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Chapter1
Introduction
Thismonographisasummaryoftheauthor’sresearchonthedeterminationof
thequadraticfunctionalsforlinearsystemswithtimedelayanditsapplicationsin
theparametricoptimizationproblem.Thismonographextendsoutcomespresented
inthemonograph[30]andincludesrecentresultsoftheauthorobtainedinlastyears.
Thisbranchofresearchwasproposedformebyprof.HenrykGóreckisupervisorof
myPhDthesis.Prof.HenrykGóreckihaswrittenveryessentialmonograph[47]and
thenmanyothersworks,forinstance[49
–
51,87]andestablishedCracowSchoolof
TimeDelaySystems.
Lyapunovquadraticfunctionalsareusedtotestthestabilityoftimedelaysystems,
incomputationofcriticaldelayvaluesfortimedelaysystems,incomputationof
exponentialestimatesforsolutionsoftimedelaysystems,incalculationofrobustness
boundsforuncertaintimedelaysystems,incalculationofaquadraticperformance
indexofqualityintheprocessofparametricoptimizationfortimedelaysystems.
ThestabilitycriteriafortimedelaysystemsareformulatedintheformofLin-
earMatrixInequalities(LMIs).AnumericalschemeforconstructionofLyapunov
functionalswasproposedbyK.Gu[53].Thismethodstartswithdiscretizationof
theLyapunovfunctional.TheschemeisbasedonLinearMatrixInequality(LMI)
technique.E.Fridman[44]introducedLyapunov-Krasovskiifunctionalsforinves-
tigationofthestabilityoflinearretardedandneutraltypesystemswithdiscrete
anddistributeddelays.Methodwasbasedonanequivalentdescriptorformofthe
originalsystemandobtaineddelay-dependentanddelay-independentconditions
intermsofLMIs.D.Ivanescuetal.[64]proceededwithdelaydependedstability
analysisforlinearneutralsystems,constructedtheLyapunovfunctionalandde-
rivedsufcientdelay-dependentconditionsintermsofLMIs.Q.L.Han[58]obtained
adelay-dependentstabilitycriterionforneutralsystemswithatime-varyingdiscrete
delay.ThiscriterionwasexpressedintheformofLMIandwasobtainedusingtheLya-
punovdirectmethod.Q.L.Han[57]investigatedrobuststabilityofuncertainneutral
systemswithdiscreteanddistributeddelays,whichwasbasedondescriptormodel
transformationandthedecompositiontechnique,andformulatedstabilitycriteriain
theformofLMIs.Q.L.Han[60]consideredthestabilityforlinearneutralsystemswith
norm-boundeduncertaintiesinallsystemmatricesandderivedadelay-dependent
stabilitycriterion.Neithermodeltransformation,northeboundingtechniquefor
crosstermsisinvolvedinderivationofthestabilitycriterion.Q.L.Han[59]devel-
opedthediscretizedLyapunovfunctionalapproachtoinvestigationofthestability
oflinearneutralsystemswithmixedneutralanddiscretedelays.Stabilitycriteria,
whichareapplicabletolinearneutralsystemswithbothsmallandnosmalldiscrete
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