Table Of ContentJürgenAppell,NguyenThiHien,LyubovPetrova,IrinaPryadko
SystemswithNon-SmoothInputs
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Jürgen Appell, Nguyen Thi Hien,
Lyubov Petrova, Irina Pryadko
Systems with
Non-Smooth
Inputs
|
Mathematical Models of Hysteresis Phenomena,
Biological Systems, and Electric Circuits
MathematicsSubjectClassification2020
Primary:93-02,34A60,46B40,47J40,47L07,93A10;Secondary:15B48,34C05,34C55,34D08,
34D30,37G15,47H04,47H30,49M37,74N30
Authors
JürgenAppell LyubovPetrova
UniversityofWürzburg VoronezhStateUniversity
DepartmentofMathematics FacultyofMechanicsandMathematics
Emil-Fischer-Str.30 Universitetskayapl.1
CampusHublandNord RU-394006Voronezh
D-97074Würzburg RussianFederation
Germany [email protected]
[email protected]
IrinaPryadko
NguyenThiHien VoronezhStateUniversity
HanoiUniversityofIndustry FacultyofMechanicsandMathematics
FacultyofFundamentalSciences Universitetskayapl.1
298CauDien RU-394006Voronezh
HaNoi RussianFederation
Vietnam [email protected]
[email protected]
ISBN978-3-11-070630-7
e-ISBN(PDF)978-3-11-070986-5
e-ISBN(EPUB)978-3-11-070993-3
LibraryofCongressControlNumber:2020952295
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|
TothelovingmemoryofourteacherandfriendBorisNikolaevichSadovsky
(1937–2013)withoutwhomthisbookandmanyotherthingscouldnotexist
Introduction
ThisbookconsistsoftwopartswhichmaybesummarizedunderthetitleSystemswith
non-smoothinputs.Thefirstpart(Chapters1and2)isconcernedwithso-calledsystems
withdiodenonlinearities(orDN-systems,forshort).Thisterminologywascreatedbya
groupofteachersandcollaboratorsoftheVoronezhStateUniversityduringthe1980th
yearsworkingonthemathematicalmodellingandnumericalinvestigationofelectric
circuits.Toexplainthetheory,methods,andapplicationsofthistopic,letusbriefly
recallthestructureofasimpleelectricalnetwork.
Considertheseriescircuit,asshowninFigure1.TheletterErepresentsasource
ofelectromotiveforcegovernedbysomefunctione = e(t).Thismaybeabatteryora
generatorwhichproducesavoltage(potentialdifference)u=u(t)thatcausesinturn
acurrenti=i(t)toflowthroughthecircuitwhentheswitchSisclosed.
ThesymbolRrepresentsaresistancetotheflowofcurrentsuchasthatproduced
byalightbulboratoaster.WhencurrentflowsthroughacoilofwireL,amagneticfield
isproducedwhichopposesanychangeinthecurrentthroughthecoil.Thechangein
voltageproducedbythecoilisproportionaltotherateofchangeofthecurrent,and
theconstantofproportionalityiscalledtheinductanceLofthecoil.
Finally,acapacitor(orcondenser),indicatedbyC,usuallyconsistsoftwometal
platesseparatedbyamaterialthroughwhichverylittlecurrentcanflow.Acapaci-
torhastheeffectofreversingtheflowofcurrentasoneplateortheotherbecomes
charged.
Figure1
Letq = q(t)bethechargeonthecapacitorattimet.Toderiveadifferentialequation
forqwerecallKirchhoff’ssecondlaw:Inaclosedcircuit,theimpressedvoltageequals
thesumofthevoltagedropsintherestofthecircuit.Now,thevoltagedropacrossa
resistanceR(measuredinOhm)attimetequalsRi(t),thevoltagedropacrossanin-
ductanceL(measuredinHenry)attimetequalsLdi/dt,andthevoltagedropacrossa
capacitanceC(measuredinFarad)attimetequalsq(t)/C.Consequently,
di 1
L +Ri(t)+ q(t)=e(t). (1)
dt C
https://doi.org/10.1515/9783110709865-201
VIII | Introduction
Thisislinearinhomogeneousfirstorderdifferentialequationforiwhichmaybe
solvedveryeasilybystandardmethods.Takingintoaccountthati = dq/dt,onemay
transform(1)intotheequation
d2q dq 1
L +R + q(t)=e(t) (2)
dt2 dt C
which is linear second order equation for q which may also be solved by standard
methods,suchasvariationofconstants.
