2-REGULARITY AND 2-NORMALITY CONDITIONS FOR SYSTEMS WITH IMPULSIVE CONTROLS

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1 Yugoslav Jounal of Opeaons Reseach 7 007), Nube, DOI: 0.98/YUJOR07049P -REGUARITY AND -NORMAITY ONDITIONS FOR SYSTEMS WITH IMPUSIVE ONTROS Naal'ya PAVOVA Nonlnea Analyss and Opzaon Depaen, Peoples Fendshp Unvesy of Russa, Moscow, Russa naashaussa@al.u Receved: July 006/ Acceped: May 005 Absac: In hs pape a conolled syse wh pulsve conols n he neghbohood of an abnoal pon s nvesgaed. The se of pas u, μ) s consdeed as a class of adssble conols, whee u s a easuable essenally bounded funcon and μ s a fne-densonal Boel easue, such ha fo any Boel se B, μb) s a subse of he gven convex closed poned cone. In hs acle he conceps of -egulay and -noaly fo he absac appng Φ, opeang fo he gven Banach space no a fne-densonal space, ae noduced. The conceps of -egulay and -noaly play a gea ole n he couse of devaon of he fs and he second ode necessay condons fo he opal conol poble, conssng of he nzaon of a cean funconal on he se of he adssble pocesses. These conceps ae also poan fo obanng he suffcen condons fo he local conollably of he nonlnea syses. The convenen ceon fo -egulay along he pescbed decon and necessay condons fo -noaly of syses, lnea n conol, ae noduced n hs acle as well. Keywods: Ipulsve conol, -egulay condon, -noaly condon.

2 50 N. Pavlova / -Regulay and -Noaly ondons. PROBEM DEFINITION onsde a conollable dynac syse dx ) = f x ), u ),) d Gd ) μ), [, ], ) x ) = x, x ) = x, ) W x, x ) = 0, μ K. 3) Hee [, ] s e, < ae gven, x s a phase vaable, whch acceps n value n he n -densonal ahecal space R, u = u, u,..., u ) R s a conol, f, GW, ae especvely n-densonal, n -densonal, and w-densonal vecofuncons n,, w ae naual nubes). The funcon W s assued o be wce connuously dffeenable. The funcon f s assued o be pece-wse connuously dffeenable, ha s, he neval [, ] s pesenable n es of a fne nube of n nevals [ τ, τ ] so, ha he escon of f a R R [ τ, τ ] s nfnely dffeenable. The se K s defned by 0 K= μ [, ]; R ) : connuous ϕ: ϕ ) K, ϕ ) dμ 0 Boel B [, ], B 0 whee K R s a gven convex closed poned cone, and K s s dual. In ohe wods, μ s a -densonal Boel easue such ha μ B) K fo all Boel subses B. An adssble conol s any pa u, μ): μ K, u [, ]. The ple x ), u ), μ )) [, ], s called an adssble pocess, f u), μ)) s an adssble conol, and x) s a coespondng soluon of equaon ), sasfyng he endpon consans. x ) = x ) f x τ), u τ), τ) dτ G τ) μ τ) τ [, ] [, ] Fo devng he suffcen condons fo local conollably of he syse )- 3), and also n he couse of devaon of he fs and he second ode necessay condons fo he opal conol poble, conssng n he nzaon of a cean funconal on he se of adssble pocesses )-3) conceps of egulay, -egulay and -noaly n consdeed pon xˆ ˆ ˆ, u ), μ )) play a gea ole. Befoe gvng sc defnons of hese conceps fo he syse )-3), we shall explan he essence of hese defnons fo he absac appng Φ, opeang fo he gven Banach space Z o R. e ẑ be a gven pon fo Z, and le he appng Φ be wce connuously dffeenable n a neghbohood of ẑ.

