Multi-City Vaccination Efforts 204

6.1.7 Pulse Vaccination Strategies for a Vector-Borne Disease 210

6.2 Discussions 217

6.2.1 Comparison of Control Schemes 219

7 A Case Study: Chikungunya Outbreak in Réunion 227

7.1 Background 227

7.2 Human–Mosquito Interaction Mechanisms 230

7.3 Chikungunya Virus Model Dynamics 233

7.4 Control via Mechanical Destruction of Breeding Grounds 235

7.5 Control via Reduction in Contact Rate Patterns 246

Contents xiii

7.6 Control Analysis: Efficacy Ratings 2487.6.1 Assessment of Mechanical Destruction of

Breeding Sites 2507.6.2 Assessment of Reduction in Contact Rate Patterns 2547.7 Discussions 256

Part IV Conclusions and Future Work

8 Conclusions and Future Directions 261

References 265

List of Symbols

Rn

Euclidean space of n-dimensions

Rmn

Vector space of m  n real-valued matrices

conv A Convex hull of a set A Rn

B r x/ Open ball of radius r > 0 centered at x 2 R n

min.Q/ Minimum eigenvalue of a symmetric matrix Q

max.Q/ Maximum eigenvalue of a symmetric matrix Q

Df x/ Jacobian matrix of f evaluated at x

C D; R m/ Continuous functions mapping D RntoRm

C1.D; R m/ Continuously differentiable functions mapping D RntoRm

C k D; R m/ Continuously differentiable functions of order k

K0 fw 2 C.RC; RC/ W w.0/ D 0; w.s/ > 0 for s > 0g

K1 fw 2 K0W w is nondecreasing in sg

K fw 2 C.RC; RC/ W w.0/ D 0 and w is strictly increasingg

C Continuous functions mappingŒ; 0  R to Rm

PC.D; Rm / Piecewise continuous functions mapping D to R m

k  k Usual sup norm: k kD sups0 k s/k for 2 PC

Speriodic Subset ofS that are periodic

Sdwell Subset ofS that admit a dwell-time

Savg Subset ofS that admit an average dwell-time

xv

xvi List of Symbols

P Finite index set of modes of switched system:P  f1; : : : ; pg

M Finite index set of modes of switched system:M  f1; : : : ; mg

N Finite index set of modes of switched system:N  f1; : : : ; ng

F Finite index set of vector fields:F  ff1; : : : ; f mg

T i t1; t2/ Total activation time of the ith mode on Œt1; t2

TC.t1; t2/ Total activation time of the modesMC M on Œt1; t2

T.t1; t2/ Total activation time of the modesM M on Œt1; t2

N t1; t2/ Number of switches activating the ith mode on Œt1; t2

1/

N.t1; t2/ Number of switches activating modes inMonŒt1; t2/

R./0 Basic reproduction number of the infectious disease model./

D./ Physical domain associated with epidemic model./

Q./DFS Disease-free solution associated with epidemic model./

Q./ES Endemic solution associated with epidemic model./

C c

C0H Cumulative number of infected humans without control

F0 Control strategy efficacy rating (F0 100C c

H =C0

H)

 Total number of vaccinations administered during a campaign

 Cost-benefit rating of a control scheme.  =.C0

H  C c

H//

Part I Mathematical Background

Chapter 1

Basic Theory

Necessary mathematical concepts are presented in this chapter Background theory

on ordinary differential equations, delay differential equations, impulsive dynamicsystems, and stochastic differential equations is detailed Stability theory is a focusthroughout this chapter, although fundamental theory is also given in each of thesetopics Section1.2draws upon the work by Hale [56] for the background materialregarding ODE systems, while the concept of partial stability is from [57] Thereader is also referred to the excellent works [71,109,123] for classical ODE theory.Sufficient background regarding delay differential equations (DDEs) is presented inSect.1.4, highlighting the material in [58] The short background in Sect.1.3onimpulsive dynamic systems details the works [76] and [15]

1.1 Preliminaries

The following notation is used throughout this monograph The set of positiveintegers is denoted by N RC denotes the set of nonnegative real numbers Rn denotes the Euclidean space of n-dimensions (equipped with the Euclidean norm

k  k) The vector space of n  m matrices with real-valued entries is denoted by

Rnm (equipped with the corresponding induced norm) A vector x 2 Rn has ith component x i2 R and can be written as

264

x1

:::

x n

37

5  x1; : : : ; x n/:

© Springer International Publishing AG 2017

X Liu, P Stechlinski, Infectious Disease Modeling, Nonlinear Systems

and Complexity 19, DOI 10.1007/978-3-319-53208-0_1

3

4 1 Basic Theory

A vector-valued function f WRn! Rmcan be similarly written as

f x/  f x1; : : : ; x n/ 

264

f1.x1; : : : ; x n/:::

f1.x1; : : : ; x n/

37

5  f1.x1; : : : ; x n /; : : : ; f m x1; : : : ; x n//:

Let I denote the n  n identity matrix and 0 denote the zero vector in Rn

(i.e.,the origin); the dimensions of these objects will be clear from the context Given

a symmetric matrix Q 2 Rnn

, letmax.Q/ and min.Q/ denote its maximum and minimum eigenvalue, respectively Let B r x/ denote the open ball of radius r > 0 centered at x 2Rn

