Research Article Solving Boundary Value Problem for a Nonlinear Stationary Controllable System with Synthesizing Control

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1 Hdaw Mathematcal Poblems Egeeg Volume 7, Atcle ID 85976, 3 pages Reseach Atcle Solvg Bouday Value Poblem fo a Nolea Statoay Cotollable System wth Sytheszg Cotol Alexade N. Kvto, Osaa S. Fyula, ad Alexey S. Eem Depatmet of Ifomato Systems, Faculty of Appled Mathematcs ad Cotol Pocesses, Sat Petesbug State Uvesty, Uvestets Pospet 35, Petegof, Sat Petesbug 9854, Russa Coespodece should be addessed to Alexade N. Kvto; alvt46@mal.u Receved Mach 7; Revsed 3 July 7; Accepted 8 August 7; Publshed 7 Septembe 7 Academc Edto: Chaudy M. Khalque Copyght 7 Alexade N. Kvto et al. Ths s a ope access atcle dstbuted ude the Ceatve Commos Attbuto Lcese, whch pemts uestcted use, dstbuto, ad epoducto ay medum, povded the ogal wo s popely cted. A algothm fo costuctg a cotol fucto that tasfes a wde class of statoay olea systems of oday dffeetal equatos fom a tal state to a fal state ude ceta cotol estctos s poposed. The algothm s desged to be coveet fo umecal mplemetato. A costuctve cteo of the desed tasfe possblty s peseted. The poblem of a teobtal flght s cosdeed as a test example ad t s smulated umecally wth the peseted method.. Itoducto ad Poblem Fomulato Tasfeg a obect fom a ow tal state to a desed potthephasespacesampotatpoblemofthe moto cotol theoy. It s qute dffcult ad has ot bee solved the geeal case. Two classc appoaches to solve t ae pogam taectoes ad tagetg taectoes. I the fome case, the obect moves alog the fxed taectoy whch caot be chaged (.e., the poblem of ts stablzato ases). Ths leads to stog estctos o the cotol system possbltes. I the latte case, the moto s cotolled costatly (wth some tme step) based o the cuet obect state ode to pefom the desed tasfe. Ths demads costat taget pot tacg wth some techcal meas, whch educes the cotol system depedece. Nowadays, the pogess computes, measuemet tools, ad othe stumets egeeg maes t possble to stat developmet ad poducto of autoomous tellectual cotol systems, whch themselves ca deteme the tasfe taectoy wth oboad computatoal ad avgatoal systems usg oly the tal (o cuet) state of the obect. Ths geatly exteds the capabltes of cotol systems, sce thee s o eed costat taget pot tacg, whch ofte caot be pactcally ealzed. Itellectual cotol systems fo may techcal obects (aeospace vehcles, gyoscopc mechasms, obotc mapulatos, etc.) ae modeled mathematcally wth olea statoay cotollable systems of oday dffeetal equatos (ODEs). The ecessay cotol acto s foud as cotol fuctos, whch povde the oto that phase coodates satsfy the gve bouday codtos. Such poblems of the mathematcal cotol theoy ae called bouday value poblems (BVPs) fo ODEs. They fom a mpotat class of poblems ad ae qute dffcult, especally whe addtoal detals ae toduced to the model, le dscete tme cosdeato, cotol sgal ad measuemet delays, pemaet petubatos, o tegal-dffeetal compoets the obect behavo. Fo the fst tme, the complete soluto of the bouday valuepoblemfoleacotollablesystemsaclassof squae-tegable cotol fuctos was poposed by Kalma et al. [. Dug the followg decades, a lot of papes ad moogaphes studyg BVPs fo lea ad olea cotolled systems of ODEs ad tegal-dffeetal equatos of a specal d have appeaed [ 33. It should be also metoed that the cotol methods based o the classc appoaches cotue to appea, especally the aeas whee models become eve moe complex, as fuzzy cotol systems (see, e.g., [34 37).

2 Mathematcal Poblems Egeeg Today, thee ae thee ma dectos of BVP study. The fst deals wth fdg the ecessay ad suffcet codtos mposedtotheghtpatofthecotolledsystemwhch guaatee the obect tasfe to the desed pot the phase space [ 5, 7, 3 6, 8, 9, 3. The secod poblem s to deteme the fal states set, wheetheobectcabetasfeedfomthetalstate[3,5 8,, 7,, 33. The thd decto coces developmet of exact o appoxmate methods fo costuctg o sytheszg pogam cotol fuctos ad the coespodg taectoes whch coect the gve pots the phase space [, 3 6,,8,,4,6. BVPs soluto s studed suffcetly well fo lea cotollable systems. Fo quas-lea (semlea) ad olea systems of the specal foms, maly teatve methods ad the modfcatos