341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 J. A. Al-Hawasy Doaa Kateb Jasim Abstract In this paper the Galerkin method is used to prove the existence and uniqueness theorem for the solution of the state vector of the triple linear elliptic partial differential equations for fixed continuous classical optimal control vector. Also, the existence theorem of a continuous classical optimal control vector related with the triple linear equations of elliptic types is proved. The existence of a unique solution for the triple adjoint equations related with the considered triple of the state equations is studied. The Fréchet derivative of the cost function is derived. Finally the theorem of necessary conditions for optimality of the considered problem is proved. Keyword: Triple linear equations of elliptic type, optimal control (vector) of continuous classical type. 1. Introduction Optimal control problems are a fundamental tool in many fields of applied mathematics and taken an important role in many aspects of life, for example in an electric power [1]. In robotics [2]. In biology [3]. In economic [4]. In medicine as [5]. In heat condition [6]. And in many others aspects. This importance encouraged researchers to study problems for the optimal control related with nonlinear ordinary differential equations [7]. Or related with different types of nonlinear partial differential equation as hyperbolic, parabolic, elliptic [8- 10]. Or related with couple of nonlinear hyperbolic, parabolic and elliptic partial differential equation [11-13]. While many others researchers studied the Numann boundary optimal control problems related with couple of nonlinear hyperbolic, parabolic and elliptic partial differential equation [14-16]. This article deals with; the existence theorem for a unique solution (continuous state vector (CSV)) for the triple linear elliptic partial differential equations (TLEPDEqs) is sated, studied and proved by using the Galerkin Method (GM) for fixed continuous classical optimal control vector (CCOCV). The existence theorem for a continuous classical optimal control vector (CCOCV) related with the TLEPDEqs is state and proved. The existence for the unique solution of the triple adjoint equations (TAEqs) which corresponds to the TLEPDEqs is studied. The Fréchet derivative (FD) of the cost function is Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.1.2380 Department of Mathematics, College of Science, University of Mustansiriyah Jhawassy17@uomustansiriyah.edu.iq hawasy20@yahoo.com The Continuous Classical Optimal Control Problems for Triple Elliptic Partial Differential Equations Article