Let cl A, int A, @A, convA denote the closure, interior, boundary, and convex hull of a set A  Rn

A set A  Rn

is compact if it is closed and

bounded Given a set A Rn

and a point x 2Rn

,dist.x; A/ D inffkz  xk W z 2 Ag:

Let f W D Rn ! Rm be a function mapping an open set D toRm f is called locally Lipschitz continuous in D if for each x 2 D, there exists a neighborhood

N  D of x and a Lipschitz constant L  0 such that

kf y/  f z/k Lky  zk; 8y; z 2 N:

of f at x is well-defined, with i; j/th entry given by

@f

i

@x j x1; : : : ; x n/

:

If m D 1 (i.e., f is a scalar-valued function), then rf x/  Df x// T 2 Rmdenotes

its gradient vector evaluated at x f is called continuously differentiable C1/ in D if for each x 2 D there exists a neighborhood N  D of x such that Df is a continuous function in N The space of continuous functions mapping D toRmis denoted by

C D; R m / The space of C1functions mapping D toRm is denoted by C1.D; R m/

Note that if f is continuously differentiable in D then it is locally Lipschitz in

V W D ! R The function V is called positive definite in D if V is continuous in D,

V 0/ D 0, and V.x/ > 0 for all x 2 Dnf0g Define the following K -class functions.

K0 fw 2 C.RC; RC/ W w.0/ D 0; w.s/ > 0 for s > 0g;

K1D fw 2 K0W w is nondecreasing in sg;

K1 fw 2 K W w.s/ ! 1 as s ! 1g:

1.2 Ordinary Differential Equations 5

1.2 Ordinary Differential Equations

Classical theory is recounted regarding ODEs (i.e., existence and uniqueness) inSect.1.2.1 Stability theory is detailed for equilibria in Sect.1.2.2 The notion ofpartial stability is given in Sect.1.2.3 ODE systems with impulsive effects areintroduced in Sect.1.3 Functional differential equations (i.e., delay differentialequations) are briefly discussed in Sect.1.4 The theory of stochastic differentialequations is highlighted in Sect.1.5

1.2.1 Fundamental Theory

Let D RnC1 be an open and connected set and f W D ! Rn be a vector-valuedfunction Consider the following system of ordinary differential equations (ODEs):

(or, more briefly, Px D f t; x/), where t 2 R is the independent variable; x 2 R n

is the dependent variable; Px  dx dt is the derivative of x with respect to t; x.t/  x1.t/; : : : ; x n t// is the state of the ODE system at time t; and

f t; x/  f1.t; x1; : : : ; x n /; : : : ; f m t; x1; : : : ; x n //; 8.t; x/ 2 D:

For a given.t0; x0/ 2 D, an initial value problem (IVP) is formulated as follows:

Px.t/ D f t; x.t//;

A solution of the IVP is defined next

Definition 1.1 Let T  R be an interval containing t0 A function  W T ! R n

is called a solution of (1.2) in T if  is C1 in T, satisfies Eq (1.1) for all t 2 T,

f.t; .t// W t 2 Tg  D, and .t0/ D x0

The solution mapping  outlined in Definition 1.1 may be written as  

.I t0; x0/ to denote its dependence on t0and x0 In this case, the solution is said topass through.t0; x0/ Assuming that f is continuous in D, the IVP (1.2) is equivalent

to the following integral equation:

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the integral is considered in the Lebesgue sense In this case, f is permitted to be measurable with respect to t (see [47] for details); this case is not pursued further inthis section Existence and uniqueness of classical solutions follows from continuityand Lipschitz continuity, as follows

Theorem 1.1 Assume that t0; x0/ 2 D If f 2 C.D; R n /, then there exists at least

through t0; x0/.

If (1.2) is an autonomous ODE system (i.e., f t; x/  f x/), the initial time t0is

not significant and can be set to zero without loss of generality

Definition 1.2 Let be a solution of (1.2) in T If e is a solution of (1.2) in eT

(i.e., a strict superset) and.t/ D e .t/ for all t 2 T, then e is a continuation of 

A solution O of (1.2) in OT is called noncontinuable if no such continuation exists;

OT is called a maximal interval of existence.

In the general case (i.e., (1.2)), the solution need not exist globally For example,the solution of the IVP

2 Such a solution is often called to have a finite escape

time (or finite blow-up time)

If (1.2) is a linear ODE system (i.e., f t; x/  Ax, for some A 2 R nn

), the uniquenoncontinuable solution is given by

That is, T D R is the maximal interval of existence A solution of the autonomous ODE (1.2) defined on a maximal interval of existence either reachesits boundary (i.e., escapes in finite time) or is defined globally

non-Theorem 1.2 Let f 2 C.D; R n / and  be a solution of (1.2) in T Then there is a

t D C1 or lim t!t.t; .t// 2 @D (i.e., it tends to its boundary) A similar result

holds for t.

Therefore, if D DRnC1then either tD C1 or k.tI x0/k ! 1 at t ! t As aresult, we get the next important corollary (where we focus on the autonomous casefor the later results in this monograph); invariance of a compact set coupled withcontinuous differentiable of the right-hand side function implies global existence of

a unique solution

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1.2 Ordinary Differential Equations 7

Definition 1.3 A set W  D is called invariant to (1.2) if every solution.I t0; x0/

of (1.2) in eT with t0; x0/ 2 W satisfies f.t; .t// W t 2 e T g  W W is called

positively invariant to (1.2) if all solutions.I t0; x0/ starting in W remain in W for all t 2 e T \ Œt0; 1/ (i.e., t0; x0/ 2 W implies f.t; .t// W t 2 e T \ Œt0; 1/g  W).