ae used [4, 5. The weaesses of such methods ae stablty to computatoal eos ad oute petubatos ad hgh computatoal tme, sce each stage cludes the soluto of a olea system of fuctoal equatos. Moeove, t s qute dffcult to fd good tal appoxmatos to the cotol fuctos ad tme fuctos of phase coodates to povde fast covegece. Ital appoxmatos ae ofte chose tutvely. Due to the metoed poblems, t s mpossble to use successve appoxmatos to solve BVPs essetally olea cotollable systems. Theoy ad methods of solvg BVPs fo olea systems of geeal fom have yet to be developed. The peset pape s based o the esults peseted [,4.I[,thecodtoofKalmatype,whchguaatees the local cotollablty fo a wde class of olea statoay systems of ODEs, s cosdeed. I [4, the poblem of local cotollablty of a olea statoay system s cosdeed. The otos of auxlay cotollable systems leazed alog the taectoy x(t) ad cotol u(t) ad auxlay cotollable system leazed about the ogal olea system equlbum pot (x ε,u ε ) ae toduced. It s pove that the local cotollablty of the ogal olea system follows fom the full cotollablty of the auxlay systems. I the cuet pape, a olea statoay system local cotollablty codto s foud, whch s aalogous to the codto peseted [, 4. The ma dffeece fom the pevous esults s that we develop a algothm fo costuctg the sytheszg cotol fucto, whch povdes the tasfe fom the og of coodates to a gve fal state fo a wde class of olea statoay systems of ODEs ude estctos o the cotol acto magtude. The algothm s qute easy to mplemet ad t has stable put ad computatoal eos. Moeove, the estmatos of the eachable fal states set ae peseted. The algothm ealzato s smple: the ogal poblem s educed to the stablzato of a olea system of the specal fom ude pemaet petubatos, ad the a tal value poblem (IVP) fo a auxlay system of ODEs s solved. The most complex ad tme-cosumg pat, whch s the auxlay system costucto ad ts stablzato, ca be made wth aalytcal methods ad ealzed wth compute algeba meas. Addtoally, the equed cotol law foud fom the auxlay system stablzato codto povdes the stablty of computatoal pocedues to put ad computatoal eos. The effectveess of the peseted algothm s demostated wth umecal soluto of a pactcal poblem. I the pape, we cosde a cotollable system of oday dffeetal equatos (ODEs): x=f(x, u), () whee x = (x,...,x ) T s a vecto cotag phase coodates ad cotol u s a vecto of the same o lesse dmeso: u=(u,...,u ) T ad. We choose the tme scale so that the cosdeed tme t [,. Theght-had sde s Notce that whee f C 4 (R R ;R ), f=(f,...,f ) T, () f (, ) =. (3) A= f (, ), x B= f (, ), u a S=, (4) S = (B, AB, A B,...,A B). The possble cotol magtude s bouded: (5) u <N. (6) Poblem. Fd a pa of fuctos x(t) C[, ad u(t) C[, that satsfy () ad the bouday codtos x () =, x () = x, x=(x,...,x ) T. We say that the pa x(t) ad u(t) s a soluto of poblems () ad (7). Theoem. If codtos (), (3), ad (4) ae satsfed fo the ght-had sde of (), the ε > such that x R : x < ε thee exsts a soluto of poblems () ad (7), whch ca be foud afte solvg, fst, a poblem of stablzg a lea ostatoay system wth expoetal coeffcets ad, secod, a tal value poblem fo a auxlay ODE system. The ma dea of the poof s to use successve chages of depedet ad depedet vaables to educe the pocess of solvg the ogal system to the poblem of stablzg a olea auxlay system of ODEs of the specal fom ude pemaet petubatos. To solve the latte, we fd a sytheszg cotol, whch povdes expoetal decease of the lea auxlay system fudametal matx. At the fal stage, we etu to the ogal vaables. (7)

3 Mathematcal Poblems Egeeg 3. Auxlay System Costucto We fd the fucto x(t) beg a pat of the soluto of () ad (7) the fom Let us deote c =x+θ c, d =θ d, x (t) =a (t) + x, =,...,. (8) I the ew vaables, system () ad the bouday codtos ae wtte as a=f(x +a,u), (9) a () = x, a () =. () We call the pa of fuctos a(t) ad u(t), whchsatsfy system (9) ad codtos (), a soluto of (9) ad (). Now cosde a poblem of fdg a(t) C[, ad u(t) C[,, whch satsfy (9), such that = m = = =, m,! =!...!, m! = m!... m!. θ [,, =,...,, (6) Usg popety () ad Taylo sees expaso of the ghthad sde of () about (x, ), we ca ewte system (3) as dc dτ =αe ατ f (x, ) +αe ατ f (x, ) c x = a () = x, a (t) as t. () Rema 3. The lmt