history: Received 13 May 2019, Accepted 11 June 2019, Publish January 2020. file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/Jhawassy17@uomustansiriyah.edu.iq, file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/Jhawassy17@uomustansiriyah.edu.iq, file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/hawasy20@yahoo.com file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/hawasy20@yahoo.com 344 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 derived; finally the theorem for necessary conditions of optimality (NCO) is stated and proved. 2 . P ro b l e m D es cri p t i o n Le t Λ b e a b o u n d e d an d o p en co n n e ct ed s u b s et i n w i t h Li p s ch i t z b o u n d a r y ∂Λ . C o n s i d er t h e C C O C V o f t h e TLEPDEqs ( 1 ) ( 2 ) ( 3 ) w i t h t h e Dirchlet b o u n d a r y co n d i t i o n , i n ∂Λ ( 4 ) w h er e ∑ ( ) , , ( ) ( ) , ( ) ( ( ) ( ) ( )) ( ( ̅)) ( t h e s ys t em ( 1 - 4 ) ) , ( ) ( ( ) ( ) ( )) ( ( )) i s t h e cl as s i c al c o n t r o l v e ct o r a n d ( ) ( ( ) ( ) ( )) ( ( )) i s a v ec t o r o f a gi v e n f u n ct i o n , f o r al l ( ) . T h e S et o f A d mi s s i b l e C o n t r o l i s ⃗⃗ ( ( )) , s u ch t h at ⃗⃗ {( ) ( ( )) |( ) ⃗⃗ } w h er e i s co n v ex s e t . T h e C o s t F u n c t i o n a l i s ( ⃗ ) ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ⃗ ⃗⃗ ( ) W h er e α i s a p o s i t i v e r e al n u m b er , ⃗⃗ i s t h e s o l u t i o n v ec t o r o f ( 1 - 4 ) co r r es p o n d i n g t o t h e c o n t i n u o u s classical control vector (CCV) ⃗ a n d ( ) i s a v e ct o r o f d e s i r ed d a t e . T h e C C O C V P ro b l e m i s t o m i n i m i z e ( ⃗ ) ( 5 ) s u b j ec t t o ⃗⃗ ( ) ⃗⃗ . Le t ⃗⃗⃗ ( ) ( ) ( ). W e d en o t e b y ( ) an d ‖ ‖ t h e i n n e r p r o d u c t an d t h e n o r m i n ( ), b y ( ⃗⃗⃗ ⃗⃗⃗ ) , ‖ ⃗⃗⃗ ‖ t h e i n n e r p r o d u ct an d t h e n o r m i n ( ) b y ( ⃗⃗⃗ ⃗⃗⃗ ) ( ) ( ) ( ) an d ‖ ⃗⃗⃗ ‖ ‖ ‖ ‖ ‖ ‖ ‖ t h e i n n e r p r o d u ct a n d t h e n o r m i n ⃗⃗⃗ an d ⃗⃗ ⃗⃗ ⃗ ( t h e d u a l o f ⃗⃗⃗ ) . 3 . W ea k Fo rmu l a t i o n o f t h e T L E PD E q s The weak form (WF) of problem (1-4) are obtained by m u l t i p l yi n g b o t h s i d es o f E q u at i o n s ( 1 - 3 ) b y , an d r es p e ct i v el y, i n t e g r at i n g t h e o b t ai n ed E q u at i o n s an d f i n al l y u s i n g t h e g en e r al i z e G r e en ' s t h e o r em f o r t h e 1 s t t er m i n t h e Left hand side ( L. H . S ) o f t h e t h r e e o b t ai n ed e q u at i o n s , t o g et ( ) ( ) ( ) ( ) ( ) ( ) ( 6 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 7 ) 341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ( ) ( ) ( ) ( ) ( ) ( ) ( 