Duality for a Control Problem Involving Support Functions

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1 Appled Matheatcs, 24, 5, Pblshed Ole Deceber 24 ScRes. Dalty for a Cotrol Proble volvg Spport Fctos. Hsa, Abdl Raoof Shah 2, Rsh K. Padey Departet of Matheatcs, Jaypee Uversty of Egeerg ad echology, Ga, da 2 Departet of Statstcs, Uversty of Kashr, Sragar, da Eal: hsa@yahoo.co Receved 4 Septeber 24; revsed 2 October 24; accepted 8 Noveber 24 Copyrght 24 by athors ad Scetfc Research Pblshg c. hs work s lcesed der the Creatve Coos Attrbto teratoal Lcese (CC BY). Abstract Mod-Wer type dalty for cotrol proble wth spport fctos s vestgated der geeralzed covety codtos. Specal cases are derved. A relatoshp betwee or reslts ad those of olear prograg proble cotag spport fctos s otled. Keywords Cotrol Proble, Spport Fcto, Geeralze Covety, Coverse Dalty, Nolear Prograg. trodcto ad Prelares Cosder the followg cotrol proble cotag spport fctos trodced by Hsa et al. [] sbect to (, ) ( ( ) + ( )) Mze f t, t, t S t K dt b a = = () ( ) g t, t, t + S t C, t, =, 2,, (2) ( ) t = htt,, t, t (3) where ) : R s a dfferetable state vector fcto wth ts dervatve ad : R cotrol vector fcto. s a sooth How to cte ths paper: Hsa., Shah, A.R. ad Padey, R.K. (24) Dalty for a Cotrol Proble volvg Spport Fctos. Appled Matheatcs, 5,

2 . Hsa et al. 2) = ab, s a real terval. 3) f : R R R, g: R R R ad h: R R R are cotosly dfferetable. 4) s( K ) ad s( C ), =, 2,, are the spport fcto of the copact set K ad C ( =, 2,, ) respectvely. Deote the partal dervatves of f where by f t, f ad f t, R deotes a -desoal Ecldea space ad [ ] f, f, f,, f, f, f,, f ft = f = f 2 2 t = where sperscrpt deote the vector copoets. Slarly we have h t, h, h ad g t, g, g. X s the space of cotosly dfferetable state fctos : R. Sch that ( a ) = ad b = ad are eqpped wth the or = + D, ad U, the space of pecewse cotos cotrol vector fctos : R havg the for or.. he dfferetal Eqato (2) wth tal codtos epressed as b = + (,, ) d ay be wrtte as H H( y) t a h s s y s st a =,, where H : X U C, R, C, R, beg the space of cotos fcto fro to R defed as H( y, ) = htt,, yt. the dervato of these optalty codto, soe costrat qalfcato to ake the eqalty costrat locally solvable [2] ad hece the Fréchét dervatve of D H(, ) = Q(, ) (say) wth respect to (, ), aely Q = Q (, ) = D H(, ), H (, ) are reqred to be srectve. [], Hsa et al. derved the followg Frtz oh type ecessary optalty for the estece of optal solto of (CP). Proposto. (Frtz Joh Codto): f (, ) s a optal solto of (CP) ad the Fréchét dervatve r Q' s srectve, the there est Lagrage ltplers α R ad pecewse sooth λ : R, : R, z: R ad ω : R sch that for all t, α f t,, + λ t g t,, + ω t + µ t h t,, + µ t =, t = α f t,, + λ t g t,, + µ t h t,, =, t = ( ) ω λ t g t,, + t t =, t µ t z t = s t K ω = s( C ), =, 2,, ω ( ) z t K, t C, =, 2,, αλ, t, t ( ) αλ, t, t, t As [3], Hsa et al. [] poted ot f the optal solto for (CP) s oral, the the Frtz oh type optal codtos redce to the followg Karsh-Kh-cker optal codtos. Proposto 2. f (, ) s a optal solto ad s oral ad Q' s srectve, there est pecewse sooth λ : R wth λ = ( λ, λ2,, λ ), : R, z: R ad ω : R, =, 2,, sch that f t,, + λ t g t,, + ω t + µ t h t,, = µ t (4) = λ µ f t,, + z+ t g t,, + t h t,, =, t (5) = ( ) ω λ t g t,, + t t =, t (6) 3526

