Automatic feedback control for one class of contact piezoelectric problems

In this paper we investigate the dynamics of solutions of the second order evolution inclusion with discontinuous interaction function which can be represented as the difference of subdifferentials. This case is actual for feedback automatic control problems. In particular, we concider mathematical...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2014
Hauptverfasser:	Zgurovsky, M.Z., Kasyanov, P.O., Paliichuk, L.S.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України 2014
Schriftenreihe:	Системні дослідження та інформаційні технології
Schlagworte:	Математичні методи, моделі, проблеми і технології дослідження складних систем
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/85460
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Automatic feedback control for one class of contact piezoelectric problems / М.Z. Zgurovsky, P.О. Kasyanov, L.S. Paliichuk // Системні дослідження та інформаційні технології. — 2014. — № 1. — С. 56-68. — Бібліогр.: 10 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-85460
record_format	dspace
spelling	irk-123456789-854602015-09-09T17:30:01Z Automatic feedback control for one class of contact piezoelectric problems Zgurovsky, M.Z. Kasyanov, P.O. Paliichuk, L.S. Математичні методи, моделі, проблеми і технології дослідження складних систем In this paper we investigate the dynamics of solutions of the second order evolution inclusion with discontinuous interaction function which can be represented as the difference of subdifferentials. This case is actual for feedback automatic control problems. In particular, we concider mathematical model of contact piezoelectric process between a piezoelectric body and a foundation and for this problem investigate the long-term behavior of state function. We deduce a priory estimates for weak solutions of studied problem in the phase spase. The theorem on the existence of a global attractor for multi-valued semiflow generated by weak solutions of the problem and the structural properties of the limit sets is prooved. The main results of the paper were applied to the investigated piezoelectric problem. Досліджено динаміку розв’язків еволюційного включення другого порядку з розривною функцією взаємодії, яка може бути представлена у вигляді різниці субдиференціалів. Цей випадок є актуальним для задач автоматичного управління зі зворотнім зв’язком. Розглянуто математичну модель контактного п’єзоелектричного процесу між п’єзоелектричним тілом та опорою, і для неї досліджено довгострокову поведінку функції стану. Введено апріорні оцінки для слабких розв’язків даної задачі в фазовому просторі. Доведено теорему про існування глобального атрактора для багатозначного напівпотоку, породженого слабкими розв’язками задачі, та про структурні властивості граничних множин. Основні результати було застосовано до досліджуваної п’єзоелектричної задачі. Исследована динамика решений эволюционного включения второго порядка с разрывной функцией взаимодействия, которая может быть представлена в виде разности субдифференциалов. Данный случай является актуальным для задач автоматического управления с обратной связью. Рассмотрена математическая модель контактного пьезоэлектрического процесса между пьезоэлектрическим телом и опорой, и для нее исследовано долгосрочное поведение функции состояния. Выведены априорные оценки для слабых решений рассматриваемой задачи в фазовом пространстве. Доказана теорема о существовании глобального аттрактора для многозначного полупотока, порожденного слабыми решениями задачи, и о структурных свойствах предельных множеств. Основные результаты были применены к исследуемой пьезоэлектрической задаче. 2014 Article Automatic feedback control for one class of contact piezoelectric problems / М.Z. Zgurovsky, P.О. Kasyanov, L.S. Paliichuk // Системні дослідження та інформаційні технології. — 2014. — № 1. — С. 56-68. — Бібліогр.: 10 назв. — англ. 1681–6048 http://dspace.nbuv.gov.ua/handle/123456789/85460 517.9 en Системні дослідження та інформаційні технології Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Математичні методи, моделі, проблеми і технології дослідження складних систем Математичні методи, моделі, проблеми і технології дослідження складних систем
spellingShingle	Математичні методи, моделі, проблеми і технології дослідження складних систем Математичні методи, моделі, проблеми і технології дослідження складних систем Zgurovsky, M.Z. Kasyanov, P.O. Paliichuk, L.S. Automatic feedback control for one class of contact piezoelectric problems Системні дослідження та інформаційні технології
description	In this paper we investigate the dynamics of solutions of the second order evolution inclusion with discontinuous interaction function which can be represented as the difference of subdifferentials. This case is actual for feedback automatic control problems. In particular, we concider mathematical model of contact piezoelectric process between a piezoelectric body and a foundation and for this problem investigate the long-term behavior of state function. We deduce a priory estimates for weak solutions of studied problem in the phase spase. The theorem on the existence of a global attractor for multi-valued semiflow generated by weak solutions of the problem and the structural properties of the limit sets is prooved. The main results of the paper were applied to the investigated piezoelectric problem.
format	Article
author	Zgurovsky, M.Z. Kasyanov, P.O. Paliichuk, L.S.
author_facet	Zgurovsky, M.Z. Kasyanov, P.O. Paliichuk, L.S.
author_sort	Zgurovsky, M.Z.