Theorem 1.3 Let f 2 C1.RnC1; Rn / and let .I x0/ be a solution of (1.2) Suppose

.t0; x0/ 2 W, then the maximal right interval of existence is OT D RC.

D! Rn , D Rn is an open connected set containing the origin, and t0 2 RC forthe remainder of this section

Definition 1.4 A vector Nx 2 D is called an equilibrium point (or equilibrium

solution) of (1.2) if f t; Nx/ D 0 for all t 2 RC

An equilibrium point Nx of (1.2) is called such since, by inspection, W RC !

Rn W t 7! Nx is a solution of (1.2) It is assumed, without loss of generality, that

f t; 0/ D 0 for all t 2 RC; if this is not true, then letting z  x  Nx yields

Pz.t/ D f t; z.t/ C Nx/  Nf.t; z/;

where Nf W t; z/ 7! f t; z C Nx/ Thus, qualitative behavior of the ODE (1.2) can be

discerned from Pz.t/ D Nf.t; z/ with the initial condition z.t0/ D x0 Nx This argument

extends to studying the stability of non-constant solutions of (1.2): supposing that

 W RC ! D is a solution of (1.2) passing through .t0; x0/, let z  x   and

 D t  t0 Then

Pz.t/ D f t; x.t//  d

dt t/ D f  C t0; z./ C . C t0// d

d. C t0/  Of.; z.//:Observe that the origin is an equilibrium point of the transformed ODE system since

dt  C t0/ D f  C t0; . C t0//

for all 2 RC Without loss of generality then, assume that Nx 0 is an equilibrium

of (1.2)

(ii) uniformly stable ifı in (i) is independent of t0, that is,ı.t0

(iii) attractive if there exists aˇ D ˇ.t0/ > 0 such that kx0k < ˇ implies that

lim

t!1 .t/ D 0I

(iv) uniformly attractive if there exists a ˇ > 0, kx0k < ˇ implies that for all

> 0, there exists a T D T / > 0 such that for all t0 2 R, k.t/k < if

t  t0C T /;

(v) asymptotically stable if (i) and (iii) hold;

(vi) uniformly asymptotically stable if (ii) and (iv) hold;

(vii) exponentially stable if there exist constantsˇ; ; C > 0 such that if kx0k < ˇthen k.t/k Ckx0k exp. t  t0// for all t  t0;

(viii) (globally attractive) (uniformly attractive) (asymptotically stable) (uniformlyasymptotically stable) (exponentially stable) ifˇ in (iii) (iv) (v) (vi) (vii) can

be made arbitrary;

(ix) unstable if (i) fails to hold

Exponential stability implies uniform asymptotic stability (and therefore totic stability) In the autonomous case, uniform stability is equivalent to stability.The stability properties of a solution can be characterized by using the method ofLyapunov functions

asymp-Definition 1.6 A function W W D ! R is said to be positive definite if W.0/ D 0 and W.x/ > 0 for all x 2 Dnf0g A function V 2 C1.RCD; R/ is said to be positive definite if V.t; 0/ D 0 and there exists a positive definite function W W D ! R such that V.t; x/  W.x/ for all t; x/ 2 RCD It is further said to be radially unbounded

if W is radially unbounded The function V is said to be decrescent if there exists a positive definite function Z W D ! R such that V.t; x/ Z.x/ for all t; x/ 2 RCD.

An auxiliary function satisfying the properties of Definition1.6is a candidateLyapunov function (so named after A.M Lyapunov) In the case that (1.2) is amodeling a mechanical system, a candidate Lyapunov function is often given by

an expression for the total energy of the system Evaluating the time-derivative of

1.2 Ordinary Differential Equations 9

Theorem 1.4 Consider (1.2) with f W RC  D  R nC1 ! Rn Let V 2

C1.RC  D; R/ and W1; W2 2 C.D; R/ be positive definite Assume that the

following conditions hold:

(ii) P V(1.2).t; x/ 0 for all t; x/ 2 RC D.

Then the trivial solution is uniformly stable If (ii) is replaced by

The trivial solution is uniformly (asymptotically) stable if there is a C1 positivedefinite, decrescent function whose derivative along the trajectories is negative

semidefinite (definite) The result becomes global if V is in addition radially

unbounded The versatility of Lyapunov’s direct method lies in the fact that ananalytic solution of (1.2) is not required The results in Theorem 1.4 can bedescribed intuitively: if PV(1.2).x/ D rV.x/  f x/ < 0 for x 2 D n f0g (i.e., in the non-autonomous case), then V is decreasing along the solution trajectory in

Œt0; 1/  D n f0g/ (i.e., V decreases along all orbits in the state space except at the origin, which is an equilibrium point); orbits therefore cut level sets of V inward

toward the origin This continues and the solution trajectory approaches the origin

as t ! 1.

Theorem 1.5 Consider (1.2) with f W RC D ! R n Let V 2 C1.RC D; R/.

condi-tions hold:

(ii) P V(1.2).t; x/ a3kxk p for all t; x/ 2 RC D.

hold, then the trivial solution is globally exponentially stable.

