On an algorithm of the dynamic reconstruction of inputs in systems with time-delay

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1 Ieraoal Joural of Advaces Appled Maemacs ad Mecacs Volume, Issue 2 : (23) pp Avalable ole a IJAAMM ISSN: O a algorm of e dyamc recosruco of pus sysems w me-delay V. I. Maksmov a, * a, Isue of Maemacs ad Mecacs, Ural Brac of e Russa Academy of Sceces, Russa; Ural Federal Uversy, Russa Receved 2 Sepember 23; Acceped ( revsed verso) 2 November 23 A B S T R A C T A algorm of e dyamc recosruco of pus sysems descrbed by dffereal equaos w me-delay s preseed. Ts algorm s sable w respec o formao oses ad compuao errors; s based o approprae modfcaos of e prcple of exremal amg, wc s kow e eory of guaraeed corol. Keywords: Recosruco, Feedback corol. MSC 2 codes: 34K29, 93B IJAAMM Iroduco We cosder e problem of e sable recosruco of a ukow pu a dyamc sysem from e resuls of accurae observaos of s rajecory. Le us expla e essece of e problem. A dyamc sysem s descrbed by a vecor olear dffereal equao w me delay. Te rajecory of e sysem depeds o a me-varyg pu, wc s erpreed wa follows as a corol. Bo e pu ad e rajecory are o kow a pror. Te pase saes of e sysem are measured durg e operao of e sysem. Te measuremes are, geerally speakg, accurae. I s requred o desg a algorm for e approxmae recosruco of e pu. Te algorm mus be dyamc ad sable. Te dyamc propery meas a e curre values of e approxmao of e correspodg coordaes are produced real me, wle e sably propery meas a e approxmaos are as accurae as possble f e measuremes are accurae eoug. We adop e followg oao: R s e -dmesoal space w e Eucldea orm ad scalar produc (, ); * Correspodg auor E-mal address: maksmov@mm.ura.ru (V. I. Maksmov)

2 54 V. I. Maksmov ( L ) 2 T; R s e Hlber space of all fucos egrable w e square of er orm ad mappg e se T o e space R (w e orm ); 2 ( L T; R ) CT ( ; R) s e Baac space of all couous fucos mappg e se T o e space R w e sup-orm ; ( CTR ; ) B s e rasposed marx B; N s e se of posve egers;, W ( T; R ) s e Baac space of all dffereable fucos wose frs dervaves belog o L ( T; R ). 2 Problem saeme ad soluo meod Te problem dscussed e prese paper ca be formulaed as follows. Tere s a dyamc sysem Σ operag o a me erval T = [, ϑ], were ϑ <+. We assume a Σ s descrbed by e sysem of dffereal equaos w me-delay x () = f( x, (), x ( )) + Bu () + F (), T, x ( + s) = x() s, s [, ], (2.) were x( ) R, u () R, ad F( ) C( T; R ) s a gve fuco, = cos > s a delay. Te rajecory x() = x( ;, x( s), u()) R, T, depeds o e al sae x () s, s, [ ], ad o e me-varyg ukow pu u( ) C( T; R ). We assume a e m fuco x() s s couously dffereable. Le Δ= { } = be a uform paro of e erval T w a sep ; we ave = + + ad m = ϑ. Te rajecory of e sysem x() s measured a e mes w a error. Te resuls of ese accurae measuremes are vecors ξ R sasfyg e equales ξ x( ), [: m ], (2.2) were (, ) s e value of formao error. I s requred o desg a algorm a recosrucs real me e corol u() a geeraes x() from e resuls of accurae measuremes of e rajecory. Before preseg a src maemacal formulao of e problem, we wll descrbe e meod of s soluo. Ts meod s based o a kow prcple of posoal corol, amely, o e prcple of auxlary models (Krasovsk ad Subbo 988; Kryazmsk ad Ospov 983). Le us formulae s prcple a form covee for us. Le Σ be descrbed by sysem (2.), were e vecor fuco f : T R R R s couous by e frs argume, sasfes e local Lpscz codo e secod ad rd argumes, ad coforms o e correspodg grow codos: f (, x, y) c( + x + y ) x, y R, c = cos >. I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23.

3 O a algorm of e dyamc recosruco of pus sysems w me-delay 55 Te soluo of sysem (2.) correspodg o e al sae x () s ad corol u() C( T; R ), as well as soluos of all sysems of dffereal equaos cosdered below, s udersood e sese of Caraéodory. Soluos of sysem (2.) are deoed by x ( ;, x( s), u()). We frs fx a famly { Δ } of paros of e erval T o alf-ope ervals [,,, + ) : m { } ( ) Δ =, = +, =, =, = ϑ. (2.3), =, +,,, m Te, coose a sysem M (called a model) wose moo w ( ), specally cose ordary dffereal equao w e al codo k,, + T, s a soluo of a w () =Φ (, w (), ξ, ξ, v ()), [, ), [: m ], (2.4) w ( ) = w. To smplfy e exposo, we assume a = ( k ), were k N. Here, v ( ) s a corol ad w ( ) s a vecor wose dmeso s equal o e dmeso of e vecor x. Te oao w () = w ( ; w, v ()) s used for e soluo of sysem (2.4) (w e al codo w ). Afer e model as bee defed (.e., equao (2.4) as bee wre), e soluo algorm s defed w e law of formg feedback corols e model. Te procedure of corollg e model s preceded by e coce of s al sae w. I accordace w e ermology adoped e eory of guaraeed corol (Krasovsk ad Subbo 988; Kryazmsk ad Ospov 983), e laws of formg e corols v ( ) e model are called sraeges ad are defed w pars S = ( Δ, U ), were Δ s defed accordg o (2.3) ad U s a fuco a maps eac poso q () () =, ξ, w ( ), [ : m ], o a vecor { } U q = v. (2.5) () ( ()) Tus, e rple ( Δ, M, U ) for every (, ) defes some algorm D o e se of measuremes ξ () Ξ( x(), ), wc forms e oupu { } Dξ () = w (), v (), (),, (2.6) I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23.