wth t ofthesolutoof(9)ad() sthesolutoof(9)ad(). We swtch fom the depedet vaable t system (9) to τ accodg to t= e ατ, τ [, +, () whee α > s a ceta costat value to be detemed. The, tems of τ, (9) ad () tae the fom +αe ατ f (x, ) d u + = αe ατ [ (x, ) c = x = x c [ + (x, ) c x u d = = + (x, ) d = u = u d +αe ατ + m =4!m! + m f x x u m u m (x, (7) dc dτ =αe ατ f (x +c,d), τ [, +, (3) c () = x, c (τ) c (τ) =a(t (τ)), d (τ) =u(t (τ)), as τ, c=(c,...,c ) T, d=(d,...,d ) T. (4) (5) Wecallthepaoffuctosc(τ) ad d(τ), whchsatsfy system (3) wth codtos (4), a soluto of poblem (3) ad (4). Wth the soluto of (3) ad (4), oe ca etu to thesolutoof(9)ad()wth()ad(5). ) c c dm d m +αe ατ + m =4!m! + m f x x u m u m d) c c dm d m, =,...,. Let us boud the age of c(τ) wth ( c, c (τ) <C, τ [, ). (8) We wll ow shft the fuctos c (τ), =,...,,aumbe of tmes. Ou am s to get a equvalet system whee all the tems the ght-had sde, whch do ot cota powes of c o d explct fom, would be of the ode O(e 4ατ x ) as τ ad x doma (6) ad (8). Atthefststage,wechagec (τ) to c () (τ) by the followg ule: c (τ) =c () e ατ f (x, ), =,...,. (9)

4 4 Mathematcal Poblems Egeeg Let D + m f + m f / x x um u m fo all =,...,. Afte substtuto of (9) to the left- ad ght-had sdes of (7), we obta the equvalet system dc () dτ = αe ατ f x (x, ) + = αe 3ατ x x (x, ) = = +α[ e ατ f (x, ) c () = x [ +e ατ = x = x (x, ) c () +α[ e ατ f (x, ) d = u [ +e ατ = x = u (x, ) d + αe ατ (x, ) c () c () x x = = +αe ατ (x, ) d x u c () + = = αe ατ (x, ) d u u d + +αe ατ = = + m =4!m! D + m f (x, ) (c () e ατ f (x, )) (c () +αe ατ e ατ f (x, )) d m d m!m! D + m f ( c, d) + m =4 (c () e ατ f (x, )) (c () e ατ f (x, )) d m d m, =,...,. It follows fom (4) ad (9) that () c () () = x +f (x, ), =,...,. () It s easy to see that, the ght-had sde of (), the tems, whch do ot cota the powes of the compoets of c o d explct fom, ae bouded wth O(e ατ x ) as τ ad x (6) ad (8). At the secod stage, we mae a chage of vaables c () (τ) =c () (τ) +e ατ φ () (x), (x) = f x (x, ), φ () = φ () () =, =,...,. () Wth espect to these ew vaables, the ogal system () ad the tal codtos () ae wtte as dc () dτ =α [ [ e 3ατ x x (x, ) = = +e 3ατ f (x, ) φ () x = e 4ατ x x (x, ) φ () + = = e 5ατ (x, ) φ () φ () = x = x +α[ e ατ f (x, ) c () = x [ e ατ x x (x, ) c () = = +e 3ατ (x, ) φ () x x = = c() +α[ e ατ f (x, ) d u = [ e ατ x u (x, ) d = = +e 3ατ (x, ) φ () d = x = u + αe ατ [ (x, ) c () = x = x [ + (x, ) d x u c () = = c ()

5 Mathematcal Poblems Egeeg 5 + = u = u (x, ) d d +αe ατ!m! D + m f (x, ) + m =4 (c () +e ατ φ () e ατ f (x, )) (c () Compaed to the pevous vaables shft, the ght-had sde tems of (3), whch do ot cota the powes of the compoets of c o d explct fom, ae ow of the ode O(e 3ατ x ) as τ ad x doma (6) ad (8). By ducto, at the thstagewth(9) (4),wehavethe ecessay shft of the fom +e ατ φ () +αe ατ e ατ f (x, )) d m d m!m! D + m f ( c, d) + m =4 c ( ) (τ) =c () +e ατ φ () (x), φ () () =, =,...,. (5) (c () +e ατ φ () e ατ f (x, )) (c () +e ατ φ () e ατ f (x, )) d m d m, =,...,. (3) c () () = x +f (x, ) φ () (x), =,...,. (4) We apply (5) 4 tmes ad collect the tems, whch ae lea wth espect to the compoets of c (4 ) ad clude the coeffcets e ατ, =,...,,adalsothetems,whch ae lea wth espect to the compoets of d ad clude the coeffcets e ατ, =,...,.Now,wehavethesystem, whch accodg to () (5) ca be wtte vecto fom as follows: dc (4 ) dτ =Pc (4 ) +Qd+R (c (4 ),d,x, τ) + R (c (4 ),d,x, τ) + R 3 (c (4 ),d,τ)+r 4 (x, c (4 ),d,τ), R =(R,...,R )T, R =(R,...,R )T, R 3 =(R 3,...,R 3 )T, R 4 =(R 4,...,R 4 )T. (6) The fuctos R cludes all the tems whch ae lea wth espect to the compoets of c (4 ) wth coeffcets e ατ, +, ad also the tems of the last sum of the ght-had sde fo whch m = ad =. Thefucto R cludes all the tems whch ae lea wth espect to the compoets of d wth coeffcets e ατ, +,adalso the tems of the last sum of the ght-had sde fo whch m = ad =.IR 3,allthetems,whchaeolea wthespecttothecompoetsofc (4 ) o d,aecotaed. Fally, the fucto R 4 cludes all the tems, whch do ot have powes of c (4 ) ad d compoets.thefuctospad Q have the fom P (x) =αe ατ (P (x) +e ατ P (x) + +e ( )ατ P (x)), P (x) = f (x, ), x P () =A, Q (x) =αe ατ (Q (x) +e ατ Q (x) + +e ( )ατ Q (x)), Q (x) = f (x, ), u Q () =B. c (4 ) () = x+f(x) φ () (x) φ ( ) (x), φ () =(φ (),...,φ() )T, =,...,4, φ () () =. (7) (8) 3. Rght-Had Sde Tems Evaluato It follows fom the costucto of (6) that, doma (6) ad (8), the followg estmatos ae tue: P (x), Q (x) as x, =,...,, =,..., ; R (c (4 ),d,x, τ) e (+)ατ L (x) c(4 ), R (c (4 ),d,x, τ) e (+)ατ L (x) d, L >, L >; R 3 (c (4 ),d,τ) e ατ L 3 ( c(4 ) + d ), Moeove, codtos () ad (3) lead to L 3 >. (9) (3) (3) R 4 (c (4 ),d,x,τ) L 4 x e 4ατ, L 4 >. (3) Estmato (3) follows fom the epesetato f (x, ) = f x (θx, ) x, θ = (θ,...,θ ) T, θ [,, =,...,. (33)