3 N. Pavlova / -Regulay and -Noaly ondons 5 Defnon. Mappng Φ s called egula noal) a he pon ẑ, f Φ ' zˆ ) = R 4) whee s he age of he lnea opeao Φ '. I s nown, ha f he appng Φ s egula a he pon ẑ, hen he plc funcon heoe holds. Besdes, fo he nzaon poble: ϕ z) n, Φ z) = 0, 5) whee ϕ s a gven sooh funcon, he agange pncple s vald fo λ 0 = ) well as he as necessay condons of he second ode. If he pon ẑ s abnoal, ha s Φ' zˆ ) R, hen he saeen of he classcal heoe of plc funcon, does no hold. Slaly, fo he nzaon poble 5) he agange pncple s no nfoave λ 0 = 0 ), and he classcal second-ode necessay condons can be false. Thus hee s a poble of fndng condons oe delcae han he condon 4), whch would guaanee local esolvably of he equaon Φ z) = y fo any z close o he pon yˆ =Φ zˆ ), and also ply subsanal necessay condons of he fs and he second ode fo he poble 5). ondons of ha ype ae -egulay obaned n []) and -noaly []. e us defne hese condons. e T zˆ) = { h Z : Φ zˆ) h = 0, Φ zˆ)[ h, h] Φ ' zˆ)}, and le h T zˆ ). Defne a lnea opeao Gzh, ˆ ): Z Ke Φ ) zˆ R accodng o: Gzh, ˆ )[ ξ, ξ ] =Φ ) zˆ ξ Φ )[, zˆ hξ ]. Defnon. The appng Φ s called -egula a he pon ẑ n he decon h, f Gzh ˆ, ) = R. 6) As s nown [], he exsence of veco h T zˆ ), along whch he appng Φ s -egula a he pon ẑ, guaanees esolvably of he equaon Φ z) = y fo any y close enough o yˆ =Φ zˆ ). Besdes ha, n he poble 5) fo any such h soe necessay condons of he fs and he second ode ae vald also n an abnoal case ha s when Φ' zˆ ) R ). e F ˆ z) be a cone, conssng of λ R, λ 0, such ha Φ ' zˆ ) λ = 0 and hee exss a subspace Π=Π λ) nsde of Z : Π Ke Φ zˆ ), cod Π ; λ, Φ zˆ ) [ zz, ] 0 z Π. z

4 5 N. Pavlova / -Regulay and -Noaly ondons We should noe ha he cone F ˆ z) can be epy. Fo exaple s obvously epy f he appng Φ s noal a he pon ẑ, snce fo 4) follows ha Φ' zˆ ) λ 0 λ 0. Besdes ha, afe jonng zeo o F zˆ ) becoes closed, bu no necessaly convex. Defnon 3. The appng Φ s called -noal a he pon ẑ, f he cone conv F ˆ z) s poned,.e. does no conan nonzeo subspaces he case F ˆ z) s no excluded, snce an epy cone s poned by he defnon). The goal of he pesen pape consss n devng he condons of -egulay and -noaly fo he consdeed dynac syse )-3). To hs end we shall pesen he syse )-3) n an absac fo. e us fx a pon ˆ, ˆ), ˆ)) n x u μ R [, ] K, so ha, ˆ xuμ, ˆ, ˆ) s an adssble pocess, and xˆ ˆ ) = x. Fo any, ), )) n x u μ R [, ] K, close enough o xˆ, uˆ ), ˆ μ )), by vue of he heoes of exsence and connuous dependence of he soluon of he auchy poble on nal condons and gh pa hee s a unque decson x) of he auchy poble. dx ) = f x ), u ),) d Gd ) μ), x ) = x, [, ]. 7) Fo he specfed x, u ), μ )), le us defne he appng w R accodng o he foula Φ x, u ), μ )) = W x, x ; x, u ), μ ))). n Φ : R [, ] Hee x; x, u), μ)), [, ] s a soluon of he auchy poble 7). To nepe he conceps of -egulay and -noaly fo he syse )-3), s necessay o deve foulas fo calculaon of devaves of he appng Φ wh espec o u ), μ )). Fo he gven ξ ) [, ], v K le us denoe by δ x ξv ) he soluon of he syse d δ ˆ ˆ ˆ ˆ xξv)) = x), u),) δxξv) d x), u),) ξ) d G) dv) 8) wh he nal condon δ x ) = 0. 9) ξv Φ ea. The devave opeao xˆ, uˆ), ˆ)): [, ] u, μ) foula Φ xˆ ˆ ˆ ˆ ˆ, u ), μ )) ξ, v) = x, x) δxξ v ), u, μ) whee δ x ξv ) s he soluon of8)-9), ξ ) [, ], v K. w μ R sasfes he