title	Automatic feedback control for one class of contact piezoelectric problems
title_short	Automatic feedback control for one class of contact piezoelectric problems
title_full	Automatic feedback control for one class of contact piezoelectric problems
title_fullStr	Automatic feedback control for one class of contact piezoelectric problems
title_full_unstemmed	Automatic feedback control for one class of contact piezoelectric problems
title_sort	automatic feedback control for one class of contact piezoelectric problems
publisher	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
publishDate	2014
topic_facet	Математичні методи, моделі, проблеми і технології дослідження складних систем
url	http://dspace.nbuv.gov.ua/handle/123456789/85460
citation_txt	Automatic feedback control for one class of contact piezoelectric problems / М.Z. Zgurovsky, P.О. Kasyanov, L.S. Paliichuk // Системні дослідження та інформаційні технології. — 2014. — № 1. — С. 56-68. — Бібліогр.: 10 назв. — англ.
series	Системні дослідження та інформаційні технології
work_keys_str_mv	AT zgurovskymz automaticfeedbackcontrolforoneclassofcontactpiezoelectricproblems AT kasyanovpo automaticfeedbackcontrolforoneclassofcontactpiezoelectricproblems AT paliichukls automaticfeedbackcontrolforoneclassofcontactpiezoelectricproblems
first_indexed	2025-07-06T12:39:42Z
last_indexed	2025-07-06T12:39:42Z
_version_	1836901290902290432
fulltext	© M.Z. Zgurovsky, P.O. Kasyanov, L.S. Paliichuk, 2014 56 ISSN 1681–6048 System Research & Information Technologies, 2014, № 1 TIДC МАТЕМАТИЧНІ МЕТОДИ, МОДЕЛІ, ПРОБЛЕМИ І ТЕХНОЛОГІЇ ДОСЛІДЖЕННЯ СКЛАДНИХ СИСТЕМ УДК 517.9 AUTOMATIC FEEDBACK CONTROL FOR ONE CLASS OF CONTACT PIEZOELECTRIC PROBLEMS M.Z. ZGUROVSKY, P.O. KASYANOV, L.S. PALIICHUK In this paper we investigate the dynamics of solutions of the second order evolution inclusion with discontinuous interaction function which can be represented as the difference of subdifferentials. This case is actual for feedback automatic control problems. In particular, we concider mathematical model of contact piezoelectric process between a piezoelectric body and a foundation and for this problem investigate the long-term behavior of state function. We deduce a priory estimates for weak solutions of studied problem in the phase spase. The theorem on the existence of a global attractor for multi-valued semiflow generated by weak solutions of the problem and the structural properties of the limit sets is prooved. The main results of the paper were applied to the investigated piezoelectric problem. INTRODUCTION AND PROBLEM FORMULATION Let us consider a mathematical model which describes the contact between a piezoelectric body and a foundation. We formulate this problem as in [1]. Let dR be a d -dimensional real linear space and dS be the linear space of second order symmetric tensors on dR with the inner product ijij ij τστσ ∑=: and the corresponding norm τττ :2 =dS , ijσ , .dij S∈τ Let us consider a plane electro-elastic material which in its undeformed state occupies an open bounded domain ,Ω dR⊂ .2=d This domain as a result of volume forces and boundary friction can contact with rigid or elastic support. Let the boundary of piezoelectric body Ω be Lipschitz continuous. Assume that the boundary Γ, on the one hand, consists of two disjoint measurable parts DΓ and NΓ , 0)Γ( >Dm and, on the other hand, consists of two disjoint measurable parts aΓ and bΓ , 0)Γ( >am (Figure). Suppose that the body is clamped on ,ΓD so the displacement field ,: dQu R→ ),,( txuu = where ),0(Ω ∞+×=Q , vanishes there. Moreover, a surface traction of density g act on ,ΓN and the electric potential R→Ω:ϕ vanishes on .