8 ) w h er e ( ) ∬ ∑ , ( ) ∬ =( ), b l en d i n g t o gat h er E q u at i o n s ( 6 ) , ( 7 ) an d ( 8 ) , o n ce get ( ⃗⃗⃗ ) ̆( ⃗⃗⃗ ) ( 9 ) w h er e ( ⃗⃗⃗ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) an d f o r f i x e d ⃗ , ̆( ⃗⃗⃗ ) ( ) ( ) ( ) ( ) ( ) ( ) The following hypotheses are useful to study the existence of unique solution for the WF (9) . H y p o t h es es : a ) ( ⃗⃗⃗ ) i s c o e r ci v e, i . e. ( ) ‖ ‖ , b ) | ( ⃗⃗⃗ )| ‖ ‖ ‖ ⃗⃗⃗ ‖ , . c ) ̆( ⃗⃗⃗ ) ⃗⃗⃗ , ⃗⃗⃗ ⃗⃗⃗ , . T h e G M i s u s ed h e r e t o f i n d t h e s o l u t i o n o f ( 9 ) , T h i s i s d o i n g t h r o u g h ch o o s i n g a f i n i t e s u b s p ac e ⃗⃗⃗⃗ ⃗⃗⃗⃗ ( ⃗⃗⃗ co n t ai n s t h e co n t i n u o u s an d p i e c ew i s e a f f i n e f u n ct i o n s i n Λ ) , h en c e t h e p r o b l e m r ed u c es t o f i n d a n ap p r o x i m a t e s o l u t i o n o f t h e f o l l o w i n g an a p p r o x i m a t i o n p r o b l em ( ⃗⃗ ⃗ ⃗⃗⃗ ) ̆( ⃗⃗⃗ ), ⃗⃗⃗ ⃗⃗⃗ ( 1 0 ) T h eo re m 3 . 1 : Fo r ev e r y f i x e d co n t r o l v e ct o r ⃗ ( ( )) , t h e W F ( 1 0 ) h a s a u n i q u e ap p r o x i m at i o n s o l u t i o n ⃗⃗⃗ . P ro o f : Le t { ⃗⃗ ⃗⃗ ⃗⃗ } b e a f i n i t e b as i s o f ⃗⃗⃗ an d l et ( ) ∑ ⃗⃗ ( ) (∑ ∑ ∑ ) ( 1 1 ) W h er e ⃗⃗ (( ) ( ) ( ) ), f o r , , , [(( ) ) ] [ ( ) ] , an d w i t h ar e u n k n o w n co n s t a n t s . B y u s i n g ∑ ⃗⃗ ⃗⃗⃗ ⃗⃗ , i n ( 1 0 ) , t o g et (∑ ⃗⃗ ⃗⃗ ) ̆( ⃗⃗ ), ( 1 2 ) w h i ch c an b e r e w r i t t en as a l i n e ar a l g eb r ai c s ys t em , i . e. ( 1 3 ) From hypothesis (a), easily once obtained the uniqueness of the solution of problem (13), which gives also the uniqueness of the solution of problem (10). 341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 T h eo re m 3 . 2 ( ), - Fo r e a ch ⃗⃗⃗ ⃗⃗⃗ t h er e ex i s t s a s e q u e n c e { ⃗⃗ } w i t h ⃗⃗ ⃗⃗⃗ f o r e ac h n , s u c h t h at ⃗⃗ → ⃗⃗⃗ s t r o n gl y i n ⃗⃗⃗ . N o w f r o m t h e W F ( 1 0 ) a n d t h eo r em ( 3 . 2 ) , o n c e g et t h at t h e r e e x i s t s a s eq u e n c e o f W F ( ⃗⃗ ⃗ ⃗⃗ ) ̆( ⃗⃗ ) , ⃗⃗ ⃗⃗⃗ ( 1 4 ) w h i ch h as a s eq u e n c e o f s o l u t i o n s { } an d t h e s eq u e n c e ⃗⃗ → ⃗⃗⃗ s t r o n g l y i n ⃗⃗⃗ . T h eo re m 3 . 3 : T h e s eq u e n c e o f s o l u t i o n s { } co n v e r g es s t r o n gl y t o t h e s o l u t i o n ( ). P ro o f : S i n ce f o r e a c h n , i s a s o l u t i o n o f ( 1 4 ) , t h en f r o m h yp o t h es es ( a & c ) , ‖ ‖ , , w i t h T h e n b y u s i n g A l a o gl u t h eo r em , t h e r e ex i s t s a s u b s eq u e n c e o f { } ( f o r s i m p l i ci t y s a y a g a i n { }) , s u ch t h at w ea k l y i n ⃗⃗⃗ . T o p