5 N. Pavlova / -Regulay and -Noaly ondons 53 Poof: Usng he heoe of connuous dependence of he soluon on paaees n he n space R [, ], n cean neghbohood U xˆ ) V uˆ) O ˆ μ) of he pon xˆ, uˆ, ˆ μ ) s an opeao n F : x, u, μ) U xˆ ) V uˆ) O ˆ μ) x ) R, whch s defned assocaes wh he ple x, u, μ ) he value of he coespondng soluon x) of he equaon ) a pon x )), and hs opeao s connuous a he pon x ˆ, ˆ, ˆ) u μ hs also ples, ha wll be connuous a all pons of he gven neghbohood). e u V uˆ), μ O ˆ μ), and le x be a soluon of he equaon ). e ξ = u uˆ, v = μ ˆ μ, x = x xˆ, so ha u = uˆ ξ, μ = ˆ μ v, x = xˆ x. Then we have x ˆ ) x ) = xˆ f xˆ τ) x τ), uˆ τ) ξ τ), τ) dτ G τ) d ˆ μ v) τ ), [, ] x ˆ ) = xˆ f xˆ τ), uˆ τ), τ) dτ G τ) dˆ μ τ). [, ] Subacng he second equaon fo he fs one and facozng f up o lnea es, we ge x ) = f xˆ τ), uˆ τ), τ) x τ) f xˆ τ), uˆ τ), τ) ξ τ ) χ τ, x τ), ξ τ))) dτ [, ] x G τ) dv τ), u whee χ = o x ξ ) when x 0, 0. ξ e us consde he funcon δ x ), whch s he soluon of 8)-9), ha s ξv 0) δ x ) = f xˆ τ), uˆ τ), τ) δ x τ) f xˆ τ), uˆ τ), τ) ξ τ)) dτ ξv x ξv u [, ] G τ) dv τ). ) e us esae he dffeence ) = x ) δxξ v ). Fo 0)-) we ge ) = f xˆ τ), uˆ τ), τ) τ)) χ τ, x τ), ξ τ)) dτ. x Fo Gonwall's nequaly [4] follows, ha hen cons χ, x, ξ),

6 54 N. Pavlova / -Regulay and -Noaly ondons and so o x ξ ). On he ohe hand x = δ x ξv, so ha o ) o δx ξ v ξ ). Fo 8)-9) and agan fo Gonwall's nequaly follows, ha δ x cons ξ V [ v]), so ha o)) o ξ V [ v]), and ξv hence o ξ V [ v]), whch eans, ha δ x ) s he an lnea pa of quany x) ha s nceen of a phase vaable x), geneaed by nceen F ξ = u uˆ X v = μ ˆ μ. Thus x ˆ ˆ ˆ, u ), μ )) ξ, v) = = δxξ v ). Ths ples he u, μ) saeen of he lea. Fo he gven ξ), η) [, ], v, θ K le us denoe by δ x ξη v θ ) he soluon of he syse of he equaons n vaaons d δ x ) ) ξηvθ f = xˆ), uˆ),) δ ˆ ˆ xξηvθ ) d x), u),)[ δ xξ v), δxηθ )] f f ) xˆ), uˆ),)[ δ ˆ ˆ xξv ), η)] x), u),)[ δxηθ), ξ)] xu xu f x ˆ), u ˆ),)[ ξ), η)] u wh he nal condon δ x ) = 0. 3) ξηvθ ξv ea. The opeao sasfes he foula Φ u, μ) xˆ, uˆ), ˆ)): [, ] [, ] w μ R Φ u, μ) xˆ, uˆ ), μ ))[ ξ, v), η, θ)] = W xˆ, xˆ )[ δ x ), δ x )] xˆ, xˆ ) δ x ), x ηθ ξv ξηvθ x whee δ ) s he soluon of )-3), ξ), η) [, ], v, θ K. x ξη v θ Poof: Slaly as n he poof of ea, we shall consde he funcon δ x ), whch s he soluon of )-3) a η= ξ, θ = v. Then, facozng f x ˆ ) x, u ˆ ) ξ, ) f x ˆ ), u ˆ ), ) up o he es of he second ode, we ge ξξvv