Analogous autonomous Lyapunov theorems are presented in Sect 4.1 in [71]while converse Lyapunov theorems are given in Sect 4.7 in [71] In the linearcase (recall the analytical solution outlined in Eq (1.3)), the following well-known

result holds: if A 2 Rnn is a Hurwitz matrix,1 then the trivial solution of (1.2)

is globally exponentially stable If there exists an eigenvalue of A with positive

real part, then the trivial solution is unstable Although this is not immediatelyapplicable to the general nonlinear setting, this result is useful thanks to theHartman-Grobman theorem; the linearization of a nonlinear autonomous ODE

system with C1 right-hand side function f W Rn ! Rn about an equilibrium point

1 All eigenvalues have negative real part.

which characterizes the solution behavior of the nonlinear ODE system locally For

example, if Df 0/ is a Hurwitz matrix, there exists a neighborhood U of the origin and M; k > 0 such that the solution   .I x0/ of the nonlinear ODE system with

stability, yet the system displays asymptotic stability characteristics To definitivelyprove asymptotic stability in these cases, the contributions of Barbashin, Krasovski,and LaSalle may be used in the form of LaSalle’s Invariance Principle First, somedefinitions are required (consider here the non-autonomous case of (1.2))

Definition 1.7 A point z 2 D is called an !-limit point of x0 if there exists a

sequence ft jg satisfying limj!1 t j D 1 and limj!1 x t j / D z The set of all such

!-limit points of x0is called the!-limit set of x0, denoted by!.x0/

Theorem 1.6 Consider (1.2) with f W D  Rn ! Rn , where D is an open and

following conditions hold:

(i) P V( 1.2 ).x/ 0 for all x 2 W;

lim

t!C1dist..t/; ˝/ D 0

The following corollary can be given

1.2 Ordinary Differential Equations 11

Corollary 1.1 Consider (1.2) with f W D  Rn ! Rn , where D is an open and

that the following conditions hold:

(i) P V(1.2).x/ 0 for all x 2 D;

trivial solution.

stable.

Stability of periodic orbits is briefly detailed by highlighting Floquet theory

as presented in Sect 2.8 in [109] This theory is concerned with the behavior ofperiodic time-varying linear ODE systems of the following form:

P

˚.t/ D A.t/˚.t/;

Recall Eq (1.3), which provides the solution of the linear ODE system Px.t/ D Ax.t/

as the matrix exponential; ˚.tI t0/  exp.A.t  t0// is the fundamental matrixsolution However, this does not generalize to (1.5) as expected Furthermore, the

eigenvalues of A do not describe the stability behavior of (1.5) as in the linearODE case

Definition 1.8 Let˚  ˚.I t0/ be the unique solution of (1.6) Then M  ˚.T; 0/

is called the monodromy matrix of (1.5) and its eigenvalues are called the Floquetmultipliers

Floquet’s Theorem is given as follows

Theorem 1.7 Let M be the monodromy matrix of (1.5) and e M  1

12 1 Basic Theory

This result is a useful tool in characterizing stability of a periodic orbit bylinearizing (1.2) about a periodic orbit (see [109] for details) Noting that thelinearization of an autonomous nonlinear system about a periodic orbit alwaysproduces a monodromy with at least one eigenvalue equal to one (see Theorem 4.19

in [109]), the following definition is made

Definition 1.9 Let' be a periodic orbit of (1.5) Then' is said to be

(i) linearly stable if all of its Floquet multipliers have magnitude less than or equal

Theorem 1.8 Let ' be a periodic orbit of a smooth vector field that is linearly

asymptotically stable (i.e., the spectrum of its Poincaré map is inside the unit circle), then it is asymptotically stable.

As remarked earlier, the stability definitions and criteria outlined above arealso applicable to the case in which (1.2) admits a time-varying solution (e.g., anisolated2periodic orbit) This is of particular importance in the present monograph

as disease-free periodic3 solutions are present in infectious disease models withpulse vaccination schemes Such a periodic orbit may attract nearby solutions(i.e., pointwise, asymptotically), leading to a physical system which exhibits anoscillatory steady state

Lastly, comparison theorems are presented which converts the study of theoriginal ODE system into that of a simpler system [77]

Theorem 1.9 Let m 2 C1.J; R/ and g 2 C.R2; R/ Suppose that

Pu.t/ D g.t; u.t//;

2 An orbit is called isolated if there exists a neighborhood containing said orbit for which there exists no other periodic orbit (This is not possible in linear ODE systems.)

3 A solution  .I x0/ of (1.2) is said to be a periodic if there exists T > 0 such that '.t C

TI x0/ D '.tI x0/ for all time t The smallest T for which this equality holds is called the period.