Γa The body Ω is lying on “support” medium, which introduce frictional effects. The interaction between the body and the Automatic feedback control for one class of contact piezoelectric problems Системні дослідження та інформаційні технології, 2014, № 1 57 support is described, due to the adhesion or skin friction, by a nonmonotone possibly multivalued law between the bonding forces and the corresponding displacements. The body forces of density f consist of force ,ef which is prescribed external loading and force sf which is the reaction of constrains introducing the skin effects, i.e. se fff += . Here sf is a possibly multivalued function of the displacement u . To describe the contact between a piezoelectric body Ω and a foundation let us consider the basic piezoelectric equations: equation of motion, equilibrium equation, strain-displacement equation, equation of electric field-potential and other constitutive relations (see [1] and references therein). We suppose that the process is dynamic. Let us set the constant mass density .1=ρ Then we have the equation of motion for the stress field and the equilibrium equation for the electric displacement field respectively: , in Div Qufu ttt γσ −=− (1) , in 0div QD = where (Ω)L∞γ ∈ is nonnegative function of viscosity; ,: dQ S→σ ( )ijσ σ= is stress tensor; ,Ω: dD R→ ),( iDD = , 1, 2i j = is the electric displacement field; )(Div , jijσσ = is the divergence operator for tensor valued functions; )(div , iiDD = is the divergence operator for vector valued. Equation (1) regulates the change in time of the mechanical state of the piezoelectric body. The stress-charge form of piezoelectric constitutive relations describes the behavior of the material and are following: , in )()( T QEu ϕεσ PA −= , in )()( QEuD ϕε BP += where dd SS →×Ω:A is a linear elasticity operator with the elasticity tensor ,)( ijklaa = 2,1,,, =lkji ; dd RS →×Ω:P is a linear piezoelectric operator represented by the piezoelectric coefficients ),( ijkpp = ;2,1,, =kji dd SR →×Ω:TP is transpose to P operator represented by )()( TT kijijk pp ==P , ,2,1,, =kji dd RR →×Ω:B is a linear electric permittivity operator with the dielectric constants ),( ijββ = ;2,1, =ji )),(()( uu ijεε = , 1, 2i j = is the linear strain tensor; )),(()( ϕϕ iEE = the electric vector field. Figure. Partition of Γ M.Z. Zgurovsky, P.O. Kasyanov, L.S. Paliichuk ISSN 1681–6048 System Research & Information Technologies, 2014, № 1 58 The elastic strain-displacement and electric field-potential relations are given by , in ))((2/1)( T Quuu ∇+∇=ε . in )( QE ϕϕ −∇= We consider the reaction-displacement law of the form: , in )),(,()),(,(),( 21 QtxuxGtxuxGtxf s ∂−∂∈− where RR →× d iG Ω: , 2,1=i are measurable in ( , )x u , convex in u for a.e. Ω∈x functionals; ),,( ⋅∂ xGi 2,1=i are their subdifferentials [2, Chapter 2]. Let 0u be the initial displacement and 1u be the initial velocity. The classical formulation of the mechanical model can be stated as follows: find a displacement field u on dR×Ω and an electric potential ϕ on R×Ω such that: , in Div Quffu tsett γσ −+=− , in 0div QD = ,in )()( T QEu ϕεσ PA −= , in )()( QEuD ϕε BP += ,in )),(,()),(,(),( 21 QtxuxGtxuxGtxfs ∂−∂∈− (2) ),,0( on Γ ),,0( on Γ0 TgnTu ND ×=×= ),,0( on Γ0 ),,0( on Γ0 TDnT ba ×=×=ϕ ,)0( ,)0( 10 uuuu t == where n denotes the outward unit normal to Γ. We now turn to the variational formulation of Problem (2). Let us consider the space 0:);Ω({ 1 =∈= vHvV dR on ).;Ω(}Γ 1 d D H R⊂ Let =H ),;Ω(2 dL R= );Ω( dR=H be a Hilbert spaces equipped with the inner products uvdxvu H ∫=〉〈 Ω , and dxτστσ :, Ω ∫=〉〈 H respectively. Then ),,( VHV be an evolution triple of spaces. Then H〉〈=〉〈 )(),(, vuvu V εε , H)(vv V ε= , Vvu ∈, is the inner product and the corresponding norm on V . Therefore ),( VV ⋅ is Hilbert space. Assume that ,2,1 ,Ω: =→× iG d i RR satisfies standard Carathéodory’s conditions, and