3 . Hsa et al. t z t = s t K (7) t ω t = s t C, =, 2,, (8),,, 2,, λ t t = (9) ω z t K, t C, =, 2,, () Usg the Karsh-Kh-cker type optalty codto gve Proposto 2, Hsa et al. [] preseted the followg Wolfe type dal to the cotrol proble (CP) ad proved sal dalty theore der the psedo covety of f ( t,.,. ) + (.) z + λ ( g ( t,.,. ) + (.) ω + µ ( t,.,. ) + (.)) dt for all z K, ad ω C,, 2,, =. (WCD): Maze sbect to (,, ) f t + µ t z t + λ g ( t,, ) + ω + µ t ( h t,, t ) dt = α ( b) a =, = f t,, + λ t g t,, + ω t + µ t h t,, + µ t =, t = λ µ f t,, + t g t,, + t h t,, =, t,,, 2,, λ = t t ω z t K, t C, =, 2,, We revew soe well kow facts abot a spport fcto for easy referece. Let Γ be a copact cove set s t Γ s defed as ( ) υ υ R. he the spport fcto of Γ deoted by { } s t Γ = a t t : t Γ, t A spport fcto, beg cove ad everywhere fte, has a sbdfferetal the sese of cove aalyss, that s, there ests z: R s y t Γ s t Γ + z t y t t for all. he sbdf- sch that feretal of s( Γ ) s gve by s( Γ ) = { z Γ : z = S( Γ) }. Let NΓ ( t ) be oral coe at a pot t Γ. he y N ( ) f ad oly f s y Γ = t y t or, eqvaletly, t Γ ( ) s the sbdfferetal of s at y. order to rela the psedocovety [], Mod-Wer type dal to (CP) s costrcted ad varos dalty theores are derved. Partclar cases are dedced ad t s also dcated that or reslts ca be cosdered as the dyac geeralzato of the dalty reslts for olear prograg proble wth spport fctos. 2. Mod-Wer ype Dalty We propose the followg Mod-Wer type dal (M-WCD) to the cotrol proble (CP): f t,, + t z t dt { } Dal (M-WCD): Maze sbect to ( b) a = = () f t,, + λ t g t,, + t ω t + µ t h t,, = µ t, t (2) = 3527

4 . Hsa et al. λ t µ µ f t,, + z+ g t,, + t h t,, = t, t (3) λ { g ( t,, ) + ω } dt (4) = ( ) µ t h t,, dt (5),, 2,, λ t t = (6) z K (7),, 2,, ω t C = (8) heore. (Weak Dalty): Asse that (A ): (, ) s feasble for (CP),,, λ,, λ, µ, z, ω,, ω s feasble for the proble (M-WCD), (A 2 ): (A 3 ): { f ( t,, ) + ( ) z } d, t for z K s psedocove, ad (A 4 ): λ g ( t,, ) + ( ) ω dt for all ω C, =, 2,, ad µ ( ( ) ( ) ) = are qascove at (, ). ad ad he sp( CD) f CP Proof: Sce t ht,, dt λ t, =, 2,,, t, g t,, + s t C, ht,, t =, t, we have = λ g( t,, ) + ω dt ( ) µ t ht,, dt Cobg these eqaltes wth (4) ad (5) respectvely, we have ( + ) ( + ) λ t g t,, t ω t d t λ t g t,, t ω t dt = = ( ) ( ) hese, becase of the hypothess (A 4 ) yelds µ t ht,, d t µ t ht,, dt ( ) λt ( g ( t,, ) + ω ) + ( ) λ g( t,, ) (9) = { ( ) } µ µ µ t h t,, + t + t h t,, dt (2) Cobg (9) ad (2) ad the sg (2) ad (3), we have {( ) ( f ( ) z) },,