r o v e , t h at t h e s eq u e n c e { } o f s o l u t i o n o f ( 1 4 ) c o n v e r g es t o a v ect o r w h i ch i s t h e s o l u t i o n o f p r o b l em ( 9 ) . F i r s t , f r o m h yp o t h e s i s ( b ) , t h e a b o v e w ea k l y co n v er g e n c es a n d s i n c e ⃗ ⃗⃗ s t r o n gl y i n ⃗⃗⃗ , t h en | ( ⃗ ) ( ⃗⃗ )| | ( ⃗ ⃗⃗ )| | ( ⃗⃗ )| ‖ ‖ ‖ ⃗ ⃗⃗ ‖ ‖ ‖ ‖ ⃗⃗ ‖ → W h i ch m e an s ( ⃗ ) → ( ⃗⃗ ) S e co n d , f r o m t h eo r e m ( 3 . 2 ) ⃗ ⃗⃗ w e ak l y i n ⃗⃗⃗ , t h e n ̆( ⃗ ) ̆( ⃗⃗ ) t o p r o v e → s t r o n g l y i n ⃗⃗⃗ , f r o m h yp o t h es i s ( 1 - a) , o n e h as ‖ ‖ ( ) ( ) ( ) ( ) ( ) ̌( )= ̌( ) ̌( ) → W h i ch co m p l e t e t h e p r o o f o f { } co n v e r g es s t r o n g l y t o w i t h r e s p ec t t o‖ ‖ . T h e u n i q u en es s o f s o l u t i o n i s o b t ai n ed ea s i l y t h r o u g h u s i n g h yp o t h es i s ( a ) . 4 . E x i s t en c e o f a C C O C V : L e m m a 4 . 1 : T h e o p e r at o r ⃗ f r o m ⃗⃗ t o ( ( )) i s Li p s c h i t z co n t i n u o u s ( LC ) , i . e. ⃗⃗⃗⃗ ̆ ⃗⃗⃗⃗ , f o r ̆ P ro o f : Le t ⃗⃗ ⃗ ( ) b e a gi v en co n t r o l v e ct o r o f t h e W F( 6 - 8 ) an d ⃗⃗ ( ) b e t h e co r r es p o n d i n g s t at e v e ct o r s o l u t i o n , w e g et n e w eq u at i o n s f o r ⃗⃗⃗ a n d ⃗⃗ , t h e n b y s u b t r a ct i n g t h e s e n ew eq u at i o n s f r o m t h ei r co r r es p o n d i n g E q u at i o n s ( 6 - 8 ) an d t h e n s u b s t i t u t i n g δ = , δ 341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 , δ = δ , = an d δ i n t h e o b t ai n e d eq u at i o n s , t o g et ( ) ( ) ( ) ( ) ( ) ( 1 5 ) ( ) ( ) ( ) ( ) ( ) ( 1 6 ) ( ) ( ) ( ) ( ) ( ) ( 1 7 ) N ex t b l en d i n g t o g et h e r t h e e q u at i o n s w h i ch o b t a i n ed b y s u b s t i t u t i n g , a n d i n ( 1 5 - 1 7 ) ) r es p e ct i v el y, t o gi v e ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 8 ) A f t e r u s i n g C a u ch - S ch w a r z i n e q u al i t y ( C - S - I) an d a p p l yi n g h yp o t h es i s ( 1 - a) , o n c e h as ‖ ⃗⃗⃗⃗ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ( 1 9 ) S i n c e ‖ ‖ ‖ ⃗⃗⃗⃗ ‖ ‖ ⃗⃗⃗⃗ ‖ a n d ‖ ‖ ‖ ⃗⃗⃗⃗ ‖ , , t h e n ( 1 9 ) b e co m es ‖ ‖ ̆‖ ⃗⃗⃗⃗ ‖ , w i t h ̆ S o ⃗ i s LC o n ( ( )) . L e m m a 4 . 2 [ 1 4 ] : T h e n o r m ‖ ‖ i s w e ak l y l o w er s em i co n t i n u o u s ( W . L. S . ) . L e m m a 4 . 3 : T h e c o s t f u n c t i o n i n ( 5 ) i s W . L. S . . . P ro o f : t h e p r o o f e a s i l y o b t ai n ed t h r o u g h ap p l yi n g l em m a ( 4 . 2 ) , t h e w e a k l y co n v e r g e o f → i n ( ) an d l em m a ( 4 . 1 ) . L e m m a 4 . 4 [ 1 4 ] : T h e n o r m ‖ ‖ i s s t r i ct l y c o n v ex . R e m a rk 4 . 