there exist )Ω(1 )( Lc i ∈ and ,0)( >iα and that ≤d id R \|\|\|\| )( duxc ii R \|\|\|\|)( )()( α+≤ for a.e. Ω∈x and any du R∈ , .),()( uxGd i i ∂∈ Moreover, )2(α is sufficiently small. Let us set the following hypotheses for the constitutive tensors: Automatic feedback control for one class of contact piezoelectric problems Системні дослідження та інформаційні технології, 2014, № 1 59 (i) ),( ijklaa = ),Ω(∞Laijkl ∈ klijijkl aa = , jiklijkl aa = , ijlkijkl aa = , ijijklijijkl xa τατττ ≥)( for a.e. Ω∈x , d ij +∈=∀ S)(ττ , ;0>α (ii) ),( ijkpp = ;)Ω(∞Lpijk ∈ (iii) ),( ijββ = )Ω(∞ββ Ljiij ∈= , 2\|\|\|\|)( dmx jiij R ζζζβ β≥ for a.e. Ω∈x , d i R∈=∀ )(ζζ , .0>βm Without loss of generality let us consider 0≡g and 0≡ef . Following [1], we present Problem ( 2 ) in the generalized formulation: ( ) ⎪⎩ ⎪ ⎨ ⎧ == ∋∂−∂+++ ,)0( ,)0( , for a.e. ,0))(()()()()( 10 _ 21 uuuu ttuJtuJtAutButu t ttt (3) where ;: VHB → ;: VVA → RHJi →: , 2,1=i are locally Lipschitz functionals, dxxuxGuJ ii ))(,(:)( Ω ∫= , .2,1=i iJ∂ is the Clarke subdifferential for )(⋅iJ , 2,1=i ; );;( VHV is evolution triple. Note that the parameters of Problem (3) satisfy following assumptions [1]: • Assumtion :)(B HHB →: be a linear symmetric such that there exists 0>β such that 2),( HH vvBv β= ;Hv∈∀ • Assumtion : )(A V is a Hilbert space; :A V V→ be a linear, symmetric and there exists 0Ac > such that 2, VAV vcvAv ≥〉〈 ;Vv∈∀ • Assumtion :)(J RHJi →: , 2,1=i be the functions such that (i) )(⋅iJ , 2,1=i are locally Lipschitz and regular (see Clarke [2]), i.e.: - for any ,, Hvx ∈ the usual one-sided directional derivative t xJtvxJ vxJ ii ti )()( lim);( 0 −+ =′ → , 2,1=i , exists, - for all ,, Hvx ∈ ,);();( vxJvxJ ii o=′ where =);( vxJi o t yJtvyJ ii txy )()( lim 0 , −+ = →→ , ;2,1=i (ii) for 2,1=i there exists 0>ic such that ; ),( ),1( HvvJlvcl iHiH ∈∀∂∈∀+≤ (iii) there exists 02 >c such that , ),( ,),( 22 2 HvvJlcvvl HH ∈∀∂∈∀+≤ λ where ( )( ) { \| ( , ) ; }i H iJ u p H p w J u w w H∂ = ∈ ≤ ∀ ∈o denotes the Clarke subdifferentials of ,)(⋅iJ 2,1=i at a point Hu∈ (see Clarke [2] for details); ),0( 1λλ ∈ , 01 >λ : 2 1 2 HVA vvc λ≥ .Vv∈∀ M.Z. Zgurovsky, P.O. Kasyanov, L.S. Paliichuk ISSN 1681–6048 System Research & Information Technologies, 2014, № 1 60 We define Hilbert space X V H= × as the phase space for Problem (3). Let .∞+<<<− Tτ∞ Definition. Let . ∞+<<<∞− Tτ The function ];,[))(),(( XTLuu T t τ∞∈⋅⋅ is called a weak solution for Problem (3) on ],[ Tτ if there exist );,(2 HTLli τ∈ , ,2,1=i ))(()( tuJtl ii ∂∈ for a.e. ( , )t Tτ∈ such that ,V∈∀ψ :),(0 TC τη ∞∈∀ +− ∫ dtttu tHt T )()),(( ηψ τ [ ] .0)()),(()),(()),(()),(( 21 =−+++ ∫ dtttltltutu HHHHt T ηψψψψ τ Theorem 1.4 from [1] provides the existance of a weak solution of Problem (3) on ],[ Tτ with initial data buau t == )( ,)( ττ (4) for any ,Va∈ .Hb∈ In the non-autonomous case the abstract existence results for such problem with nonmonotone skin effects are presented in [1]. The long-time behavior of all weak solutions for this problem with continuous interaction function is investigated by Ball in [3]. The solution dynamics for autonomous model when 02 ≡J is studied in [4], [5]. The particular scalar situation is considered in [6]. Here we consider the case of multidimensional laws with discontinuous interaction function which can be represented as the difference of subdifferentials, that is actual for feedback automatic control problems. The main purpose of this paper is to investigate the long-term behavior of state function, to study the structural properties of the limit sets and to deduce sufficient conditions that direct the system to the desired asymptotic level. PROPERTIES OF SOLUTIONS We consider a class of functions ).];,([ XTCW T ττ = To simplify our conclusions from Assumptions ( )A , ( )B we suppose that ,),(),( ,, ,,),( 2 HHVVVV vBuvuvAuvvAuvu =〉〈=〉〈= β ., ),(2 VvuvBvv HH ∈∀=β (5) Let us set ),()()( 21 uJuJuJ −= .Hu∈ Lebourgue’s mean value theorem [2, Chapter 2] yields the existence of constants 3 4, 0c c > and :),0( 1λμ∈ . 