1.2 Ordinary Differential Equations 13

If in Theorem 1.9, m 2 C1.J; R n / and g 2 C.R nC1; Rn/, the conclusion of

the theorem remains valid under the additional assumption that g t; u/ is monotone increasing in u (i.e., for each i 2 f 1; : : : ; ng, g i t; u/ g i t; v/ whenever

quasi-u v and u iD vi)

Theorem 1.10 Let m 2 C1.J; R n / and g 2 C.R nC1; Rn / such that (1.7) holds

Pu.t/ D g.t; u.t//;

The comparison theorems outlined above hold under relaxed assumptions on the

function m (e.g., continuity) but have been restated in these useful forms for present

purposes

1.2.3 Partial Stability

Extending the stability concepts outlined in Definition 1.5, stability analysis ofODE systems with respect to a set of the state variables is known as partialstability Arising in applications requiring performances of certain state variables,this concept was first formulated by A.M Lyapunov and later by V.V Rumyantsev.Consider again the ODE system (1.2) and let the components of the state variable x

be partitioned into the following groups (based on the nature of the problem):

1 The set of basic variables fy1; : : : ; y m g  fx1; : : : ; x n g, 1 < m < n, whose

stability properties are of interest

2 The remaining variables denoted by fz1; : : : ; z p g, p D n  m, called the

uncontrollable variables

By construction, fy1; : : : ; y m g [ fz1; : : : ; z p g D fx1; : : : ; x ng and, without loss of

generality, suppose that a solution x  x.I t0; x0/ of (1.2) can be written as x  y1; : : : ; y m ; z1; : : : ; z p /  y; z/ Here the uncontrollable variables are not of interest

in any stability analysis, however, the dynamics of the basic variables are related tothe dynamics of the uncontrollable variables As a result, the analysis of the partialstability of (1.2) requires an analysis of the behavior of all state variables Considerthe autonomous case of (1.2) (i.e., f t; x/  f x/) and assume that f 0/ D 0.

Definition 1.10 Let  .I x0/  y.I x0/; z.I x0// be the unique solution ofsolution of (1.2) inRCand let x0 y0; z0/ Then the trivial solution of (1.2) is said

to be

(ii) exponentially y-stable if there exist constants ˇ; ; C > 0 such that if ky0k < ˇ

then ky.tI x0/k < Cky0k exp. t/ for all t  0;

(iv) globally asymptotically (exponentially) y-stable if it is asymptotically

(expo-nentially) stable andˇ is arbitrary,

(v) unstable if (i) fails to hold

variables and one uncontrollable variable) Partial .y1; y2/-stability of the trivialsolution of (1.2) implies that for any

This stability notion is not concerned with the behavior of the uncontrollable

solution mapping z1.I x0/ Moreover, the initial condition is in a ı-cylinder (with

1; only the initial values of the basicvariables are tied to

1.3 Impulsive Systems

An impulsive differential equation (IDE) system is a natural way to model theevolution of a system which experiences instantaneous changes in the system state,

called impulsive effects Let D Rn be open and connected, let f W R  D ! R n,

and let g k W D ! R n for each k 2 N Given t0 2 T and x0 2 D, an IVP in IDEs is

given by the following dynamic system:

Px.t/ D f t; x.t//; 8t … fT kg;

k x.t//; 8t 2 fT kg;

(1.10)

where h!0Cx t  h/ and fT k g  fT k 2 T W k 2 Ng is the set of

impulsive times (also called impulsive moments) The impulsive times necessarily

1.3 Impulsive Systems 15

satisfy t0< T1and T k1< T k for each k 2 N For t ¤ T k, (1.10) evolves as an ODE

system and at t D T k, an impulsive effect is applied A solution of (1.10) is defined

as follows (see, e.g., [15])

Definition 1.11 Let T  R contain t0 A function W T ! R nis a solution of (1.10)

in T if it satisfies the following:

(i)  is C1from the right in T;

(ii)  satisfies P.t/ D f t; .t// for all t 2 T n fT kg;

Theorem 1.11 If f 2 C1.R  D; R n /, g k is continuous in D and x C g k x/ 2 D for

Extending the comparison results outlined earlier (e.g., see Theorem1.10), thenext comparison result applies to impulsive systems [77]

Theorem 1.12 Let m W J ! R n

C.Rn; Rn / for each k Suppose that

The IVP in IDEs (1.10) experiences impulses at the fixed times fT kg A moregeneral formulation with variable impulsive moments is given by the system

Px.t/ D f t; x/; 8t … fT k x/g;

k x.t//; t 2 fT k x/g;

(1.12)

where T k x/ < T kC1.x/ and lim k!1 T k x/ D 1 The impulsive moments depend

here on the solution; solutions initialized at different points may therefore havedifferent points of discontinuity More details on IDEs, including variable impulsivemoments, stability, and Lyapunov function methods, are given in [15,76]

16 1 Basic Theory

1.4 Delay Differential Equations

In this part, background material from delay differential equations is presented Forthe remainder of this section, let > 0 be a given real number and for notational

convenience denote by C  C.Œ; 0; R n/ to be the space of continuous functionsmappingŒ; 0 to Rn Equip the space Cwith the sup-norm: given 2 C, let

kk sup

s0 k.s/k:

It follows that C is a Banach space4 Given t0; t f 2 R, such that t0 < t f, and

x 2 C.Œt0 ; t0C t f; Rn /, let x t 2 C denote the mapping x t s/  x.t C s/ for

where0 j t/  for j D 1; 2; : : : ; p It also includes types of integro-differential

equations; for example,

where g are suitably defined functions and > 0 represents an upper bound onthe distribution of delays For theory devoted to systems of integro-differentialequations (including those with unbounded delay), see [78] Given t0 2 R and aninitial function02 C, the IVP associated with (1.13) is given by

Px.t/ D f t; x t/;

4 Complete normed vector space.

1.4 Delay Differential Equations 17

Definition 1.12 Given t f > t0, a function W Œt0 ; t f/ ! Rn

is called a solution

of (1.14) inŒt0; t f / if  2 C.Œt0; t f/; Rn/,  satisfies Eq (1.13) for all t 2 Œt0; t f/,

fx t W t 2 Œt0; t f /g  D, and x.t0C s/ D 0.s/ for all s 2 Œ; 0.

