Solving a Class of Non-Smooth Optimal Control Problems

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1 I.J. Itelliget Systes d Applictios, 213, 7, ulished Olie Jue 213 i MECS ( DOI: /iis Solvig Clss of o-sooth Optil Cotrol roles M. H. oori Sdri E-il: Mth.oori@yhoo.co H. R. Erfi Correspodig uthor E-il: Erfi@usc.c.ir A.V. Kyd E-il: Avyd@yhoo.co M. H. Frhi E-il: Frhi@Mth.u.c.ir Fculty of Mtheticl Scieces, Ferdowsi Uiversity of Mshhd, Mshhd, Ir Astrct I this pper, we first propose ew geerlized derivtive for o-sooth fuctios d the we utilize this geerlized derivtive to covert clss of o-sooth optil cotrol prole to the correspodig sooth for. I the ext step, we pply the discretiztio ethod to pproxite the otied sooth prole to the olier progrig prole. Filly, y solvig the lst prole, we oti pproxite optil solutio for i prole. Idex ers Geerlized Derivtive, o-sooth Optil Cotrol, o-lier rogrig I. Itroductio Cosider the followig o-sooth optil cotrol prole: Miiize x ( ) suect to x ( t ) g ( x ( t )) f ( u ( t )), t [, ], x (), x ( t ) X, u ( t ) U, t [, ]. where x (.) :[, ] X is the stte vrile, u(.) :[, ] U is the cotrol vrile d,. Moreover ssue tht fuctio f (.) is sooth d fuctio g (.) (1) is o-sooth (or odifferetile) ut piecewise cotiuous. his id of optil cotrol proles ppers i y fields of scieces such s thetics, physics, ecooics d egieerig. I geerl, o-soothess rises vries i wide rge d oe c cosider the followig typicl pplictio res: echics (cotct d frictio proles), electricl egieerig (circuits with switchig d/or piecewise lier eleets), hydrulics (oe-wy vlves), theticl progrig (ic optiiztio suect to iequlity costrits), d theticl fice (pricig of derivtives with erly exercise opportuities) (see [1]). Also, y icl systes risig i pplictios re o-sooth. here is ture literture descriig y differet pproches to the stu of o-sooth ics such s copleetrity systes, differetil iclusios d Filippov systes (see [2]). I the pst, lrge collectio of vrious proles of o-sooth systes hs ee ivestigted withi severl fields. hese efforts lre resulted i susttil literture i the correspodig fields. My reserchers coied their efforts i resolvig the chlleges of o-sooth systes (see [1]). But, ethods for uericlly solvig optil cotrol proles re divided ito two ctegories: Idirect ethods d direct ethods. Idirect ethods re sed o the vritiol foru ltio, resultig i ult iplepoit oudry vlue prole. I s uch s ultiple-poit oudry vlue prole i geerl cot e solved lyticlly, oe hs to rely o uericl ethods. O the other hd, i direct ethods discretized stte vriles d cotrol vriles re treted s the desig vriles i the olier progrig ethod, d the perforce idex is directly iiized y hvig the stte equtios icluded i costrit coditios. Furtherore, direct ethods llow rther stright-forwrd tretet of iequlity coditios, d the solutios re ore roust to iitil solutio guesses. hese properties of the direct ethods hve recetly ttrcted ttetio for solvig coplex optil cotrol proles (see [3]). Despite existece of these ethods, solvig of osooth optil cotrol proles is difficult d ofte ipossile ct. o solve these proles, the