1 : T h e co s t f u n ct i o n ( ) i s s t r i ct l y c o n v ex b y u s i n g L em m a ( 4 . 4 ) . T h eo re m 4 . 1 : If ( ) i s co er ci v e a n d ⃗ i s co n v ex , t h en t h er e ex i s t s C C O C V f o r t h e p r o b l em ( 5 ) . P ro o f : ⃗⃗ i s c o n v ex s i n c e ⃗ i s co n v ex w i t h ( ) , an d ( ) i s co e r ci v e t h e n t h e r e ex i s t a m i n i m i z at i o n s eq u en c e * + ⃗⃗ s u ch t h at ( ) ⃗⃗ ⃗⃗ ( ⃗ ) Therefore ‖ ‖ , ( 2 0 ) T h e n , t h e s eq u en c e * + h as a s u b s e q u en ce f o r s i m p l i ci t y s a y a g ai n* + s u c h t h at → w e ak l y i n ( ( )) , ( b y u s i n g t h e A l o g l u t h eo r em ) . B u t t h eo r e m 3 . 1 , t e l l u s t h at t h e s e q u en ce o f p r o b l em s ( 9 ) h as t h e s eq u e n c e o f s o l u t i o n s{ }. T o p r o v e{ } , i s b o u n d ed i n ⃗⃗⃗ , t h e h yp o t h es es ( a a n d c ) , an d t h e C - S - I, ar e u s ed t o g et t h at : 341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ‖ ‖ ( ) ̆( ) ( 2 1 ) ≤‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ +‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ≤ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ≤ ϖ‖ ‖ W h er e ( ) ( ) ( ) ( ) t h e n ‖ ‖ , f o r e ac h n , w i t h B y A l a o gl u t h e o r em t h er e ex i s t s a s u b s eq u en c e o f { } ( f o r s i m p l i c i t y s a y a g ai n { }) s u ch t h at w e a k l y i n ⃗⃗ S i n c e f o r e ac h n , s at i s f i es t h e w e ak f o r m ( 9 ) , t h en ( ⃗⃗⃗ ) ̆ ( ⃗⃗⃗ ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 ) T o s h o w t h at ( 2 2 ) co n v e r g es t o ( ⃗⃗⃗ ) ̆( ⃗⃗⃗ ) ( 2 3 ) F i r s t , s i n c e , → ( ). T h en b y u s i n g t h e C - S - I a n d h yp o t h es i s ( b ) , o n c e g et s : | ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )| ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ → S ec o n d , t h e r i g h t h a n d s i d e ( R . H . S ) o f ( 2 2 ) c o n v e r g es t o t h e R . . H . S o f ( 2 3 ) , s i n c e → ( ( )) , w h i ch gi v es ( 2 2 ) c o n v e r g es t o ( 2 3 ) . Bu t ( ) i s W . L. S . , w i t h ( ( )) , t h en ( ) ( ) ⃗⃗ ⃗⃗ ( ⃗ ), w h i ch gi v e s ( ) ⃗⃗ ⃗⃗ ( ⃗ ) i . e . , i s a c co cv . O n e c an e as i l y a p p l i e s r e m ar k 4 . 1 , t o g et t h e u n i q u e n e s s o f . 5 . T h e N e c es s a r y C o n d i t i o n s f o r O p t i m a l i t y T h eo re m 5 . 