2 )( ),1(\|)(\| 4 22 3 HucuuJucuJ HH ∈∀−−≥+≤ μ (6) According to [7, Lemma 4.1, p. 78], [7, Lemma 3.1, p. 71] and [1, Theorem 1.4] the following existence result holds. Automatic feedback control for one class of contact piezoelectric problems Системні дослідження та інформаційні технології, 2014, № 1 61 Lemma 1. For any ,T<τ ,Va∈ Hb∈ Cauchy problem (3), (4) has a weak solution ).;,(),( XTLuu T t τ∞∈ Moreover, each weak solution T tuu ),( of Cauchy problem (3), (4) on the interval ],[ Tτ belongs to the space )];,([ XTC τ and ).;,( 2 VTLutt τ∈ Let us consider the next denotations: Xba T ∈=∀ ),(τϕ we consider T tT uu ))(),({()(, ⋅⋅=ττ ϕD \| T tuu ),( is a weak solution of (3) on ],[ Tτ , ,)( au =τ })( but =τ . Lemma 1 implies that .)];,([)(, T T WXTC τττ τϕ =⊂D Note that translation and concatenation of weak solutions are also the weak solutions. Lemma 2. If ∞+<<< Tτ0 , ,X∈τϕ ),()( , ττ ϕϕ TD∈⋅ then =⋅)(ψ )()( , ττ ϕϕ sTss −−∈+⋅= D s∀ . If ,Tt <<τ ,X∈τϕ )()( , ττ ϕϕ tD∈⋅ and )()( , τϕψ TtD∈⋅ , then ⎩ ⎨ ⎧ ∈ ∈ = ],[),( ],,[),( )( Ttss tss s ψ τϕ θ belongs to ).(, ττ ϕTD Proof. The proof is trivial. Let Xba T ∈= ),(ϕ and ).()( 2 1)( 21 2 aJaJX −+= ϕϕV (7) Then we have the next lemma. Lemma 3. Let ,∞+<<<∞− Tτ ,X∈τϕ ).())(),(()( , ττ ϕϕ T T tuu D∈⋅⋅=⋅ Then RT →],[: τϕoV is absolutely continuous function, and for a.e. ),( Tt τ∈ 2)())(( Ht tut dt d βϕ −=V . Proof. Let ∞+<<<∞− Tτ , TT t Wuu τϕ ∈⋅⋅=⋅ ))(),(()( be an arbitrary weak solution of (3) on ).,( Tτ As ),;,())(( 2 HTLuJ i τ⊂⋅∂ 1, 2i = then from [7, Lemma 4.1, p. 78] and [7, Lemma 3.1, p. 71] we get that the function 22 )()( VHt tutut +→ is absolutely continuous, and for a.e. :),( Tt τ∈ [ ] =+=+ HtttVHt tutAutututu dt d ))(),()(()()( 2 1 22 ,))(),(())(),(()( 21 2 HtHtHt tutltutltu +−−= β (8) where ))(()( tuJtl ii ∂∈ , 2,1=i for a.e. ),( Tt τ∈ and ).;,()( 2 HTLli τ∈⋅ As )];,([)( 1 HTCu τ∈⋅ and RHJi →: , 2,1=i are regular and locally Lipschitz, due to [5, Lemma 2.16] we obtain that for a.e. ),( Tt τ∈ there exist ),)(( tuJ dt d i o .2,1=i Moreover, ),,())(( 1 TLuJ dt d i τ∈⋅o 2,1=i and for a.e. M.Z. Zgurovsky, P.O. Kasyanov, L.S. Paliichuk ISSN 1681–6048 System Research & Information Technologies, 2014, № 1 62 ,),( Tt τ∈ )),(( tuJp i∂∈∀ ,2,1=i Hti tuptuJ dt d ))(,())(( =o , .2,1=i In particular, for a.e. ),( Tt τ∈ ,))(),(())(( Htii tutltuJ dt d =o .2,1=i Taking into account (8) we finally obtain the necessary statement. The lemma is proved. Lemma 4. Let 0>T . Then any weak solution of Problem (3) on ],0[ T can be extended to a global one defined on ),0[ ∞+ . For any X∈0ϕ and )( 0ϕϕ D∈ the next inequality holds :0 >∀t , )(2 )0( 2 )( 1 1432 1 312 μλ λ ϕ μλ λ ϕ − + + − + ≤ ccc t XX (9) where for an arbitrary X∈0ϕ let )( 0ϕD be the set of all weak solutions (defined on ),0[ ∞+ ) of problem (3) with initial data .)0( 0ϕϕ = Proof. The statement of this lemma follows from Lemmas 1–3, conditions (5), (6) and from the next estimates: ),())(),(()( , , , τττ ϕϕϕτ T T tuuXT D∈⋅⋅=⋅∀∈∀<∀ ],[ Tt τ∈∀ 22 1 3 3 )()( 2 12 HtV uu c c ττ λ +⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ++ =≥≥ ))((2))((2 tϕτϕ VV 4 22 1 22 2)()(1))((2)()( ctututuJtutu HtVHtV −+⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −≥++= λ μ . The lemma is proved. Now let us provide the continuity property for the weak solutions of the main problem in the weak topologies of the phase and the extended phase spaces. Theorem 1. Let ,T<τ T nn Wτϕ ⊂⋅ ≥1)}({ be an arbitrary sequence of weak solutions of (3) on ],[ Tτ such that τϕτϕ →)(n weakly in ,X ∞+→n , and let ],[}{ 1 Tt nn τ⊂≥ be a sequence such that ,0ttn → ∞+→n . Then there exists )(, ττ ϕϕ TD∈ such that up to a subsequence )()( 0ttnn ϕϕ → weakly in ,X ∞+→n . Proof. Let ,Tτ < T nnnn Wuu τϕ ⊂⋅′⋅=⋅ ≥1))}(),(()({ be an arbitrary sequence of the weak solutions of (3) on ],[ Tτ , and ],[}{ 1 Tt nn τ⊂≥ such that . , , in weakly )( 0 ∞+→→→ nttX nn τϕτϕ (10) According to Lemma 4 we have that 1)}({ ≥⋅ nnϕ is bounded on ).;,( XTLW T ττ ∞⊂ Therefore there exists a subsequence 11 )}({)}({ ≥≥ ⋅⊂⋅ nnknk ϕϕ such that Automatic feedback control for one class of contact piezoelectric problems Системні дослідження та інформаційні технології, 2014, № 1 63 ,,);,(instarweakly ,,);,(instarweakly ,,),(a.e.forin)()( ,,],[a.e.forin)()( ,,);,(in ,,);,(instarweakly ,,);,(instarweakly ,,);,(instarweakly ,,);,(instarweakly * 2 2 * 2 , * ∞+→→ ∞+→′→′ ∞+→∈′→′ ∞+→∈→ ∞+→→ ∞+→→ ∞+→′′→′′ ∞+→′→′ ∞+→→ ∞ ∞ ∞ ∞ kVTLAuAu kHTLuBuB kTtVtutu kTtHtutu kHTLuu kHTLll kVTLuu kHTLuu kVTLuu kn kn kn kn kn iikn kn kn kn τ τ τ τ τ τ τ τ τ (11) where );,(2, HTLl in τ∈ be such that ,)()()()()( 2,1, FtAutltltuBtu nnnnn =+−+′+′′ )),(()(, tujtl niin ∂∈ .2,1 ,1 ),,( for a.e. =≥∈ inTt τ Since ij∂ , 2,1=i is demiclosed, the following inclusion holds: ))(()( ⋅∂∈⋅ ujl ii , 2,1=i , where .)(),(: , T Tt Wuu τττ ϕϕ ⊂∈= D For a fixed Vh∈ formula (11) implies that the sequence of real functions )),(( hu kn ⋅ is uniformly bounded and equicontinuous one. According to (9), (11) and the density of V in H we obtain that )()( 0tutu kk nn ′→′ weakly in H and )()( 0tutu kk nn → weakly in V as .∞+→k The theorem is proved. Theorem 2. Let ,Tτ < T nn Wτϕ ⊂⋅ ≥1)}({ be an arbitrary sequence of weak solutions of (3) on ],[ Tτ such that τϕτϕ →)(n strongly in ,X ∞+→n , then up to a subsequence )()( ⋅→⋅ ϕϕn in ),];,([ XTC τ ∞+→n . Proof. The proof follows from [4, Theorem 2] and Lemma 3. MAIN RESULTS Now let us examine the long-time behavior of all weak solutions of the main problem as time .∞+→t For this purpose let us define the m-semiflow G as .0 )},()(\| )({),( 00 ≥∈⋅= ttt ξξξξ DG (12) Denote the set of all nonempty (nonempty bounded) subsets of X by )(XP ))(( Xβ . Note that the multivalued map )(: XPXR →×+G is a strict m-semiflow, i.e. (see Lemma 2): dI),0( =⋅G (the identity map), )),(,(),( xstxst GGG =+ +∈∈∀ RstXx ,, . Further G∈ϕ will mean that )( 0ξϕ D∈ for some .0 X∈ξ M.Z. Zgurovsky, P.O. Kasyanov, L.S. Paliichuk ISSN 1681–6048 System Research & Information Technologies, 2014, № 1 64 We recall, that the m-semiflow G is asymptotically compact if for any sequence G∈≥1}{ nnϕ 1)}0({ ≥nnϕ is bounded, and for any sequence :}{ 1≥nnt ∞+→nt as ∞→n , the sequence 1)}({ ≥nnn tϕ has a convergent subsequence. Let us consider a family )( 00 uXu DK ∈+ = U of all weak solutions of inclusion (3) defined on ),0[ ∞+ . Note that +K is translation invariant one, i.e. +∈⋅∀ K)(u , 0≥∀h +∈⋅ K)(hu , where ,)()( shusuh += 0≥s . On +K we set the translation semigroup 0)}({ ≥hhT , )()()( ⋅=⋅ huuhT , ,0≥h +∈Ku . In view of the translation invariance of +K we conclude that ++ ⊂ KK)(hT as 0≥h . On +K we consider a topology induced from the Fréchet space .);(loc XRC + Note that 0 );( in )()( loc >∀⇔⋅→⋅ + MXRCffn ),];,0([ in )(Π)(Π XMCff MnM ⋅→⋅ where MΠ is the restriction operator to the interval ],0[ M [8, p.179]. We denote the restriction operator to ),0[ ∞+ by .Π+ Let us consider autonomous inclusion (3) on the entire time axis. Similarly to the space );(loc XRC + the space );(loc XRC is endowed with the topology of local uniform convergence on each interval RMM ⊂− ],[ (cf. [8, p. 180]). A function );();(loc XRLXRCu ∞∩∈ is said to be a complete trajectory of inclusion (3) if Rh∈∀ ++ ∈⋅ K)(Π hu [8, p. 180]. Let K be a family of all complete trajectories of inclusion (3). Note that Rh∈∀ , K∈⋅∀ )(u K∈⋅)(hu . We say that the complete trajectory K∈ϕ is stationary if zt =)(ϕ for all t R∈ for some .Xz∈ Following [9, p. 486] we denote the set of the rest points of G by ).