As in the ODE system case, Definition (1.12) can be equivalently formulated as

function x ensures continuity of x t ; if x 2 C.Œt0 ; t0C ˛; Rn/ for some ˛ > 0 then

the mapping t 7! x t is continuous inŒt0; t0C ˛ As in Theorem1.1for the ODEsystem, existence follows from continuity of the right-hand side function

Theorem 1.13 If f is continuous in R  D, then there exists at least one solution

of (1.14) with initial data t0; 0/.

As in Theorem 1.1, uniqueness of the solution requires Lipschitz continuity:

given t 2 T, f t; / is Lipschitz in W  D if there exists L  0 such that

kf t; 1/  f t; 2/k Lk 1 2k; 8 1; 22 W:

Theorem 1.14 If f is continuous in R  D and f t; / is Lipschitz continuous for

of (1.14) with initial data t0; 0/.

Assume that f 2 C1.RC  D; R n /, f t; 0/ D 0 for all t 2 RC, and (1.14)has a unique solution in Œt0  ; 1/ Stability concepts for the DDE (1.14) areanalogous to the ODE case in Definition1.5(e.g., see page 130 in [58]) and can

be examined using a Lyapunov functional approach (an extension of Lyapunov

stability in ODE theory) Consider an auxiliary function V 2 C1.R  C; R/

(e.g., a candidate Lyapunov functional) The time-derivative of V along trajectories

Theorem 1.15 Given any bounded subset W  D, assume that f maps R  W

(ii) @V @t t; / C @V

@x f t; / c3.k 0/k/ for all t; / 2 R  D.

18 1 Basic Theory

Stability of (1.14) can also be demonstrated via Lyapunov functions, which isthe basis for Razumikhin-type theorems The main idea is as follows: supposethat (1.14) has a unique solution x  x I t0; 0/ in Œt0 ; 1/ Suppose that initiallythe solution is contained inside a neighborhood of.0/ but reaches its boundary at

t D t> t0; i.e., fx t/ W t 2 Œt0; t/g  B..0/; r/ for some r > 0 and

kPx.t/k  0:

This case requires consideration of initial data satisfying such properties and the

analysis then is concerned with investigating t 7! x t.0/ In this way, the

time-derivative of a function V 2 C1.R  Rn; R/ along (1.14) can be defined as

; (ii) @V @t t; 0// C @V

@x f t; 0// c3.k 0/k/ if V.t C s; .s// < q.V.t; 0///

1.5 Stochastic Differential Equations

A brief introduction to stochastic ODEs is presented here, with the work of Mao[106] as its basis Let.˝; F ; P/ be a complete probability space; i.e., a set of events

˝, a -algebra of ˝ denoted by F , and a probability measure P W F ! Œ0; 1 such that P ˝/ D 1 and, for any disjoint sequence fA i W i  1g  F ,

Definition 1.13 Let fX k W k 2 Ng and X be R n-valued random variables

(i) If there exists a P-null set˝0 2 F such that for all ! … ˝0 the sequence

fX k !/g converges to X.!/, then fX k g is said to converge to X almost surely;

limk!1 X k D X almost surely (a.s.).

1.5 Stochastic Differential Equations 19

to converge to X in probability.

(iii) If X k and X belong to L p (i.e., pth moment has finite value) and lim k!1 EŒkX k

Xkp  D 0, then fX k g is said to converge to X in the pth moment.

Convergence in the pth moment or almost surely imply convergence in

prob-ability Referring the reader to Chap 1 in [106] for a full background treatment

on the subject matter, we proceed by defining a stochastic process under the Itôinterpretation, with a filtration fF tgt0 satisfying the usual conditions, as follows:

let B W RC ! Rm be an m-dimensional Brownian motion (i.e., mean zero and variance dt) defined on said probability space Consider the following stochastic

where f W Rn ! Rn and g W Rn ! Rnmand the integrals are understood as Itô

integrals A solution x is called an Itô process or stochastic process with stochastic differential dx (see Definition 2.1 in [106] for details) The functions f and g are assumed to be Borel measurable functions and must be L1and L2, respectively, when

composed with a solution x of (1.17) Fundamental theory regarding existence anduniqueness of solutions is given in Sect 2.3 in [106]

Supposing that f 0/ D 0 and g.0/ D 0, the direct method of Lyapunov outlined

in Sect.1.2.2 can be extended to stochastic ODEs Here, stability in probability,moment stability, and almost sure stability are extensions of the classical stabilitynotions (Definition1.5)

Definition 1.14 Let  .I x0/ be a solution of (1.17) inRC Then the trivialsolution is said to be

(i) stable in probability if for every

0k < ı implies that

P

(otherwise it is stochastically unstable);

(ii) stochastically asymptotically stable if it is stochastically stable and for every

0k < ˇ implies that

P

nlim

t!1 k.t/k D 0o

is stochastically asymptotically stable Further, if V is radially unbounded and

LV( 1.17 ).x/ < 0 for all x 2 R n n f0g, then the trivial solution is stochastically

asymptotically stable in the large.