2 Solvig Clss of o-sooth Optil Cotrol roles 17 geerlized derivtive plys iportt role. Before of this pper, we hve proposed two ew pproches [4, 5, 6] for geerlized derivtive of o-sooth fuctios d i dditio, we used it for o-sooth ordiry differetil equtios, o-sooth optiiztio proles d syste of o-sooth equtios (see [7, 8]). he dvtges of our geerlized derivtives with respect to the other pproches except siplicity d prcticlly re s follows: i. he geerlized derivtive of o-sooth fuctio y our pproch does ot deped o the o-soothess poits of fuctio. hus we c use this GD for y cses tht we do ot ow the poits of o-differetiility of the fuctio. ii. he geerlized derivtive of o-sooth fuctios y our pproch gives good glol pproxite derivtive s o the doi of fuctios, wheres i the other pproches the GD re clculted i oe give poit. iii. he geerlized derivtive y our pproch is defied for o-sooth piecewise cotiuous fuctios, wheres the other pproches re defied usully for loclly Lipschiptz or covex fuctios. I this pper, i first step, we defie ew geerlized derivtive where it hs the ove-etio dvtges. he, we utilize this geerlized derivtive to covert the o-sooth optil cotrol prole (1) to the sooth for, d oti pproxite optil solutio. he structure of this pper is s follows. Sectio 2 proposes ew geerlized derivtive for o-sooth fuctios. I Sectio 3, the i o-sooth optil cotrol prole is coverted to sooth prole. I Sectio 4, the otied sooth prole is pproxited to discrete prole. I Sectios 5, uericl exple is preseted for efficiecy of our pproch d i Sectio 6, the coclusio of our pproch is give. II. A ovel Geerlized Derivtive We egi with the followig le. Le II.1: Let (.) : e ouded d itegrle fuctio. We hve ( ), x y y x x (2) where ( x, y ) e 2 ( y x ) 2 roof: See pge 124 d 125 of [9]., 2 ( x, y ). (3) By ttetio to the ove Le d (2), for y fuctio (.) : which is itegrle o itervl [, ] d zero o \[, ], we hve ( ), [, ]. x y y x x (4) ow, we hve the followig theore: heore II.2: Let g (.) :[, ] e ouded d differetile fuctio. We hve ( ), [, ] L x y g y g x x (5) where 2 L( x, y ) ( x, y ), ( x, y ). (6) y d (.,.) stisfies (3). roof: We defie the fuctio g (.) : g ( x ), x [, ] g( x ), otherwise. It is trivil tht fuctio g (.) is ouded d itegrle. So y itegrtig y prts d Le II.1, for y x [, ], we hve L( x, y) g( y) li L( x, y) g( y) li ( x, y) g( y) y li ( x, ) g( ) ( x, ) g( ) s ( x, y) g( y) li ( x, ) g( ) ( x, ) g( ) = li ( x, y) g( y) g( x) g( x). li ( x, y) g( y) li ( x, y) g( y) ow, we cosider the followig prole: Let g (.) e piecewise cotiuous o-sooth (C) fuctio. Fid fuctio (.) such tht for ( ), [, ] L x y g y x x (7)

3 18 Solvig Clss of o-sooth Optil Cotrol roles where L (.,.) is defied y (4). ote tht if g (.) is cotiuous differetile fuctio, the the uique solutio of (7) is (.) g (.). ow, we defie the followig geerlized derivtive for the C fuctios: Defiiti o II.4: Let g (.) e C fuctio o [, ] d (.) is the solutio of (7). he geerlized derivtive of fuctio g (.) is deoted y x g (.) d defied s x g (.) (.). Rer II.5: ote tht y heore II.2, if g (.) is dg sooth (or differetile) fuctio the x g (.) dx. his shows the vlidity d stility of this type of geerlized derivtive. ow cosider (7) d ssue tht ( x ) ( x ), x [, ] where (.),,1,... is totl set for spce of piecewise cotiuous fuctios o [, ]. By this ssuptio, we hve the followig prole: Let g (.) e C fuctio. Fid coefficiets,,1,2,... such tht for ll x [, ] L ( x, y ) g ( y ) ( x ), (8) where L (.,.) is defied y (6). For covertig the ifiite diesiol (8) to the fiite prole, we ssue tht d M re two sufficietly ig uers d write (8) s follows: L ( x, y ) g ( y ) ( x ), x [, ]. (9) M Here, we defie the followig optiiztio prole for the solvig ove equtio:,,1,2,..., L ( x, y) g( y) Miiize dx. ( x) Let M e ig turl uer d ssue (1) y x,,1,2,...,. (11) We utilize the trpezoidl pproxitio to covert the itegrl i (1) to the fiite su. We oti the followig olier progrig (L) prole: Miiize,=,1,2,..., i= w i = w L (x,y )g(y ) M i - (x ) = i where the weights w,,1, 2,..., re s follows: w w, w,,1,2,...,. 2 By ssuptio z w L ( x, y ) g ( y ) ( x ), i M i i (12) for i,1,2,..., the L (12) e coverted to the correspodig lier progrig (L) prole s follows: i i zi, i Miiize suect to w z (13) z ( x ) w L ( x, y ) g ( y ), i i M i z ( x ) w L ( x, y ) g ( y ), s i M i z, i,1,..., i where g (.) : is give C fuctio d, d M re sufficietly ig uers. But, For otiig etter geerlized derivtive we re goig to cosider soe costrits for (13). For this gol, we utilize the followig le: Le II.6: Let g (.) e cotiuously differetile fuctio o itervl [, ]. he there exists such tht for ll l 1,2,..., 1d l 1, l 1, g ( x ) ( ) ( ) g x l g x, where poits x, x1,..., x re defied y (11). roof: his is result of derivtive s defiitio. ow, we ssue g ( x ) ( x ), i,1,2,..., i i (14) d dd the costrit (14) to (13). We oti the followig L prole:

4 Solvig Clss of o-sooth Optil Cotrol roles 19 i i zi, i Miiize suect to w z (15) z ( x ) w L ( x, y ) g ( y ), i i M i z ( x ) w L ( x, y ) g ( y ), i i M i ( x ) g ( x ) g ( x l ), ( x ) g ( x ) g ( x l ), z, i,1,...,, l 1,2,..., 1, l 1, l 1. i By solvig the ove L prole, we oti the optil solutios z i, i,1,..., d,,1,...,. So we hve x s g ( x ) ( x ), x [, ]. (16) I the ext sectio, we covert the o-sooth optil cotrol prole (1) to the sooth for y usig ove GD. III. Soothig rocess Cosider the osooth optil cotrol prole (1). Let x g (.) is the geerlized derivtive of osooth fuctio g (.) defied y (16). We ssue X is the set of o-soothess poits of fuctio g (.). We lso ssue is coutle set. So y Rers dg II.5, x g (.) (.) o set X \. hus x (.) x g ( x (.)) dx dg x(.) ( x(.)) lost everywhere (.e) o X. Further, dx for ll t [, ], we hve t t dg x ( z ) ( ( )) ( ) ( ( )) x g x z dz x z x z dz dx t d g ( x ( z )) dz g ( x ( t )) g ( x ()) dz g ( x ( t )) g ( ). So y ove reltio the osooth optil cotrol prole (1), is coverted to the followig sooth for: Miiize x ( ) (17) suect to t x( t) g( ) x( z) g( x( z)) dz f ( u( t)), x(), x( t) X, u( t) U, t [, ] x IV. Discretiztio rocess I this stge, we pproxite the sooth optil cotrol prole (17), to the discrete for. For this gol, select poits t,,1,2,..., where is sufficietly ig u er, d ssue x ( t ) x,,1,2,...,. By these, we pproxite the velocity x (.) i poits t,,1,2,..., s follows: x ( t ) x x 1, x ( t ) x 1 x,,1,2,..., 1 Moreover, y usig the trpezoidl pproxitios, we pproxite the itegrl ter i (17) s follows: t x x x ( z ) g ( x ( z )) dz w x ( t ) g ( x ( t )), for,1,...,, where the weights w,,1,2,..., re s follows: 1 1 w, w w 1, 2. w w, w, 2,3,..., 2 So the discrete for of sooth optil cotrol prole (1) is s follows: Miiize suect to x (18) 1 1 x x g ( ) w x x g ( x ) x 1 f ( u ),,1,..., 1 x x 1 g ( ) w x x 1 x g ( x ) w x x g x f u 1 x ( ) ( x, x X, u U,,1,...,. By solvig the ove sooth L prole, we oti the followig pproxite optil solutios for the osooth optil cotrol prole (1). x ( t ) x, u ( t ) u,,1,..., We defie the poitwise error for ove pproxite optil solutio s follows: E( t ) x ( t ) g( x ( t ) f ( u ( t )),,1,...,. ), (19)

5 2 Solvig Clss of o-sooth Optil Cotrol roles V. uericl Result Cosider the followig o-sooth optil cotrol prole: Miiize x ( ) (2) suect to x( t) x( t).5 u( t), t [, ], x().3, x( t) 1, u( t) 1, t [, ], where 2. Here, we ssue g ( x ) x.5, ( x ) cos( x ), x [,1] d 69, 2, M 1. We solve the L prole (15) d oti the geerlized derivtive of fuctio g ( x ) x.5 s 69 g ( x ) cos( x ), x [,1] x which is show i Fig. 1. Further, y solvig the correspodig sooth L (18), we oti the pproxite optil stte d optil cotrol for the osooth optil cotrol prole (2) s follows: x ( t ) x, u ( t ) u,,1,...,2, where they re illustrted i Figs. 2 d 3, respectively. Here the oective fuctio is x (2) d the error of otied pproxite optil stte d cotrol, correspodig to the reltio (19), is show i Fig. (4). he upper oud for poitwise error is Fig. 1: he grph of geerlized derivtive Fig. 2: he pproxite optil stte