6 6 Mathematcal Poblems Egeeg Rema 4. Let us deote the th colum of Q as q.costuct the matx S ={q,p q,...,p q,q,p q,...,p q,...,q, (34)...,P q }. Hee, fo each =,..., s the maxmal umbe of colums of the fom q,...,p q such that all the vectos q, P q,...,p q, q, P q,...,p q, q,...,p q ae lealy depedet. It follows fom (4) ad (7) that thee exsts ε >so that a S =fo ay x R satsfyg x < ε. We ow cosde the system dc (4 ) dτ 4. Auxlay Lemma =Pc (4 ) +Qd. (35) Lemma 5. Let codtos () ad (4) be satsfed fo system (). The, ε >, ε<ε such that x R : x < ε thee exsts the cotol d(τ) of the fom d (τ) =M(τ) c (4 ) (36) that povdes a expoetal decease of the fudametal matx (35). Poof. We use Kasovs s lea ostatoay systems stablzato method [3 the poof. Let L, =,...,,bethe th colum of Q.Costuctthematx S ={L,L,...,L,L,...,L,...,L,...,L }, L =PL dl dτ, =,...,, =,...,. (37) Hee, fo each s the maxmal umbe of colums L,...,L such that the vectos L, L,...,L, L,...,L, L,...,L ae lealy depedet. We show that a S =. (38) Let L, =,...,,betheth colum of the matx αe ατ Q. Cosde the matx S 3 ={L, L,...,L, L,...,L,...,L,...,L }, L =αe ατ P L dl dτ, =,...,, =,...,, (39) whee ae detemed the same way as fo S. Codtos (), (7), ad (9) povde the oto that S S 3 as x. Ths leads to the exstece of ε >: ε <ε such that x R : x < ε a S = a S3. (4) Now, tag to accout Rema 4 ad equalty (4), we ca pove by cotadcto that x R : x < ε a S 3 =. (4) Fom (4) ad (4), t follows that codto (38) s satsfedthedoma x < ε.moeove,duetothe stuctue of S 3, the estmato S =O(eατ ), τ, (4) s tue that doma. Wth the use of (38), we chage the vaable c (4 ) accodg to the expesso As a esult, we get the system of ODEs: c (4 ) =S (τ) y. (43) dy dτ =S (PS ds dτ )y+s Qd. (44) Accodg to [38 fo the fst tem of the ght-had sde of (44) we have S (PS ds dτ )={e,...,e, φ (τ),...,e + + +,...,e + +, φ (τ)}, (45) whee e sazeovectooflegth wth the oly ut at the th place, φ =( φ,..., φ,..., φ,..., φ,,...,) T, (46) ad φ s the coeffcet of L + expaso to the sum of the vectos L, L,..., L, L,...,L, L,...,L (otce that = =); that s, L + = φ (τ) L φ (τ) L. (47) The secod tem s = = S Q={e,...,e +,...,e γ+ }, γ = Cosde the stablzato of the system dy dτ ={e,...,e, φ }y + e d, y =. (48) =(y,...,y ) T, =,...,, (49)

7 Mathematcal Poblems Egeeg 7 whee e s a zeo vecto of legth wth the oly ut at the th place ad φ =( φ,..., φ ) T. Let y =ψ.thephasevaablesof(49)aecoectedto ψ(τ) ad ts devatves wth the equaltes y =ψ, y y =ψ () +φ ψ, =ψ () +φ ψ () +( dφ dτ +φ )ψ, y =ψ ( ) + (τ) ψ ( ) + + (τ) ψ () + (τ) ψ.. (5) The dffeetato of the last equalty (5) togethe wth (49) leads to ψ ( ) +ε (τ) ψ ( ) + +ε (τ) ψ=d, =,...,. (5) The fuctos (τ),..., (τ), ε (τ),...,ε (τ) (5) ad (5) ae lea combatos of the fuctos φ (τ), =,...,, =,...,, ad the devatves. Rema 6. Fom the stuctue of the matces P ad Q, t follows (see (7) ad epesetato (47)) that the fuctos φ (τ),...,φ (τ) ad the devatves, as well as (τ),..., (τ), ε (τ),...,ε (τ),aebouded.otheelemetsofthe colums φ, =,...,, obey the estmato O(e ( )ατ ), τ. Let d = = (ε (τ) γ )ψ ( ), =,...,, (5) whee γ, =,...,,aechosetomaetheootsλ,...,λ of the equato λ +γ λ + +γ =, =,...,, (53) satsfy the codtos λ =λ (54) f =, λ < ( + ) α fo =,..., =,... Dueto(5)ad(54)adRema6,thecotold = δ T y, =,...,, povdes the expoetal decease of system (44) solutos. Swtchg bac to the ogal vaables (5) ad usg (43), we have d =δ T S c (4 ), =,...,, (55) whee δ = (ε (τ) γ,...,ε (τ) γ ); T s the equalty (5) matx, whch meas that y = T ψ, ψ = (ψ ( ),...,ψ) T ; S s the matx whch cossts of the coespodg ows of S. The desed cotol ca be wtte the