(GZ We remark that }.0)()()( , \| ),0{()( _ 21 _ ∋∂−∂+∈= uJuJuAVuuZ G Assumptions )(A and )(J provide that the set )(GZ is bounded in .X Lemma 3 implies the existence of a Lyapunov type function [9, p.486] for m-semiflow G. We consider construction presented in Ball [9], Melnik and Valero [10]. We recall that the set A is said to be a global attractor for G if: (i) ),,( AGA t⊂ ;0≥∀t (ii) A is attracting set, i.e. ),( , ,0)),,((dist XCtCt β∈∀∞+→→AG (13) where XEeDd edED −= ∈∈ infsup),(dist is the Hausdorff semidistance; (iii) for any closed set HY ⊂ satisfying (13), we have .Y⊆A The global attractor is invariant if ,),( AGA t= .0≥∀t Provide the main result of this paper. Automatic feedback control for one class of contact piezoelectric problems Системні дослідження та інформаційні технології, 2014, № 1 65 Theorem 3. The m-semiflow G has the invariant compact in the phase space X global attractor A . For each K∈ψ the limit sets ztXz j →∈= )( \|{)( ψψα for some sequence },∞−→jt ztXz j →∈= )( \|{)( ψψω for some sequence }∞+→jt are connected subsets of )(GZ on which V is constant. If )(GZ is totally disconnected (in particular, if )(GZ is countable), the limits ),(lim tz t ψ ∞−→ − = )(lim tz t ψ ∞+→ + = exist and −z , +z are the rest points; furthermore, )(tϕ tends to a rest point as ∞+→t for every +∈Kϕ . Proof. According to Theorems 1, 2 and [3, Theorem 2.7] we need to provide that m-semiflow G is asymptotically compact. Let ),,( nnn vtG∈ξ )(XCvn β∈∈ , 1≥n , ,∞+→nt .∞+→n Let us check the precompactness of 1}{ ≥nnξ in .X In order to do that without loss of the generality it is sufficiently to extract a convergent in X subsequence from 1}{ ≥nnξ . From Lemma 4 we obtain that there exist such 1}{ ≥knk ξ and X∈ξ that ξξ → kn weakly in ,X XXn a k ξξ ≥→ , .∞+→k Show that Xa ξ≤ . Let us fix an arbitrary .00 >T Then for rather big 1≥k )),(,(),( 00 kkkk nnnn vTtTvt −⊂ GGG . Hence, ),,( 0 kk nn T βξ G∈ where ∈ knβ ),( 0 kk nn vTt −∈ G and ∞+< ≥ Xn k k β 1 sup (see Lemma 4). From Theorem 1 for some 1 1{ , } { , } j j k kk k j n n kξ β ξ β≥ ≥⊂ , XT ∈ 0 β we obtain: . , in weakly ),,( 000 ∞+→→∈ jXT TkT j βββξ G (14) From the definition of G we set: 1≥∀ j ,))(),(( 00 T jjk TuTu j ′=ξ ,))0(),0(( T jjk uu j ′=β ,))(),(( 0000 TTuTu ′=ξ ,))0(),0(( 000 T T uu ′=β where ),];,0([),( 0 XTCuu T jjj ∈′=ϕ ),;,0( * 02 VTLu j ∈′ ),;,0( 0 HTLl j ∞∈ )),(()( ,0)()()()()( , _ 2,1, tuJtltltltAutuBtu jiijjjjjj ∂∈=−++′+′′ ),0(a.e. for 2,1 0Tti ∈= . Let for each ],0[ 0Tt ∈ )).(),(( 2 ))(())(()( 2 1:))(( 21 2 tututuJtuJttI jjjjXjj ′+−+= βϕϕ Then, in virtue of [5, Lemma 2.16], [7, Lemma 4.1, p.78] and [7, Lemma 3.1, p.71], )),(())(( ))(( ttI dt tdI jj j ϕβϕβ ϕ H+−= for a.e. ,),0( 0Tt ∈ where M.Z. Zgurovsky, P.O. Kasyanov, L.S. Paliichuk ISSN 1681–6048 System Research & Information Technologies, 2014, № 1 66 )).(),(( 2 1))(())(),(( 2 1))(())(( 2,21,1 tutltuJtutltuJt jjjjjjj +−−=ϕH From (9), (14) we have that there exists :0 _ >R 0≥∀ j ],0[ 0Tt∈∀ .)()( _ 222 Rtutu VjHj ≤+′ Moreover, , ),;,0( in weakly 020 ∞+→→ jVTLuu j , ),;,0( in weakly 020 ∞+→′→′ jHTLuu j , ),;,0( in 020 ∞+→→ jHTLuu j , ),;,0( in weakly 02, ∞+→→ jHTLll iij (15) , ),;,0( in weakly * 020 ∞+→′′→′′ jVTLuu j . , in )()( ],0[ 00 ∞+→→∈∀ jHtutuTt j For any 0≥j and ],0[ 0Tt ∈ ,))(())0(())(( )( 0 dseseItI st j t t jj −−− ∫+= ββ ϕϕϕ H in particular .))(())0(())(( )( 0 0 0 0 0 dseseITI sT j T T jj −−− ∫+= ββ ϕϕϕ H From (15) and [5, Lemma 2.16] we have . ,))(())(( )( 0 0 )( 0 0 0 0 0 ∞+→→ −−−− ∫∫ jdsesdses sT T sT j T ββ ϕϕ HH Therefore, =+≤ −−− +→+→ ∫ dseseITI sT T T j j j j )( 0 0 0 0 0 0 ))(())0((lim))((lim ββ ∞∞ ϕϕϕ H . ,))(())0(())0((lim))(( 00 00000 TT jj eTIeIITI ββ ∞ ςϕϕϕϕ −− +→ +≤⎥⎦ ⎤ ⎢⎣ ⎡ −+= where ς does not depend on .00 >T On the other hand, from (15) we have ≥ +→ ))((lim 0TI jj ϕ ∞ )).