7 N. Pavlova / -Regulay and -Noaly ondons 55 x ) = f xˆ τ), uˆ τ), τ) x τ) f xˆ τ), uˆ τ), τ) ξ τ) f ˆ ), ˆ ), )[ ), )] ˆ ), ˆ xx x τ u τ τ x τ x τ fxu x τ u τ), τ)[ x τ), ξ τ)] f ˆ ), ˆ uu x τ u τ), τ)[ ξ τ), ξ τ)] ς τ, x τ), ξ τ))) dτ [, ] x G τ) dv τ), u 4) o x ) ) ς = ξ when x 0, 0. ξ e us esae he dffeence R ) = x ) δ x ) δ x ), whee δ xξξvv ) s he soluon of )-3) unde η = ξ, θ = v, ha s ξv ξξvv δx ˆ ˆ ˆ ˆ ξξvv ) = fx x τ), u τ), τ) δxξξvv τ) fxx x τ), u τ), τ)[ δxξ v τ), δxξ v τ)] 5) f xˆ τ), uˆ τ), τ)[ δ x τ), ξ τ)] f xˆ τ), uˆ τ), τ)[ ξ τ ), ξ τ)] dτ. xu ξv uu Fo ), 4) and 5) we have R ) = f ˆ ), ˆ ), ) ) ˆ ), ˆ x xτ uτ τ Rτ fxx xτ u τ), τ)[ x τ), x τ)] f ˆ ), ˆ xx x τ u τ), τ)[ δxξv τ), δxξv τ)] f ˆ ), ˆ xu x τ u τ), τ)[ x τ) δxξ v τ), ξ τ)] ςτ, x), τ ξτ ))) dτ. Fo Gonwall's nequaly follows ha hen R ς, x, ξ) x δxξv 3 x δxξv ξ, whee,, 3 ae soe consans. Usng he nequaly deved n he poof of ea x δx ξ o ξ V [ v]), we ge R ) ) [ ]) ). o x ξ o ξ V v Bu x = R δ x δ x so ξv ξξvv, R o R δ xξv δ xξξvv ξ V [ v]) ). Fo ), 5) and fo Gonwall's nequaly follows ha δ x δ x δ x ξ ξ, ξξvv 4 ξv 5 ξv 6

8 56 N. Pavlova / -Regulay and -Noaly ondons δ x ξ V [ v]), ξv 7 whee 4, 5, 6, 7 ae soe consans. Then Hence and follows Thus ) ) ξ [ ]) ξ [ ] R o R V v V v. ξ ) R o R ) o V [ v]) o ξ V [ v]), ξ ) R o V [ v]). F u, μ) xˆ, uˆ ), ˆ μ ))[ ξ, v), η, θ)] = δ x ), whee δ ) s a soluon ξηvθ x ξη v θ of he syse of he equaons n vaaons ) wh he nal condon 3). The saeen of he lea follows fo hee.. -REGUARITY ONDITION Defnon 4. e h [, ], g K. Fo he poble )-3) a he pon xˆ ˆ ˆ, u ), μ )) he -egulay condon n he decon hg, ) s sasfed f y R ξ, ξ [, ], v, v K: W xˆ, xˆ ˆ ˆ ˆ ˆ ) δxξ v ) x, x)[ δxhg ), δxξ v )] x, x) δxh ξ gv ) = y, x ˆ ˆ, x ) δ x ξ v ) = 0. Fo eas and obvously follows ha -egulay n es of defnon 4 eans -egulay of he noduced appng Φ a pon wˆ = xˆ ˆ ˆ, u), μ)) n he decon h = h, g). The followng lea gves he ceon fo -egulay of he syse )-3). ea 3. Fo he poble )-3) a he pon xˆ ˆ ˆ, u ), μ )) -egulay condon n w w he decon hg, ) holds ue f and only f, hee s no R, 0, q R, such ha fo funcons ψ, ψ, whch ae soluons of he auchy poble ψ ˆ ˆ = x ), u ), ) ψ, 6) x