6 Solvig Clss of o-sooth Optil Cotrol roles 21 Fig. 3: he pproxite optil cotrol Fig. 4: he grph of error fuctio VI. Coclusio I this wor, we showed tht the o-sooth optil cotrol proles re coverted to the sooth for y usig ew prcticl geerlized derivtive. Moreover, we showed tht the sooth optil cotrol proles were pproxited to the olier progrig prole y usig the discretiztio ethod. Acowledgeets he reserch ws supported y grt fro Ferdowsi Uiversity of Mshhd (o. MA89187KAM).

7 22 Solvig Clss of o-sooth Optil Cotrol roles Refereces [1] Cliel M. K., d ieier H., Alysis d Cotrol of o-s ooth Dyicl Systes, It. J. Roust olier Cotrol, 27, 17: [2] Di Berrdo M., Bifurctios i o-sooth Dyicl Systes, SIAM Review, 28, 5(4): [3] Goto. d Kwle H., Direct optiiztio ethods pplied to olier optil cotrol prole, Mthetics d Coputers i Siultio, 2, 51 : [4] Kyd, A.V., oori Sdri, M.H. d Erfi, H. R., A ew Defiitio for Geerlized First Derivtive of o-sooth Fuctios, Applied Mthetics, 211, 2(1): [5] oori Sdri M.H., Kyd A.V., d Erfi H.R., A ew rcticl Geerlized Derivtive for o-sooth Fuctios, he Electroic Jourl of Mthetics d echology, 213, i press. [6] Erfi H. R, oori Sdri M.H., d Kyd A.V., A ew Approch for the Geerlized First Derivtive d Extesio It to the Geerlized Secod Derivtive of osooth Fuctios. I.J. Itelliget Systes d Applictios, 213, i press. [7] Erfi H. R, oori Sdri M.H., d Kyd A.V., A uericl Approch for o-sooth Ordiry Differetil Equtios. Jourl of Virtio d Cotrol, 212, Forthcoig pper. [8] Erfi H. R., oori Sdri M.H. d Kyd A. V, A ovel Approch for Solvig osooth Optiiztio roles with Applictio to osooth Equtios, Jourl of Mthetics, Hidwi pulictio, i press. [9] G.F. Roch, Gree s Fuctio: Itroductory heory with Applictios, V ostrd Reihold Copy, Lodo, 197. Hid Rez Erfi, h.d. i cotrol d optiiztio fro fculty of theticl scieces, Ferdowsi uiversity of Mshhd, Ir. His reserch iterests iclude optil cotrol, optiiztio, osooth lysis d cotrol of osooth icl systes. Ali Vhidi Kyd, h.d. i cotrol d optiiztio fro Leeds Uiversity, Egld. He is full professor t the fculty of theticl scieces, Ferdowsi Uiversity of Mshhd, Ir, his reserch iterests re ily o optil cotrol d optiiztio, fuzzy theory, osooth lysis d pplictio of cotrol i edicie. Mohd Hdi Frhi,, h.d. i cotrol d optiiztio fro Leeds Uiversity, Egld. He is full professor t the fculty of theticl scieces, Ferdowsi Uiversity of Mshhd, Ir d his reserch iterests re ily o optil cotrol d optiiztio. How to cite this pper: M. H. oori Sdri, H. R. Erfi, A.V. Kyd, M. H. Frhi,"Solvig Clss of o-sooth Optil Cotrol roles", Itertiol Jourl of Itelliget Systes d Applictios(IJISA), vol.5, o.7, pp.16-22, 213. DOI: /iis Authors rofiles Mohd Hdi oori Sdri, h.d. i cotrol d optiiztio fro fculty of theticl scieces, Ferdowsi Uiversity of Mshhd, Ir. His reserch iterests iclude sooth d osooth optil cotrol, cotiuous d discrete optiiztio d icl systes.