NowsupposethatthecapacitorisreplacedbyanidealdiodeD,i.e.,anelement
whichleadsthecurrentinthedirectionofthearrow(i.e.,fromtheanodetothecath-
ode),butnotinthereversedirection.
Figure2
Inthiscaseoneisledtoaadifferentialinclusion(alsocalledmultivalueddifferential
equationbysomeauthors)oftheform
di
dt ∈E(t)−δi(t)−NQ(i), (3)
withE(t) = e(t)/Landδ = R/L.Theimportantpointisthattheinclusion(3)contains
theso-callednormalconeNQ(i)ofthesetQ = [0,∞)atthepointi.Sincethetheory
ofconesisfundamentalinthemathematicalmodellingofDN-nonlinearities,wetake
thelibertytorecallthebasicnotions.
GivenaclosedconvexsetQ ⊆ ℝmandapointx ∈ Q,thenormalconeofQatxis
definedby
m
NQ(x):={y∈ℝ :⟨y,z−x⟩≤0forallz ∈Q}, (4)
where⟨⋅,⋅⟩denotestheusualscalarproductinℝm.Similarly,thetangentconeofQat
xisdefinedby
m
TQ(x):={z ∈ℝ :⟨z,y⟩≤0forally∈NQ(x)}. (5)
Introduction | IX
ThesetsNQ(x)andTQ(x)formthenapairofmutuallyadjointcones.Itisillumi-
nating to study the analytical properties of the multivalued maps x → NQ(x) and
x → TQ(x);forexample,themapNQ isclosedandmaximalmonotone,butthemap
T isingeneralnotclosed.Wewillstudytheseandrelatedquestionsinsomedetail
Q
inSection1.2;thus,thissectionmaybeconsideredasafriendlyintroductionintothe
theoryofHilbertspaceswithconesfornon-specialistsinthefield.
Now,theconnectionwiththeproblemstatedaboveisasfollows.Inthemostgen-
eralsetting,adifferentialinclusionoftheform
ẋ(t)∈f(t,x(t))−NQ(x(t)), (6)
wheref :[a,b]×Q→ℝmisagivenfunctionwithcertainregularityproperties,isthen
calledsystemwithdiodenonlinearity(DN-system).Interestingly,theinclusion(6)may
beequivalentlyreformulatedintheform
ẋ(t)=τx(t)f(t,x(t)), (7)
whereτx(y)denotesthemetricprojectionofy∈ℝmontothetangentconeTQ(x),i.e.,
theuniqueelementofbestapproximationtoyintheclosedconvexsetTQ(x).
Equation(7)ismuchmorecomplicatedthanthefamiliardifferentialequation
ẋ(t)=f(t,x(t)), (8)
becauseitcontainstheprojectionoperatorτwhichdependsonthepositionx(t)at
timet.AlargepartofChapter1willinfactbedevotedtothequestionwhichresults
fromthetheoryoftheordinarydifferentialequation(8)carryovertotheDN-problem
(7).Itturnsoutthatonehastodevelopalargemachineryofsophisticatedconstruc-
tionstoanswerthisquestion.Oneofthemostimportantresultsisanexistenceand
uniquenesstheoremfor(7),subjecttoasuitableinitialcondition,whichissimilarto,
butnotaconsequenceof,theclassicalCauchy–Peanotheorem.
The theory gives particularly interesting and precise results under two restric-
tions:thedimensionism = 2(i.e.,weareconsideringonlyplanarDN-systems),and
thenonlinearityf in(7)doesnotdependont (i.e.,weareconsideringonlytheau-
tonomous case). Under these and some additional hypotheses (too technical to be
stated here), one may then prove that the equation (7) (or, equivalently, the inclu-
sion(6))hasauniqueclosedtrajectorywhichisorbitallystableinasensetobemade
precise. This result may be considered as an analogue to the well-known Poincaré-
Bendixsontheoremonω-limitsetsofplanardynamicalsystems.
While Chapter 1 is basically concerned with the theoretical background of DN-
systems,Chapter2dealswithapplications.Themostimportantapplicationrefersof
course to electric networks. In Section 2.1 we will show that, under certain natural
physicalhypothesesonthepathsjoiningtwoinputsofadiodicconverter,thecircuit
mayberepresentedasDN-system,andsotreatedbymeansofthetheorydeveloped
inthefirstchapter.