9 N. Pavlova / -Regulay and -Noaly ondons 57 ψ ˆ ˆ = x ), u ),) ψ E) ψ, 7) ψ ) = xˆ, xˆ ), 8) ψ x W x ) x, x ) x, x ) q, 9) = δ ˆ ˆ ˆ ˆ hg x x he followng ae a place: x ˆ), u ˆ),) ψ) = 0 ebesque-a.e., 0) x ˆ), u ˆ),) ψ) E) ψ) = 0 ebesque-a.e., ) ψ ), G) υ 0 υ K, [, ], ) ψ ), G) υ 0 υ K, [, ], 3) ψ ˆ ˆ ), G) υ) = 0 μ a.e. 4) ψ ˆ ˆ ), G) υ) = 0 μ a.e. 5) ˆ whee ˆ ) d μ υ = ) d ˆ s he Radon Ncody devave. μ Hee f f E) = xˆ), uˆ),) δ ˆ ˆ xhg ) x), u),) h), x xu f f E ˆ ˆ ˆ ˆ ) = x), u),) δxhg ) x), u),) h). xu Poof: By vue of he heoe of sepaably fo convex ses, he -egulauy condon s volaed f and only f w R, 0: xˆ ˆ, x) δxξ v ) W xˆ, xˆ)[ δ x ˆ ˆ hg ), δxξ v )] x, x) δxh ξ gv ), = 0 6) ξ [, ], v, K; ξ [, ], v K: xˆ, xˆ ) δ x ) = 0. ξv e us nepe condon 6). Fo hs pupose we shall consde he lnea opal conol poble

10 58 N. Pavlova / -Regulay and -Noaly ondons W xˆ, xˆ ) δ x ) xˆ, xˆ )[ δ x ), δ x )] ξ v hg ξv x ˆ ˆ, x ) δ x hξ gv ), n, 7) d δ x )) = xˆ), uˆ),) δ x ) d xˆ), uˆ),) ξ ) d G) dv ), ξv ξv 8) d δ x )) = xˆ), uˆ),) δ x ) d xˆ), uˆ),) ξ ) d G) dv ), ξv ξv 9) d δ x )) = xˆ), uˆ),) δ x ) d E ) δ x ) d E ) ξ ), hξgv hξgv ξv 30) δ x ) = 0, 3) ξ v δ x ) = 0, 3) ξv x ˆ, x ˆ) δ x ξ v ) = 0, hξgv 33) δ x ) = 0. 34) In hs poble conol vaables ae ξ ), v ), ξ ), v )), whle phase vaables - ae δ x ), δ x ), δ x )). ξ v ξv hξgv Ponyagn's funcon Halonan) H and he no agangan l of he poble 7)-34) ae gven by H x, ξ, ξ, v, v,, ψ, ψ, ψ3) = = xu ˆ, ˆ, ) δ ˆ ˆ xξ v ψ xu,, ) ξ, ψ xu ˆ, ˆ, ) δ ˆ ˆ xξ,, ) v xuξ, ψ xu ˆ, ˆ,) δxh ξgv E ) δxξv E ) ξ, ψ3, l δ x ), δ x ), δ x ), q) = xˆ, xˆ ) δ x ) ξ v ξv hξgv ξ v W xˆ, xˆ)[ δ x ˆ ˆ hg ), δxξv )] x, x) δxh ξ gv ), x ˆ ˆ, x ) δ x ξ v ), q. x

11 N. Pavlova / -Regulay and -Noaly ondons 59 w Hee q R, and ψ, ψ, ψ 3 ae n-densonal colun vecos. Accodng o he Ponyagn's axu pncple hee exs a veco q and soluons ψ, ψ, ψ 3 of he auchy poble Such ha ψ ˆ ˆ = x ), u ), ) ψ, x ψ ˆ ˆ = x ), u ),) ψ E) ψ, ψ ˆ ˆ 3 = x ), u ), ) ψ3, x ψ ) = ˆ σ, xˆ ), x ψ δ W W x ) σ, x ) σ, x ) q, = ˆ ˆ ˆ ˆ hg x x ψ ) = ˆ σ, xˆ ), 3 x x ˆ), u ˆ),) ψ) = 0 ebesque-a.e., x ˆ), u ˆ),) ψ) E) ψ3) = 0 ebesque-a.e., ψ ), G) υ 0 υ K, [, ], ψ ), G) υ 0 υ K, [, ], ψ ˆ ), G) υ) = 0 ˆ μ a.e. ψ ˆ ), G) υ) = 0 ˆ μ a.e. ˆ whee ˆ ) d μ υ = ) s he Radon Ncody devave. d ˆ μ The lea's saeen edaely follows fo he las elaons.