5 hs, de to the psedocovety of f ( t,.,. ) + (.) z dt for z ( ) K ples f t,, + zd t f t,, + zdt Sce z S( K) ( ) yeldg, the above eqalty gves { + } + { } f t,, t K f t,, t zt dt sp( CD) f CP heore 2. (Strog Dalty): f (, ) λ R wth. Hsa et al. s a optal solto of (CP) ad s oral, the there est pecewse sooth : λ = λ, λ2,, λ, z: R ad sch that (,,,,,,, 2,, λ λ zω ω ω ) s feasble for (M-WCD) ad the correspodg vales of (CP) ad 2 (M-WCD) are eqal. f also, the hypotheses of heore hold, the (,, λ,, λ, z, ω, ω,, ω ) s optal solto of the proble (M-WCD). Proof: Sce (, ) s a optal solto of (CP) ad s oral, t follows by Proposto 2 that there est pecewse sooth λ : R, =, 2,,, µ : R, z: R ad ω : R ( =, 2,, ). satsfyg for all t, the codtos (4)-() are satsfed. he codtos (4)-(6) together wth (9) ad () 2 ply that (,, λ,, λ, z, ω, ω,, ω ; µ ) s feasble for (M-WCD). Usg z = s( K), we obta, ( + + ( )) = + f t,, S t K f t,, t zt dt he eqalty of the obectve fctoals of the probles (CP) ad (M-WCD) follows. hs alog wth the 2,, λ,, λ, z, ω, ω,, ω for (M-WCD) follows. he followg gves the Magasara type strct coverse dalty theore: heore 3. (Strct Coverse Dalty): Asse that (A ): (, ) s a optalty solto of (CP) ad s oral; ˆ ˆ ˆ, ˆ, λ,, λ, ˆ µ, zˆ, ˆ ω,, ˆ ω s a optal solto of (M-WCD), hypotheses of heore, the optalty of (A 2 ): (A 3 ): { f ( t,, ) + ( ) z } dt strctly s psedocove for all z K, ad (A 4 ): g ( t,, ) + ( ) ω dt for all ω C, =, 2,, ad µ ( ( ) ) = qas cove., ˆ, ˆ, =.e. ( ˆ ˆ ) he, s a optal solto of (CP). Proof: Asse that (, ) ( ˆ, ˆ) ad ehbt a cotradcto. Sce (, ) (CP). By heore 2 there est ( λ, µ, z, ω,, ω ) wth (,,,,,,,, λ λ zω ω ) s a optal solto of (M-WCD). ad hs Sce (, ) t ht,, dt are s a optalty solto of 2 λ = λ, λ,, λ sch that { f( t,, ) + z } d t= { f( t, ˆ, ˆ) + t ˆ zt ˆ } dt (2) s feasble for (CP) ad ( ˆ ˆ ˆ, ˆ, λ,, λ, ˆ µ, zˆ, ˆ ω,, ˆ ω ) ˆ ( ) ˆ λ + ω λ ( ( ˆ ˆ) + ˆ ˆ ω ) for (M-WCD), we have t g t,, t t d t t g t,, t t dt ( ) ˆ µ t ht,, t d t ˆ µ t ht, ˆ, ˆ t ˆ dt 3529

6 . Hsa et al. hese, becase of the hypothess (A4) ply the erged eqalty ˆ ˆ λ ( g (, ˆ, ˆ) ˆ ) ˆ (, ˆ, ˆ) ˆ t + ω t + µ t h t µ = ˆ + ˆ λ gˆ (, ˆ, ˆ) ˆ (, ˆ, ˆ t + µ t h t) dt = hs, by sg the eqalty costrats (2) ad (3) of (M-WCD) gves By the hypothess (A 2 ), ths ples ( ˆ) ( ˆ ˆ) ( ˆ) ( ( ˆ ˆ ) ˆ ) f t,, + f t,, + z t dt ( + µ ˆ ) > ( ( ˆ ˆ) + ˆ µ ˆ ) = + µ ˆ f t,, zt d t f t,, t zt d t f t,, t zt dt (sg (2)). Coseqetly, we have > t zˆ t dt t z t dt Sce z = S( K) for z K ad zˆ s( K) for z s( z ) dt > s( K) dt hs caot happe. Hece ( ) = ( ˆ ˆ) 3. Coverse Dalty,,. he proble (M-WCD) ca be wrtte as the follows: ψ, µ, z, λ,, λ, ω,, ω Maze: sbect to where a =, b = ( ) ( ) θ,, λ,, λ, µ, ω, ω 2,, ω =, ttt t t t t t t t θ 2 λ λ µ ω ω 2 ω = ttt,, zt, t,, t, t, t, t,, t, t = ω λ t g t,, + t t dt ( ) µ t h t,, t dt z K, t,,, 2,, ω = t C t λ, t, =, 2,, 2 θ = θ (,,,, λµ, ) = + λ + ω + µ + µ = tz f t g t t h t 2 2 θ = θ ( tz,,,, λµ, ) = f + z+ λ g + µ h Cosder 2 θ.,.,., λ., ω2 (., ), ω (., ) µ (.) ad ( t, (., ) (., ) (., ) z(.) (.)) ˆ K, ths yelds, θ λ µ as defg a ap- 353