Definition 1.15 Let  .I x0/ be a solution of (1.17) inRC Then the trivialsolution of (1.17) is said to be pth moment exponentially stable if there exist ; C > 0 such that for any x02 Rn,

EŒk.t/k p  Ckx0kp

exp. t/; 8t 2 RC:

Theorem 1.18 Let V 2 C2.Rn; RC/ Assume that there exist positive constants

c1; c2;  > 0 such that the following conditions hold:

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Chapter 2

Hybrid and Switched Systems

The theory of hybrid and switched systems is reviewed here, with an emphasis onstability theory The material presented in this section is mainly based on [84,85];the interested reader is also referred to the hybrid systems literature (e.g., [45,136,

153]) Hybrid dynamical systems [45,136,153] are governed by a combination ofcontinuous and discrete dynamics Switched systems [23,84,144], a type of hybridsystem, evolve according to mode-dependent continuous/discrete dynamics (e.g.,modeled as differential equations) and experience abrupt transitions between modesaccording to a logic-based switching rule Switched systems most often arise in twocontexts [31]: (1) a natural system that experiences sudden changes in its dynamicsbased on, for example, environmental factors; and (2) when switching control isused to stabilize a continuous system The following examples illustrate the mixture

of continuous and discrete dynamics displayed by these types of systems:

1 Multi-controller architecture: Suppose a user is tasked with achieving a desired

performance behavior for a complex dynamic process and no continuous back control exists Given a family of controllers, each of which is designedfor a particular task in the implementation, it may be possible to control theprocess by switching between controllers As the system evolves, a decisionmaker determines which controller is active in the closed-loop system In thisway, the decision maker acts as a logic-based switching supervisor; the governingdynamics are naturally modeled as a switched system (see Fig.2.1)

feed-2 Air conditioner (AC) unit: Consider a climate-control system designed so that when the temperature T increases to some preset threshold T D Thot, the AC

is automatically turned on This precipitates a decrease in the temperature and,

once another threshold T D Tcoldis reached, the AC is automatically turned off.The temperature then changes according to the ambient temperature The larger

the difference between Thotand Tcold, the less number of switches present in thesystem See Fig.2.2for a simple illustration

3 Vehicle with manual transmission: The motion of a vehicle along a fixed path

at time t can be described by its position x t/ and velocity v.t/ A simplified

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X Liu, P Stechlinski, Infectious Disease Modeling, Nonlinear Systems

and Complexity 19, DOI 10.1007/978-3-319-53208-0_2

21

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22 2 Hybrid and Switched Systems

Fig 2.1 Process system with

supervisory switching control Controller 1

Controller 2 Controller 3

Process

Environment Supervisor

x

uu

uu

Fig 2.3 Vehicle with manual

1st gear 2nd gear 3rd gear

version of the model has two control inputs [136]: the current throttle angle, u.t/, and the current gear engaged, g.t/ At any time t, g.t/ 2 f1; 2; 3g (i.e., the mode

of the switched system); changing gears requires an abrupt action by the driverand represents a switch between modes This situation is depicted in Fig.2.3.Switched systems can lead to interesting behavior, such as the instability of aswitched system comprised solely of stable continuous subsystems [85] and theswitched and impulsive control of unstable continuous subsystems that leads to

a stable switched system [54,55] Due to its advantages in improving transientresponse and providing an effective mechanism to deal with highly complexsystems and systems with large uncertainties, hybrid control has also receivedgrowing interest [54] Further, even though the complex system behaviors mayfollow unpredictable patterns, impulsive and switching control is an effectivemethod in achieving stabilization of complex systems using only small controlimpulses in different modes [54] Having conditions which guarantee stability is

a substantial part of the switched systems literature Some common techniques toshow stability of these systems are the switched invariance principle [12,61,62]and common/multiple Lyapunov function techniques [22,23,31,124] There isliterature on families of subsystems that are triangularizable [112], as well asthose that commute [119] Work has been done on the control of discrete switchedsystems [28], the stabilization of nonlinear switched systems using control [115],and criteria for the instability of switched systems under arbitrary switching [135]

A general overview of hybrid and switched systems and its literature can be seen in[31,32,85,136]

2 Hybrid and Switched Systems 23

The mathematical framework of a switched system is characterized in thismonograph by a set of continuous modes (i.e., ordinary differential equations) and alogical rule orchestrating switching between said modes Consider a finite index set

is a smooth function (i.e., C1in an open and connected set D Rn containing thezero vector) The set of autonomous ODE systems

Definition 2.1 Associated with the switching rule are the switching times ft kg,

assumed to satisfy t k1 < t k for each k 2 @  f1; : : : ; n s g, where n s2 N [ fC1g

is the total number of switches (which may be finite or infinite) This is a piecewiseconstant function (assumed continuous from the right) that takes on values from afinite index setP Denote the set of all such switching rules by S

Note that the initial time t0 has been shifted to zero, without loss of generality

(simply redefine a new time as h D t  t0 and a new switching time sequenceaccordingly) The corresponding intervalsŒt k1; t k /, k 2 @, are called the switching

intervals;.t/ D i k 2 P for all t 2 Œt k1; t k / That is, the mode i kis active on theintervalŒt k1; t k/, so that (2.2) evolves according to