12 60 N. Pavlova / -Regulay and -Noaly ondons e us defne on ses H and he agangan l by: H xu,,, ψ) = ψ, f xu,, ), lx, x, λ) = λ, Wx, x). 3. -NORMAITY ONDITION n l n R R R R and R n R he Halonan funcon w Hee λ R, and ψ s he n-denzonal veco-colun. e ˆ xuμ, ˆ, ˆ) be he gven adssble pocess. Defnon 5. The pocess ˆ xuμ, ˆ, ˆ) sasfes Ele-agange equaon, f he veco λ 0 exss such ha fo a veco-funcon ψ, whch s he soluon of he auchy poble ψ = H xˆ), uˆ),, ψ))/ x, ψ ) = l xˆ, xˆ, λ)/ σ, 35) he followng holds: ψ ) = l xˆ, xˆ, λ)/, H xˆ), uˆ),, ψ ))/ u = 0 ebesque-a.e., ψ), G) υ 0 υ K, [, ], ψ ), G) ˆ υ) = 0 ˆ μ a.e. ˆ Hee ˆ ) d μ υ = ) d ˆ s he Radon Ncody devave, xˆ μ = x ˆ ˆ ˆ ), x = x ). e us denoe by Λ ˆ xuμ, ˆ, ˆ) he se of vecos λ whch coespond o he gven exeal ˆ xuμ, ˆ, ˆ) by vue of Ele-agange equaons. Fo he foulaon of he second ode condons fo he pocess ˆ xuμ, ˆ, ˆ) we shall consde he followng syse of he equaons d δx)) = xˆ), uˆ),) δx) d xˆ), uˆ),) δu) d G) d δμ)). 36) Hee δu [ ˆ, ], δμ T K μ), and he soluon of he equaon n vaaons should sasfy he condons: xˆ, xˆ ) δx ) = 0, xˆ, xˆ ) δx ) = 0. 37) n e λ Λ xˆ, uˆ). On he space X = R [, ] of pons ς, δu, δμ ) we shall defne he quadac fo Ω λ by he foula l Ω ˆ ˆ λ ς, δu, δμ) = x, x, λ)[ δx ), δx )), δx ), δx ))] x, x) H ˆ, ˆ,, )[ ), )), ), ))]. x u ψ δ x δ u δ x δ u d xu, )

13 N. Pavlova / -Regulay and -Noaly ondons 6 Heenafe δ x s he soluon of he syse of he equaons n vaaons 36) wh he nal condon δ x ) = ς coespondng o δu, δμ ). e χ denoe he lnea subspace of X, whch consss of hose ς, δu, δμ ), such ha ς = δ x. e be a naual nube and le Λ ˆ, ˆ, ˆ =Λ xuμ) denoe he se of hose λ Λ xu ˆ, ˆ, ˆ μ), fo whch he ndex of naowng he fo Ω λ on he subspace χ does no exceed. Defnon 6. Adssble pocess ˆ xuμ, ˆ, ˆ) s called -noal f he cone s poned. Fo egande's condon [4] follows ea 4. Fo soe le he cone λ Λ xu ˆ, ˆ, ˆ μ), such ha convλ xˆ, uˆ, ˆ μ) Λ ˆ xuμ, ˆ, ˆ) be non epy. Then hee exss H x ˆ), u ˆ),, ψ )) 0 ebesque-a.e. e us apply -noaly concep o lnea conollable syses. Assue ha f xu,, ) a0 x, ) ua, ), x a = pecewse sooh veco funcons. Then = = whee 0, a ae gven dx ) = a0 xd,) ua xd,) Gd ) μ), 38) x ) = x, x ) = x, 39) W x, x ) = 0. 40) H xu,,, ψ) = ψ, a x, ) u ψ, a x, ). 4) 0 = onsde he pocess xˆ, uˆ) xˆ ˆ ˆ ˆ ) = x, x ) = x). Whou loss of genealy we shall assue ha u ˆ ) 0. The coespondng syse of he equaons n vaaons s he followng: d δx)) = A ) δxd ) B ) δud ) Gd ) δμ)), xˆ, xˆ ) δx ) = 0, xˆ, xˆ ) δx ) = 0. 4) whee δu [ ˆ, ], δμ T μ), A and B ae defned by foulas: K a ˆ 0 x ), ) ) =, ) = ),..., )), = ˆ ), ). A B b b b a x