(),(( 2 ))(())(()(lim 2 1 0000002001 2 0 TuTuTuJTuJT Xjj ′+−+≥ +→ βϕ ∞ Therefore, we obtain: 022 2 1 2 1 T X ea βςξ −+≤ .00 >∀T Thus, . X a ξ≤ The Theorem is proved. Automatic feedback control for one class of contact piezoelectric problems Системні дослідження та інформаційні технології, 2014, № 1 67 APLICATION Let us apply main Theorem 3 to Problem (2). Corollary. Under listed above assumptions on parameters of Problem (2) all statements of Theorem 3 for m-semiflow G defined in (12) hold. In particular, for any Vu ∈ such that HuA ∈ there exist such functionals 1G and 2G such that Assumption )(J holds and .}{)( uZ =G CONCLUSIONS For one class of feedback automatic control problems in sence of the global attractor theory the dynamics of solutions is investigated. In particular, we concider the mathematical model of contact piezoelectric problem with discontinuous interaction function which can be represented as the difference of subdifferentials. A priory estimates for weak solutions of studied problem in the phase spase are deduced. This contributes to obtain the existence of the weak solutions and their properties. The existence of global attractor for generated multi-valued semiflow is proved. The structural properties of the limit sets are studied. These results are applied to the considered piezoelectric problem. Thus, it became possible to forecast the long-term behavior of state function and to direct the investigated system to the desired asymptotic level. This research was partially supported by Grants of the President of Ukraine GP/f44/076, GP/F49/070 and Grant of NAS of Ukraine 2273/13. REFERENCES 1. Liu Z., Migórski S. Noncoercive Damping in Dynamic Hemivariational Inequality with Application to Problem of Piezoelectricity // Discrete and Continuous Dynamical Systems Series B. — 2008. — doi:10.3934/dcdsb.2008.9.129. 2. Clarke F.H. Optimization and Nonsmooth Analysis. — NY: Wiley, Interscience, 1983. — 380 p. 3. Ball J.M. Global attaractors for damped semilinear wave equations // DCDS. — 2004. — Vol. 10. — P. 31–52. 4. Zgurovsky M.Z., Kasyanov P.O., Zadoianchuk N.V. Long-time behavior of solutions for quasilinear hyperbolic hemivariational inequalities with application to piezoelectricity problem // Applied Mathematics Letters. — 2012. — Vol. 25. — P. 1569–1574. 5. Zgurovsky M.Z., Kasyanov P.O., Kapustyan O.V., Valero J., Zadoianchuk N.V. Evolution Inclusions and Variation Inequalities for Earth Data Processing III. Berlin: Springer-Verlag, 2012. — 330 p. 6. Gorban N.V., Kapustyan V.O., Kasyanov P.O., Paliichuk L.S. On Global Attractors for Autonomous Damped Wave Equation with Discontinuous Nonlinearity // Continuous and Distributed Systems: Theory and Applications / M.Z. Zgurovsky, P.O. Kasyanov, L.S. Paliichuk ISSN 1681–6048 System Research & Information Technologies, 2014, № 1 68 V.A. Sadovnichiy, M.Z. Zgurovsky (Eds.). — Springer-Verlag, 2014. — P. 221–237. 7. Temam R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics. — NY: Springer-Verlag, 1988. — 500 р. 8. Vishik M., Chepyzhov V. Trajectory and Global Attractors of Three-Dimensional Navier-Stokes Systems // Mathematical Notes. — 2002. — doi:10.1023/ A:1014190629738. 9. Ball J.M. Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations // Journal of Nonlinear Sciences. — 1997. — doi:10.1007/s003329900037. 10. Melnik V.S., Valero J. On attractors of multivalued semi-flows and differential inclusions // Set-Valued Analysis. — 1998. — 6, № 1. — P. 83–111. Received 18.11.2013 From the Editorial Board: the article corresponds completely to submitted manuscript.

Automatic feedback control for one class of contact piezoelectric problems

Institution

Ähnliche Einträge