Px.t/ D f i k x.t//; 8t 2 Œt k1; t k/;

and the modes switch at t D t k where the active mode changes from.t

k/ WDlimh!0C.t k  h/ 2 P to .t k / 2 P (Note that .t k1/ D .t

k / and .t

k/ ¤

.t k/ in general.) The switching times may be time-dependent (e.g., pre-specified

event times), state-dependent (i.e., t k  t k x/), or a mixture of both; other types of

switching rules, such as Markovian switching, are detailed in [85] Unlike the ODEsystem (1.2), Eq (2.2) admits a set of solutions parameterized by both the initialcondition and switching rule Formally, a solution of (2.2) is considered here as apiecewise smooth function.I x0/ satisfying the following [12]

Definition 2.2 Let t f 2 RC[ fC1g and  2 S with switching times ft k g  Œ0; t f

and mode sequence fi k g  P, a function  W Œ0; t f ! Rn is called a solution

of (2.2) if the function  .I x0; / is a piecewise smooth function satisfying

f.t/ W t 2 Œ0; t f g  D and is an integral curve of the switching vector field;

.t/ D x0C

Z t

0 f .s/ .s//ds; 8t 2 Œ0; t f:

24 2 Hybrid and Switched Systems

Equation (2.2) is often written in the alternative form

Px.t/ D f i k x.t//; t 2 Œt k1; t k /; k 2 @;

x 0/ D x0;where .t/ D i k 2 P for t 2 Œt k1; t k/ The switched system (2.2) has an

equilibrium point Nx 2 D (also called a common equilibrium point) if f i Nx/ D 0 for all i 2 P As it is possible to shift such a point to the origin by setting y  x Nx,

it is assumed, without loss of generality, that f i 0/ D 0 for all i 2 P in the remainder

of this section The stability definitions for the trivial solution of (2.2) are analogous

to the classical ones outlined earlier for ODE systems (see, e.g., [12])

At this point, it is appropriate to define some fundamental objects associatedwith (2.2) More specifically, the set of switching rules exhibiting a nonvanishingdwell-time and the total activation time or switches into a mode (or set of modes)

Definition 2.3 LetSdwell  S denote the set of all switching rules  which have nonvanishing dwell-times; for any x0, there exists > 0 such that the switching

times ft kg associated with  satisfy

N P  jft k 2 Œt1; t2/ W .t k / 2 Pgj

Detailed in the remaining parts of this section, the majority of the literature onswitched systems stability can be categorized into one of the following problems[85]:

1 Finding conditions guaranteeing asymptotic stability of the trivial solution underarbitrary switching rules

2 Identifying classes of switching rules under which the trivial solution is totically stable

asymp-3 Constructing switching rules guaranteeing asymptotic stability of the trivialsolution

For the remainder of this chapter, it is assumed that @ DN (i.e., n sD C1)

2.1 Stability Under Arbitrary Switching 25

2.1 Stability Under Arbitrary Switching

A brief overview of preservation of stability under arbitrary switching is givenhere For more details regarding stability under arbitrary switching, the reader isencouraged to see Chap 2 of [84], from which this material is adapted A first

observation is that if the jth mode of (2.2) is unstable,.t/  j for all t immediately

gives instability; all modes of (2.2) being stable is thus a necessary condition forstability under arbitrary switching However, this condition is not sufficient, asdemonstrated in the following example adapted from [84]

Both A1 and A2are Hurwitz matrices; the trivial solution is globally exponentiallystable for each mode in isolation Suppose that is constructed according to the

following rule: if x1x2< 0, engage mode 1; if x1x2  0, engage mode 2 (Note thatthis switching rule construction is state-dependent.) The trivial solution of (2.2) isunstable in this scenario (see Fig.2.4for an illustration)

The existence of a so-called common Lyapunov function is a sufficient conditionfor asymptotic stability of the trivial solution of (2.2) The main idea is that therate of decrease of a Lyapunov function along (2.2) is unaffected by the switching;asymptotic stability is uniform with respect to the switching rule

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Theorem 2.1 Let V 2 C1.D; RC/ and let W 2 C.D; RC/ be a positive definite and

switching.

Observe that Eq (2.4) is the Lie derivative of V with respect to each mode i 2 P

of the switched system The level sets of V are cut inward by trajectories of (2.2)regardless of the mode sequence and switching times

LaSalle’s Invariance Principle from classical ODE theory (i.e., Theorem1.6) fails

to hold for the switched system (2.2) under arbitrary switching However, Liu et al.[12] provided an invariance principle for switched systems possessing a so-calledweak common Lyapunov function Before presenting the switched invariance result,the following definitions are needed

Definition 2.5 A set˝  D is said to be weakly invariant with respect to (2.2)

if for each x0 2 D, there exists an index i 2 P and constant b > 0 such that the

solutionf i  f i I x0/ of the ith mode (in isolation) satisfies  f i t/ 2 ˝ for either

1Recall this means that W x/ ! 1 as kxk ! @D.

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1 All eigenvalues have negative real part.

which characterizes the... partialstability Arising in applications requiring performances of certain state variables,this concept was first formulated by A. M Lyapunov and later by V.V Rumyantsev.Consider again the ODE system. ..

12 Basic Theory

This result is a useful tool in characterizing stability of a