fom of (36), whee M (τ) =δ T S (δ T S,...,δ T S ) T. (56) Deote the fudametal matx of system (5) closed wth cotol (5) va Ψ(τ) (Ψ() = I, a detty matx). It s obvous that the elemets of Ψ(τ) ae expoetal fuctos of egatve agumet o the devatves. Cosde system (35) wth cotol (55): dc (4 ) dτ =D(τ) c (4 ), D (τ) =P(τ) +Q(τ) M (τ). (57) Itoduce a bloc-dagoal matx T(τ). Itsdagoalblocs ae matces T, =,...,. The, coespodg to (43) ad (5), the fudametal matx Φ(τ), Φ () = I, of system (57) has the fom The estmato Φ (τ) =S (τ) T (τ) Ψ (τ) T () S (). (58) Φ (τ) Ke ατ e λτ, λ>, K>, (59) follows fom (58), the stuctue of matces S (τ) ad Ψ(τ), ad Rema 6. Let ε=ε. The, the coectess of the lemma becomes clea fom (59). Besdes, o base of (4), (55), ad (58) ad Rema 6, we have Φ (τ) Φ (t) K e λ(τ t) e ( )αt, M (τ) =O(e ατ ), 5. Ma Theoem Poof t τ, τ [, ), K > ; τ. System (6) wth cotol (55) has the fom dc (4 ) dτ =D(τ) c (4 ) +R (c (4 ),M(τ) c (4 ), x, τ) +R (c (4 ),M(τ) c (4 ), x, τ) +R 3 (c (4 ),M(τ) c (4 ),τ) +R 4 (x, c (4 ),M(τ) c (4 ),τ). (6) (6)

8 8 Mathematcal Poblems Egeeg Chage of the vaables z (τ) =Φ (τ) c (4 ) τ () + Φ (τ) Φ (t) e3αt c (4 ) =ze 3ατ, c (4 ) () =z() leads to the followg epesetato: (6) [R (e 3αt z, M (t) e 3αt z, x, t) +R (e 3αt z, M (t) e 3αt z, x, t) +R 3 (e 3αt z, M (t) e 3αt z, t) (67) dz =C(τ) z dτ +e 3ατ R (e 3ατ z, M (τ) e 3ατ z, x, τ) +R 4 (x, e 3αt z, M (t) e 3αt z, t) dt, fo τ [,τ ). +e 3ατ R (e 3ατ z, M (τ) e 3ατ z, x, τ) +e 3ατ R 3 (e 3ατ z, M (τ) e 3ατ z, τ) +e 3ατ R 4 (x, e 3ατ z, M (τ) e 3ατ z, τ), C (τ) =D(τ) +3αE. (63) Now, fom (66) ad (67), usg (3), (3), (3), (36), (6), ad (64) afte chagg c to z, the followg estmatos doma (6) ad (8) ae tue: z (τ) Ke βτ Φ (τ )z(τ ) Letusshowthatallsolutosof(63)wthtalvalues (6) that stat a suffcetly small eghbohood of zeo decease expoetally. Let Φ (τ), Φ () = I, be the fudametal matx of the system dz/dτ = C(τ)z. The,o the bass of (59), (6), ad (6), we have Φ (τ) Ke βτ, z (τ) τ + e β(τ t) K (Le αt z (t) +L 4 x e αt )dt, τ Ke βτ c(4 ) () fo τ [τ, ); (68) Φ (τ) Φ (t) K e β(τ t) e ( )αt, Let us choose α to povde β=λ 3α. (64) β>. (65) The soluto of (63) wth tal values (8) ca be peseted as τ + e β(τ t) K (Le αt z (t) +L 4 x e αt )dt, fo τ [,τ ), L>. The costat L depeds o (6) ad (8). We apply the well-ow esult fom [39 to equaltes (68) ad obta z (τ) Ke μτ Φ (τ )z(τ ) z (τ) =Φ (τ) Φ (τ )z(τ ) τ + Φ (τ) Φ τ (t) e3αt τ +K e μ(τ t) L 4 x e αt dt, τ fo τ [τ, ), μ=β K Le ατ ; (69) [R (e 3αt z, M (t) e 3αt z, x, t) z (τ) Ke μ τ c(4 ) () +R (e 3αt z, M (t) e 3αt z, x, t) +R 3 (e 3αt z, M (t) e 3αt z, t) (66) τ +K e μ(τ t) L 4 x e αt dt, fo τ [,τ, μ =β K L. +R 4 (x, e 3αt z, M (t) e 3αt z, t) dt, fo τ [τ, ), Usg (65), we fx τ satsfed. > so that the equalty μ > s

9 Mathematcal Poblems Egeeg 9 Now, let us boud the choce of α>wth the codto α<μ. The, afte the tegato the ght-had sdes of (69), we have z (τ) Ke μτ Φ (τ ) z(τ ) +K e ατ L 4 x, z (τ) K 3 c(4 ) () +K 4L 4 x, fo τ [τ, ), fo τ [,τ. (7) Note that all K >. O the bass of (8) ad (3), two last estmatos ca be wtte as a sgle equalty the doma x < ε: z (τ) K 5 e ατ x, τ [, ), K 5 >. (7) The costat K 5 depeds o doma (6) ad (8). Wth the use of (36), (6), (6), ad (7), we estmate d(τ) : d (τ) M (τ) e 3ατ K 5 e ατ x K 6 e ( )ατ x, K 6 >, fo τ [, ). (7) Now, we ca set the value ε fom the theoem fomulato, to be m{ε, C /K 5,N/K 6 }. Ths shows that the soluto of system (63) wth tal values (6) ad (8) does ot leave the aea z < C fo x < ε ad deceases expoetally. Besdes, by (7), the coespodg fucto d(τ) obeys estcto (6). Moeove, f we substtute the soluto to (6) ad (36) ad etu to the vaables c(τ) accodg to (9) ad () (5), we get the soluto of (3) ad (4). The, we wte eveythg the ogal vaables wth (5) ad () ad fally (8). Now, accodg to Rema 3, passg to the lmt as t gves the soluto of the ogal poblem () ad (7). Ths poves the theoem. The algothm of solvg poblem () ad (7) cossts of the followg steps: () Costucto of the auxlay system (6); pefomed wth aalytcal methods. () Stablzato of system (35); the sytheszg cotol (55) s foud; pefomed wth aalytcal methods. (3) Soluto of the IVP (6) ad (8) wth cotol (55); ts soluto s substtuted to (55) ad thus the fuctos c (4 ) (τ) ad d(τ) ae foud; pefomed wth umecal tegato methods. (4) Retu to the ogal vaables accodg to () (5), (9), (5), (), ad (8); pefomed wth aalytcal methods. Rema 7. Cosde the system x=f(x, u) +F, (73) whee x=(x,...,x ) T, x R ; u=(u,...,u ) T, u R ; t [,,, F = (F,...,F ) T, F R. F s a costat petubato. Fom the theoem follows Coollay 8. Coollay 8. If codtos () (4) ae satsfed fo system (73), the thee exsts ε>such that, fo ay x R ad F R f x < ε ad F < ε, a soluto of (73) ad (7) exsts a ca be foud afte stablzg a lea ostatoay system wth expoetal coeffcets ad solvg the tal value poblem fo a auxlay system of ODEs. 6. Example of the Algothm Applcato To demostate the effectveess of the poposed method, we cosde a poblem of tasfeg a mateal pot (a satellte) movg a cetal gavtatoal feld to a desed ccula obt wth et powe. Accodg to [3, the systems devatos fom the pescbed ccula moto ad codto (7) tae the fom x =x, x = ( +x ) + (χ +x 3 ) ( +x ) 3 +a u, x 3 =(x + )a ψ u, x () =, x () = x, u <N. =,, 3, (74) (75) Phase coodates x =, x =, adx 3 = χ χ show the devato fom the ogal ccula obt wth adus ; s the adal velocty; χ s the geealzed mometum ad χ = ( ) / ; a ad a ψ ae the elatve velocty vecto poectos oto the adal ad tagetal dectos, espectvely (they ae costat); m s the mass, m s ts chage ate ad the cotol u = m/m; = M,whee s the uvesal gavtatoal costat; M s the mass of Eath. The vecto of phase coodates x=(x,x,x 3 ) T.Thecotolu s scala. System (3) ad codtos (4) fo the poblem have the fom dc dτ =e ατ c, wth dc dτ =e ατ g(c,c 3 )+e ατ a d, dc 3 dτ =e ατ (c + x + )a ψ d, (76) g(c,c 3 )= ( +c + x ) + (χ +c 3 + x 3 ) ( +c + x ) 3, (77) c () = x, c (τ) as τ, =,,3. (78)

10 Mathematcal Poblems Egeeg I the followg, we assume x=(x,,) T.Itsecessayto mae seve tasfoms of type (5) to solve poblem (76) ad (78): c =z e ατ g(x ), c =z + e ατ g(x ), To stablze system (8), we costuct the matx S = {L,L,L 3 },wthl = Q, L = PL (d/dτ)l, L 3 = PL (d/dτ)l ;thats, L =( αe ατ αe ατ (b + e ατ h) ) z =w 6 e 3ατ g c (x )g(x ), z =w + 4 e 4ατ g c (x )g(x ), α e ατ L =( α e ατ (e ατ a 3 b+e 3ατ a 3 h+) ) α e ατ (b + 3e ατ h) (8) w =υ e 5ατ g, w =υ +e 6ατ g, υ =u e 7ατ g, g= ( g (x c )) g(x ) g 4 c (x )g (x ), g= 7 ( g (x c )) g(x )+ g 4 c (x )g (x ), g= 7 { 7 ( g 3 (x c )) g (x ) + 3 g 48 c 3 (x )g 3 (x ) + g (x 4 c ) g c (x) g (x )}. (79) Let a =.ThematcesP ad Q ad the system aalogous to (35) loo le dc dτ =Pc+Qd, c=(υ,u,c 3 ) T, P=αe ατ Q= a a 3 αe ατ αe ατ b+ α e 3ατ a ψ b=( + x )a ψ,,, (8) L 3 α 3 e ατ (e ατ a 3 b+e 3ατ a 3 h+3) =( α 3 e ατ (e ατ a +3e ατ a 3 b+7e 3ατ h+) ) α 3 e ατ (b + 9e ατ h) wth h=a ψ /. Aftetoducgc=Sy,y=(y,y,y 3 ) T, we get dy dτ =φ (τ) y 3 +d, dy dτ =y +φ (τ) y 3, dy 3 dτ =y +φ 3 (τ) y 3. (8) The chage y 3 = ψ(τ) educes (8) to a lea equato of the thd ode: ψ (3) φ 3 (τ) ψ () ( dφ 3 dτ +φ (τ))ψ () ( d φ 3 dτ + dφ 3 dτ +φ (τ))ψ=d. (83) The vaables y, y,ady 3 ae coected to ψ (), ψ (),adψ wth Let y=tψ, Ψ=(ψ (),ψ (),ψ) T, T= φ 3 ( dφ 3 dτ +φ 3) φ 3 d=ψ (3) (6+φ 3 (τ))ψ (). (84) a = g c (x ), a 3 = g c 3 (x ). (+ dφ 3 dτ +φ (τ))ψ () (6+ d φ 3 dτ + dφ 3 dτ +φ (τ))ψ. (85)

11 Mathematcal Poblems Egeeg Afte substtuto of d(τ) to (83), we get the equato wth,,ad 3 as oots of the chaactestc polyomal. The etu to the ogal vaables gves d=γ(τ) T (τ) S (τ) c, Γ=( (6+φ 3 ), (+ dφ 3 dτ +φ ), (6+ d φ 3 dτ + dφ dτ +φ )). (86) It s obvous that (86) povdes a expoetal decease of solutos of (8). At the fal stage, we solve the IVP fo the system obtaed fom (76) afte chagg the phase coodates accodg to (79) wth cotol (86). The, we etu to the ogal vaables. The tal values fo the Cauchy poblem ae υ () = x g(x ) g (x 4 c )g(x ) g, u () =g(x )+ g (x 6 c )g(x )+g+g, c 3 () =. 