7 . Hsa et al pgs Q : X Λ W W W V B ad Q : X Z Λ V B respectvely where Λ s the space of pecewse sooth λ, V s space of pececewse sooth µ, W s the space of pecewse of sooth W, =, 2,,, B ad B are Baach spaces. θ = (,, λµω,,, ω,, ω ) ad θ = (,, λµ,, z) wth λ = ( λ,, λ ). Here soe restrctos are reqred o the eqalty costrats. For ths, t sffces that f the Frechet dervatves Q = ( θ,,,,,, θ θλ θµ θ θ ω ω ) ad = θ, θ, θ, θ, θ,, θ, have weak * closed rage. ( ) Q λ µ ω ω heore 4. (Coverse Dalty): Asse that (A ): f, g ad h are twce cotosly dfferetable. λµ,,,, z, ω,, ω s a optal solto of (CP). (A 2 ): (A 3 ): 2 ad Q have weak * closed rages. σ t A, σ t dt = σ t =, t Q (A 4 ): σ t R, ad for soe f + λ t g + µ t h f + λ t g + µ t h A = f λ t g µ t h f λ t g µ t h (A 5 ): ) he gradet vectors λ g + ω ad 2) he gradet vectors µ h + µ ad t h (A 6 ): µ ( a) = = µ ( b). Proof: Sce (,, λ,, λ, µ, z, ω,, ω ) λ are learly depedet, or µ are learly depedet. s a optal solto of (M-WCD), by Proposto there ests α R, γ R, ad β R, ad pecewse sooth fctos φ : R, 2 φ : R, δ : R, =, 2,,, sch that ( ) ( ), 2 f + t f + t g + t h + t f + t g + t h α φ λ µ φ λ µ + γ λ g + ω + β µ t h µ t = t ( ) ( g) h, t α f + z + φ t f + λ t g + µ t h + φ t f + λ t g + µ t h 2 ( ) + γ λ + β µ = ( ) 2 φ t g + ω t + φ t g + γ g + t ω t + δ t =, t (24) h h ( h ) 2 (22) (23) φ φ + φ + β = (25) 2 φ K, α t + t N z t (26) t + t t N, =, 2,,, t (27) φ λ γλ ω C ( ( )) γ λ t g + t ω t dt = (28) β µ t h dt = (29) δ t λ t =, t (3) ( ( t )) α,,, γβδ (3) ( 2 ) α, t, t,,, t, t φ φ γβδ (32) 353

8 . Hsa et al. Mltplyg (24) by λ, t, =, 2,, ad sg over ad the tegratg sg (28), we have whch ca be wrtte as, or ( ) 2 ( ) φ t λ t g + ω t + φ t λ t g dt =, t ( λ ( g + ω )) 2 ( φ φ ) t = (33), d λ g Mltplyg (25) by µ ad the tegratg ad sg (29), we have ( ) 2 t= b φ µ φ µ µ φ µ φ hs ples t t h + t t h dt t t + t t dt = t= a ( φ ( ) ( )) µ 2 µ φ µ t t h t + t t h dt = 2 ( φ φ ) h + µ h µ µ t, t dt = Usg the eqalty costrats (2) ad (3) of the proble (M-WCD) (22) ad (23), we have ( ) ( ) ( ) ( t g ) ( h ) ( f g h ) ( ) ( γ α ) λ ( g + ω ) + ( β α ) µ h + µ 2 + φ t f + λ t g + µ t h + φ t f + λ t g + µ t h = γ α λ + β α µ + φ + λ + µ 2 + φ t f + λ t g + µ t h = Cobg (35) ad (36), we have λ t g + ω µ h + µ γ α + ( β α) µ h λ t g ( ) λ f + λ t g + µ t h f + λ t g + t h φ t + dt 2 = f λ t g t h f t g t h φ hs by preltplyg by φ, φ, ad the tegratg, we have ( ) 2 λ t g + ω t µ 2 h + µ γ α ( φ t, φ t ) d t+ ( β α) ( φ, φ ) dt λ h λ t g λ f 2 + λ t g + µ t h f + λ t g + t h φ t + ( φ, φ ) dt 2 = f + λ t g + t h f + t g + t h φ Usg (33) ad (34), we have (34) (35) (36) 3532