1 : C o n s i d e r t h e co s t f u n c t i o n ( 5 ) , a n d t h e T A E q s ( ) eq u at i o n s o f t h e s t at e E q u at i o n s ( 1 - 4 ) a r e gi v en b y : ( 2 4 ) ( 2 5 ) ( 2 6 ) 341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ( 2 7 ) T h e n t h e Fr é ch et d e r i v at i v e o f i s ( ( ) ⃗⃗⃗⃗ ) ( ⃗⃗⃗⃗ ) P ro o f : W r i t i n g t h e T A E q s ( 1 9 - 2 2 ) b y t h ei r W F, t h e n ad d i n g t h em an d t h en s u b s t i t u t i n g ⃗⃗ ⃗⃗⃗⃗ i n t h e r es u l t i n g e q u at i o n t o ge t t h e f o l l o w i n g W F ( t h e p r o o f o f t h e ex i s t en c es o f a u n i q u e s o l u t i o n f o r t h i s W F i s s i m p l e r t h an t h e p r o o f o f t h eo r em ( 3 . 1 ) ) : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (28) N o w , s u b s t i t u t i n g t h e s o l u t i o n s a n d i n ( 6 ) s e p a r at el y, t h e n s u b t r a ct i n g t h e o b t a i n e d 1 s t e q u at i o n f r o m t h e 2 n d o n e, f i n al l y s et t i n g t o o b t ai n ( ) ( ) ( ) ( ) ( ) ( 2 9 ) S am e s t ep s c an b e u s ed i n E q u at i o n ( 7 ) f o r t h e s o l u t i o n s an d w i t h , ( i n E q u at i o n ( 8 ) f o r t h e s o l u t i o n a n d w i t h ), t o g et r e s p ec t i v el y ( ) ( ) ( ) ( ) ( ) ( 3 0 ) ( ) ( ) ( ) ( ) ( ) ( 3 1 ) Blending together the above triple equations, then subtracting the obtained equation from (28), to get ( ) ( ) ( ) ( ) ( ) ( ) ( 3 2 ) Now, (5) , once get ( ⃗⃗⃗⃗ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ‖ ⃗⃗⃗⃗ ‖ ‖ ⃗⃗⃗⃗ ‖ ( 3 3 ) F r o m ( 3 2 ) an d ( 3 3 ) , o n c e g et ( ⃗⃗⃗⃗ ) ( ) ( ⃗⃗⃗⃗ ) ‖ ⃗⃗⃗⃗ ‖ ‖ ⃗⃗⃗⃗ ‖ ( 3 4 ) f rom lemma (4.1), once obtain ‖ ⃗⃗⃗⃗ ‖ ‖ ⃗⃗⃗⃗ ‖ ( ⃗⃗⃗⃗ ) ⃗⃗⃗⃗ (35) where ( ⃗⃗⃗⃗ ) ( ⃗⃗⃗⃗ ) ( ⃗⃗⃗⃗ ) → as ⃗⃗⃗⃗ → T h e n f r o m t h e d ef i n i t i o n o f FD o f , an d ( 3 4 - 3 5 ) , o n c e g e t ( ( ) ⃗⃗⃗⃗ ) ( ⃗⃗⃗⃗ ) Theorem 5.2 : The CCOCV of (1- 4) is: ( ) with ⃗ and ⃗ . Proof: If i s C C O C V o f ( 1 - 4 ) , t h en ( ) ⃗⃗ ⃗⃗ ( ⃗ ), ⃗ ( ( )) , 311 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 i . e . , ( ) ⃗ ⃗⃗⃗⃗ ⃗ T h u s N C O i s ( ) ( ⃗ ), ⃗ ( ( )) . 6 . C o n cl u s i o n : T h e ex i s t en c e a n d u n i q u e n e s s t h e o r em f o r t h e s o l u t i o n ( C S V ) o f t h e T LE P D E q s i s s t at ed an d p r o v e d s u c ce s s f u l l y b y u s i n g t h e G M w h en t h e C C C V i s gi v e n . A l s o , t h e ex i s t en c e t h e o r em o f a C C O C V g o v er n i n g b y t h e T LE P D E q s i s p r o v e d . T h e ex i s t en c e a n d u n i q u en es s s o l u t i o n o f t h e T A E q s r el at ed w i t h t h e t r i p l e o f t h e s t at e e q u a t i o n s i s s at e d a n d s t u d i ed . T h e d e r i v at i o n o f t h e FD i s gi v e n . Fi n al l y t h e N C O o f t h i s p r o b l e m i s p r o v e d . R ef er en ce s 1. Nguyen, D.B.; Scherpen, J.M.A.; Bliek, F. 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