7. Numecal Smulato (87) Fo umecal smulato, we choose aothe poblem cotol of a sgle-l mapulato. Accodg to [4, the system of equatos descbg the moto of a mapulato ude petubatos has the fom x = x, x = a x a s x +u+μt, (88) whee x s the mapulato devato agle the vetcal axs, x s the deflecto agle chage ate, a = αl m, m =m +M/3, a =gl (m +M/)m, g s the gavty facto, α s the vscous fcto coeffcet, m stheloadmass, L s the mapulato legth, ad M s ts mass. The vecto of phase coodates x=(x,x ) T.Thecotolu s scala. We cosde the bouday codtos x () = x, x () =. (89) System (3) ad codtos (4) fo poblem (88) ad (89) have the fom dc dτ =αe ατ c, dc dτ = αe ατ a s c αe ατ a c +αe ατ d +μαe ατ ( e ατ ), (9) c () = x, c () =, c (τ), c (τ) as τ. (9) Tosolve(9)ad(9),wepefomoetasfomatoof fucto c (τ): c (τ) =c () (τ) μe ατ, (9) whch leads to ew fuctos c () (τ). The system taes the fom dc dτ =αe ατ c () +αμe ατ, dc () dτ = αe ατ a s c αe ατ a c () αe ατ a μ +αe ατ d αμe ατ. The lea pat of (93) s dc dτ =αe ατ Pc+αe ατ Qd, c=(c,c () )T, P=, a a Q=. (93) (94) Afte solvg the stablzato poblem (9), we obta the cotol law d (τ) =M(τ) c, c=(c,c () )T, 6αeατ +4α e ατ +e ατ a α (95) M (τ) =( α ( 3 3α + αe ατ a )e ατ α ). Next, we solve the Cauchy poblem fo (93) ude (95) wth tal data c () = x, c () () = μ. Fally,weetuto ogal vaables x, x, u,adt. Fo umecal smulatos, we choose x =ad, α =., α =.5, L=metes, M=g, m =g, ad μ=..fguesadshow the cotol u(t) ad the coespodg taset plots of the phase coodates x (t), x (t), adx 3 (t) wth espect to the ogal depedet vaable t. 8. Numecal Smulato Dscusso Numecal smulato cossts of the followg stages: () The auxlay system (93) costucto, pefomed wth aalytcal methods ad ealzed wth compute algeba T

12 Mathematcal Poblems Egeeg 9. Coclusos t The aalyss of the theoem poof shows that the most dffcult ad tme-cosumg pat of the algothm mplemetato ca be poceeded wth aalytcal methods of compute algeba pacages. The esults of the umecal smulato of teobtal flght pove that the method ca be used fo costucto ad smulato of vaous techcal obects cotol systems. u x (t) x (t) Fgue : Tme chage of the phase vaables. u Fgue : Tme chage of the cotol u(t). () System (93) stablzato ad cotol law (95) detemato, pefomed wth aalytcal methods ad ealzed wth compute algeba (3) Chage of vaables fom c to c, pefomedwth compute algeba (4) Soluto of the IVP (9) ad (9) wth cotol (95) c vaables, pefomed wth some umecal tegato method (5) Retu to the ogal vaables accodg to (8), (), ad (5) alog wth the IVP soluto. It follows fom the smulato esults that (a) cotol at the tal stage demads the geatest eegy put whch s detemed by the tal stage (x, x ); (b) multple tal data smulatos show that, fo the gve tasfe tme of secod, the eachable aea of tal states s gve by x π/ad x ; (c) the umecal smulato ca be pefomed wth a aveage pesoal compute. t Coflcts of Iteest The authos declae that thee ae o coflcts of teest egadg the publcato of ths pape. Refeeces [ R.E.Kalma,P.L.Falb,adM.A.Abb,Topcs Mathematcal System Theoy, McGaw-Hll Boo Co., New Yo, NY, USA, 969. [ E.B.LeeadL.Maus,FoudatosofOptmalCotolTheoy, R.E. Kege Publshg, Malaba, Ida, d edto, 967. [3N.N.Kasovs,Theoy of Moto Cotol, Naua,Moscow, Russa, 968. [4 I. F. Veeshchag, Methods of Studyg The Flght Regmes of a Vaable Mass Devce, Pem State Uvesty, Pem, Russa, 97. [5 V. I. Zubov, Lectues o Cotol Theoy, Textboo, Naua,La, Moscow, d edto, 975. [6 D. Aeyels, Cotollablty fo polyomal systems, Aalyss ad optmzato of systems, Pat (Nce, 984), vol.63oflect. Notes Cotol If. Sc., pp , Spge, Bel, 984. [7 A. P. Kshcheo, Cotollablty ad eachablty sets of olea cotol systems, Aademya Nau SSSR. Avtomata Telemehaa, o. 6, pp. 3 36, 984. [8 N. L. Lepe, A geometcal appoach to studyg the cotollablty of secod-ode blea systems, Automato ad Remote Cotol,vol.45,o.,pp.4 46,984. [9 S. Walcza, A ote o the cotollablty of olea systems, Mathematcal Systems Theoy. A Iteatoal Joual o Mathematcal Computg Theoy, vol.7,o.4,pp , 984. [ K. Balachada, Global ad local cotollablty of olea systems, Isttuto of Electcal Egees. Poceedgs. D. Cotol Theoy ad Applcatos,vol.3,o.,pp.4 7,985. [ V. A. Komaov, Estmates of eachable sets fo lea systems, MathematcsoftheUSSR-Izvesta,vol.5,o.,pp.93 6, 985. [ V. P. Pateleev, O cotollablty of tme-depedet lea systems, Dffeetal Equatos, vol.,o. 4, pp , 985. [3 H. S. Q, O the cotollablty of a olea cotol system, Computes & Mathematcs wth Applcatos, vol.,o.6,pp (985), 984. [4 V. I. Koobov, The almost-complete cotollablty of a lea statoay system, Uaa Mathematcal Joual, vol. 38, o., pp , 986. [5 Z. Bezad, Global ull cotollablty of petubed lea peodc systems, Isttute of Electcal ad Electocs Egees. Tasactos o Automatc Cotol, vol. 3, o. 7, pp , 987.