9 . Hsa et al. 2 ( φ φ ) λ, f + λ t g + µ t h f + λ t g + t h φ t t d t = f λ t g t h f t g t h φ hs becase of hypothess (A 4 ) ples 2 Usg φ φ t = = t, t, gves hs, becase of hypothess (A 5 ) ples Asse α =, (37) gves γ β. φ 2 φ t = = t, t ( ) ( ) γ α λ t g + ω t + β α µ t h + µ t =, t ( γ α ) ( β α ) =, = (37) = = fro (24) t follows, t. ( α, φ ( t ), φ 2 ( t ), γβδ,, ( t )) =, t, δ = Coseqetly we have cotradctg (32). Hece α >, γ > ad β >. he relatos (26) ad (27) gves ad ( ω ), =, 2,, t N z yeldg z = s K ad Fro (24), we have ad Fro (25), we have ad he feasblty of (, ) Cosder K t N C t ω t = S t C, =, 2,,. ω g + t t, =, 2,,, t (38) g ω λ + = (39) h =, t (4) µ t h =, t (4) for (CP) follows fro (38) ad (4). ( ( )) f( t,, ) + t z+ λ g( t,, ) + t ω + µ t ( ht,, ) dt = f t,, + S t K dt t z t = s( t K) alog (39) ad (4)). hs alog wth the geeralzed covety hypotheses ples that (,,,,, zλ λ, µω,,, ω ) (by sg optal solto of (M-WCD). 4. Specal Cases Let for t, where B( t ) ad D, (, 2,, ) ( ) 2 = s a = be postve sedefte atrces ad cotos o. he t B t t s t K {, } K = B t z t z t B t z t t 3533

10 . Hsa et al. ad where ( ) 2 t D t t = s t C, =, 2,, { ω ω ω, } C = D t t t D t t t Replacg the spport fcto by ther correspodg sqare root of a qadratc for, we have: Pral (CP ): Mze ( ) 2 f t,, + t Btt dt sbect to = = b a 2 g t,, + t D tt, t, =, 2,, t = h t,,, t (M-WCD ): Maze + sbect to f t,, t B t z t dt = = b a f t,, + λ g t,, + D t ω t + µ t h t,, = µ t, t = = λ ( ω ) g t,, + t D t t dt ( ) µ t h t,, t dt λ z t B t z t, t ω t D t ω t, t, =, 2,, he above par of odfferetable dal cotrol proble has ot bee eplctly reported the lteratre bt the dalty aogst (CP ) ad (M-WCD ) readly follows o the les of the aalyss of the precedg secto. 5. Related Nolear Prograg Probles f the te depedecy of the proble (CP) ad (M-WCD) s reoved, the these probles redce to the followg proble (NP), ts Mod-Wer dal (M-WND): Pral (NP ): Mze f(, ) + S( K) sbect to Dal (M-WND ): Maze sbect to g, + S D, =, 2,, f, + z h(, ) = 3534

11 . Hsa et al. f, + λ g, + ω, + µ h, = = λ µ f, + z+ g, + h, = = λ ( g ( ) S( D )), + µ h, λ, =, 2,,, z K ω C, =, 2,, he above olear prograg probles wth spport fctos do ot appear the lteratre. However, f f(, ) ad S( K ) are replaced by f ( ) ad S( K ) respectvely (NP ), the probles redced to followg stded by Hssa et al. [4]. (P ): Mze f ( ) + S( K) sbect to (M-WCD): Maze sbect to 6. Coclso f + t z g + S C, =, 2,, λ ω µ f + z+ g + + h = λ, =, 2,,, z K ω C, =, 2,, Mod-Wer type dalty for a cotrol proble havg spport fctos s stded der geeralzed covety assptos. Specal cases are dedced. he lkage betwee the reslts of ths research ad those of olear prograg proble wth spport fctos s dcated. he proble of ths research ca be revsted ltobectve settg. Refereces [] Hsa,., Ahad, A. ad Shah, A.R. (24) O a Cotrol Proble wth Spport Fctos. (Sbtted for Pblcato) [2] Crave, B.D. (978) Matheatcal Prograg ad Cotrol heory. Chapa ad Hall, Lado. [3] Mod, B. ad Haso, M. (968) Dalty for Cotrol Proble. SAM Joral o Cotrol, 6, [4] Hsa,., Abha ad Jabee, Z. (22) O Nolear Prograg wth Spport Fcto. Joral of Appled Matheatcs ad Coptg,,

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