14 6 N. Pavlova / -Regulay and -Noaly ondons Then Fo any λ Λ xˆ, uˆ) le H xˆ ), uˆ ),, ψ λ ) ) Dλ ) =, H xˆ ), uˆ ),, ψ λ) ) λ ) =. xu l Ω ˆ ˆ λ ς, δu) = x, x, λ) δx ), δx )), δx ), δx )) x, x ) λδ δ λδ δ ) D x, x x, u d, whee δ x s he soluon of he syse of he equaons n vaaons 4) wh he nal condon δ x ) = ς. Assue ha axes A), B), ), D) and all he devaves can have on [, ] jups only n fne nube p of pons τ,...,. τ p ea 5. Suppose ha fo he soe he cone Λ ˆ, ˆ, ˆ xuμ) s no epy. Then λ Λ ˆ, ˆ, ˆ xuμ), such exss, ha condons ae sasfed ) fo he soluon ψ of adjon equaon 35), coespondng o he veco λ, he followng leads ) H ˆ), ˆ),, )) 0 { x u ψ = τ,..., τ p}, H x ˆ), u ˆ),, ψ )) = 0 { τ,..., τ p}, H x ˆ), u ˆ),, ψ )) 0 { τ,..., τ p}, nd w ς, δu, υ) 43) 44) 45) 3) Hee q nd Δ B λ ) τ nd ) B ). λ = 46) l w ς, δu, υ) = xˆ ˆ, x, λ) δx ), δx )), δx ), δx )) x, x ) λξξ ξυ υυ) D, P, Q, d,

15 N. Pavlova / -Regulay and -Noaly ondons 63 =, = ) /), P B Dλ λ λa Q B DλB λab B Aλ λb λb Δ Mτ = M τ 0) M τ 0) s he jup of he ax M a pon τ, and nd Ψ denoes he ndex of he quadac fo of cean syec ax Ψ. e us noe ha, as can be seen fo he poof gven below, all aces n 46) ae syec. Poof: Unde he hypohess of he heoe hee exss such λ Λ ˆ, ˆ x u), ha nd Ωλ χ. e us conve he fo Ω λ accodng o Hooch by vue of syse 4). Fo convenence, n he sequel we shall o he subscp λ s ax funcons, D, and quadac fo Ω. e us noduce new vaables υ = δ u, υ ) = 0, ξ = x Bυ. Funcons ξ and υ n belong o spaces W, and W especvely and sasfy coelaons ξ = A ) ξ AB B ) υ, xˆ, xˆ ) ξ ) = 0, xˆ, xˆ ) ξ ) B ) υ )) = 0. n Hee W, s he space of n-densonal funcons, whch have pecewse-pschzan fs-ode devave, and W s he space of -densonal pecewse- pschzan funcons. Then q Ω ςδ, u, δμ) =Ω ςδ, u, υ) = w ςδ, u, υ) Δ B) υτ ), υτ ) q ξτ ), Δ υτ ) ξ ), Δ ) υ ) ) B ) υ ), υ ). = τ = τ Hee l w ς, δu, υ) = xˆ ˆ, x, λ) δx ), δx )), δx ), δx )) x, x ) ξξ ξυ υυ υδ ) D, P, Q, V, u d,,, ) /). P = B D A V = B B Q = B DB AB B A B B 47) e us noe, ha Hooch's conveson does no change he ndex of he quadac fo. By he vue of necessay condons of fneness of ndex of he fo w on χ he followng holds V ) = 0 ; Q ) = Q ), Q ) 0. 48)

16 64 N. Pavlova / -Regulay and -Noaly ondons The fs of hese condons s called Hooch's condon, and he second one s a genealzaon of egende's condons. Fo condons 48) follows, ha all he aces eneng 46) ae syec. In [3] he followng nequaly s poved: q nd Ω nd w ςδ, u, υ) nd Δ B) nd ) B ). = τ q Snce nd Ω, las nequaly ples nd Δ B) nd ) B ) = τ. Dec dffeenaon of 4) yelds he foulas: d H V ) = x ˆ), u ˆ),, ψ )), ud u d H Q ) = x ˆ), u ˆ),, ψ )). u d u By he vue of 47), las wo elaons ply 44)-45). REFERENES [] Auynov, A.V., Exeu ondons: Abnoal and Degeneae Pobles M., Facoal, 997. [] Auynov, A.V.,"Iplc funcon heoe as agangan pncple ealzaon. Abnoal pons", Maheacal olleco, 9) 000) 3-6. [3] Auynov, A.V.,Yachovch, V., "-noal pocesses of conollable dynacal syses", Dffeenal Equaons, 388) 00) [4] Ioffe, A.D., Thoov, V.M., "Theoy of exeal pobles M", Scence, 974.