13 Mathematcal Poblems Egeeg 3 [6 K. Balachada, Cotollablty of petubed olea systems, IMA Joual of Mathematcal Cotol ad Ifomato, vol. 6, o., pp , 989. [7 D. L. Kodat yev ad A. V. Lotov, Exteal estmates ad costucto of attaablty sets fo cotolled systems, USSR Computatoal Mathematcs ad Mathematcal Physcs,vol.3, o., pp , 99. [8 S. A. Asagalev, O the Theoy of the Cotollablty of Lea Systems, Automato ad Remote Cotol, vol. 5, o., pp. 63 7, 99. [9 K. Balachada ad P. Balasubamaam, Null cotollablty of olea petubatos of lea systems, Dyamc Systems ad Applcatos,vol.,o.,pp.47 6,993. [ S. N. Popova, Local attaablty fo lea cotol systems, Dff. Equatos,vol.39,o.,pp.5 58,3. [ A. N. Kvto, O a cotol poblem, Dffeetal Equatos,vol. 4, o. 6, pp , 86, 4. [ Y. I. Bedyshev, O the costucto of the attaablty doma a olea poblem, Joual of Compute ad Systems Sceces Iteatoal,vol.45,o.4,pp. 6,6. [3 V. I. Koobov, Geometc cteo fo cotollablty ude abtay costats o the cotol, Joual of Optmzato Theoy ad Applcatos,vol.34,o.,pp.6 76,7. [4 J. M. Coo, Cotol ad oleaty, AMS, 7. [5 K. Sathvel, K. Balachada, J.-Y. Pa, ad G. Devpya, Nullcotollabltyofaoleadffusosystemeacto dyamcs, Kybeeta, vol. 46, o. 5, pp ,. [6 A. N. Kvto, O oe method of solvg a bouday poblem fo a olea ostatoay cotollable system tag measuemet esults to accout, Automato ad Remote Cotol, vol.73,o.,pp. 37,. [7 S.V.Emel yaov,a.p.kshcheo,add.a.fetsov, Cotollablty eseach o affe systems, Dolady Mathematcs, vol. 87,o.,pp.45 48,3. [8 K. Balachada ad V. Govdaa, Numecal cotollablty of factoal dyamcal systems, Optmzato. A Joual of Mathematcal Pogammg ad Opeatos Reseach, vol.63, o. 8, pp , 4. [9 B. Radhasha, K. Balachada, ad P. Auola, Cotollablty esults fo factoal tegodffeetal systems Baach spaces, Iteatoal Joual of Computg Scece ad Mathematcs,vol.5,o.,pp.84 97,4. [3 G. Ath ad K. Balachada, Cotollablty of mpulsve secod-ode olea systems wth olocal codtos Baach spaces, Joual of Cotol ad Decso, vol., o. 3, pp.3 8,5. [3 K. Balachada ad S. Dvya, Relatve cotollablty of olea eutal factoal Voltea tegodffeetal systems wth multple delays cotol, Joual of Appled Nolea Dyamcs,vol.5,o.,pp.47 6,6. [3 S. Dvya ad K. Balachada, Costaed cotollablty of olea eutal factoal tegodffeetal systems, Joual of Cotol ad Decso,vol.4,o.,pp.8 99,7. [33 H. Th ad L. V. He, O eachable set estmato of twodmesoal systems descbed by the Roesse model wth tmevayg delays, Iteatoal Joual of Robust ad Nolea Cotol,pp.,7. [34S.Tog,Y.L,adS.Su, Adaptvefuzzyoutputfeedbac cotol fo swtched ostct-feedbac olea systems wth put oleates, IEEE Tasactos o Fuzzy Systems, vol. 4, o. 6, pp , 6. [35 S. Tog, Y. L, ad S. Su, Adaptve fuzzy tacg cotol desg fo uceta o-stct feedbac olea systems, IEEE Tasactos o Fuzzy Systems, vol.4,o.6,pp , 6. [36 Y. L ad S. Tog, Adaptve Fuzzy Output-Feedbac Stablzato Cotol fo a Class of Swtched Nostct-Feedbac Nolea Systems, IEEE Tasactos o Cybeetcs, vol. 47, o. 4, pp. 7 6, 6. [37 Y. L, S. Tog, L. Lu, ad G. Feg, Adaptve output-feedbac cotol desg wth pescbed pefomace fo swtched olea systems, Automatca. A Joual of IFAC, the Iteatoal Fedeato of Automatc Cotol,vol.8,pp.5 3,7. [38 E. Ya. Smov, Stablzato of Pogammed Moto, CRCPess,. [39 E. A. Babash, Itoducto to the theoy of stablty, Goge, The Nethelads, Woltes-Noodhoff, 97. [4 V. N. Afaasev, V. B. Kolmaovsy, ad V. R. Nosov, AMathematcal Theoy of Desgg Cotol Systems, Vysshayashola Publ,Moscow,3dedto,3.

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