4 56 V. I. Maksmov accordg o feedback prcple (2.4),(2.5). Here, Ξ( x(), ) deoes e se of all pecewse cosa fucos ξ (, ) ξ () = ξ, = [, ),,,, + sasfyg equales (2.2) for =, Δ. Te corol v ( ) s also pecewse cosa: v () = v,,. (2.7) Te algorm D (for fxed ) works by e followg sceme. A paro Δ=Δ = { } = ( =,, m= m ) of e erval T ad a auxlary sysem,.e., a model M, are cose ad fxed before e al me. Te algorm D s decomposed o m decal seps. Te sep,, s performed o e me erval [, + ). Te followg operaos are carred ou durg s sep. A me, e oupu x( ) s measured (w a error);.e., a vecor ξ w propery (2.2) s foud. Te, a corol model (2.4) s foud by rule (2.5), (2.7) ad e memory s modfed;.e., e segme w () = w ( ; ( ), w, v ), [, + ], of e rajecory of e model s formed sead of w ( ). Te procedure sops a me ϑ. Te problem of e dyamc recosruco cosss desgg a famly of algorms D = ( Δ, M, U ) (2.3) (2.5), (2.7), (, ), suc a v () u() C( T; R ) as. (2.8) A famly of algorms D = ( Δ, M, U) (2.3) (2.5), (2.7), (, ), w propery (2.8) s called a recosrucg famly. Te problem queso cosss desgg a recosrucg famly of algorms. Problems of recosrucg of ukow caracerscs of a dyamcal sysem, roug measuremes of a par of s pase coordaes are embedded o e eory of verse problems of dyamcs. Ts eory s esvely developed a e prese me. Oe of approaces o solvg smlar problems based o meods of e eory of posoal corol (Krasovsk ad Subbo 988) was suggesed (Kryazmsk ad Ospov 983) ad developed (Ospov ad Kryazmsk 995; Maksmov 22; Ospov e al. 2; Blzorukova e al. 22; Maksmov 24; Maksmov 27). I e prese paper followg e researces s feld, a algorm of dyamcal recosruco of a corol of a medelay sysem s desged. Ts algorm s dyamcal ad work e real me mode. I s sable w respec o formaoal oses ad compuaoal errors. Te algorms suggesed e works ced above realze e recosruco process e mea-square merc. I s paper, a solvg algorm for recosrucg ukow pu e uform merc s preseed. We wll cosder a me-delay sysem. For oer dyamcal algorms recosrucg ukow caracerscs ( e L2 -merc) of e me-delay sysem, see (Blzorukova e al. 22; Maksmov 24; Maksmov 27). m I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23.

5 O a algorm of e dyamc recosruco of pus sysems w me-delay 57 3 Te solvg algorm I wa follows, we assume a e followg codo s vald. Codo. Te marx B s odegeerae. As meoed above, o solve e problem, we mus defe a famly of algorms D, (, ), cossg of (a) a famly of paros Δ of e me erval T of form (2.3); (b) some auxlary sysem (model) M of form (2.4); (c) e law of formg e corol e model by e feedback prcple U (2.5), (2.7). A frs, we fxed a famly of paros of e erval T : m { } Δ =, =, = ϑ, = + ( ), =, m,, +, (see (2.3)). Noe a, vrue of e fac a e fuco f s Lpscz, s possble o gve a umber M >, for wc e followg equales x () M foraa.. T, (3.) f( x, ( ), x ( )) f(, ξ, ξ ) M( + + ω( )) for = [, ) (3.2) k + are rue. Here, =,, ω ( ) s e modulo of couy of e fuco f(, x( ), x( )), T,. e., { f x x f x x } ω ( ) = sup (, ( ), ( )) (, ( ), ( )) : [ ϑ, ]. As e model M, we ake a lear sysem descrbed by e followg ordary dffereal equao w () = f(, ξ, ξ ) + Bv () foraa.. = [, ), (3.3) [ : m ], m= m, w e al codo k + w () = x (). Le a fuco ( ) :(, ) (, ) be fxed. Te law of formg e corol e model s defed as follows: () U ( q ( )) = v = B[ w ( ) ξ ] for ( ), =. (3.4) Le e corol v ( ) be defed equao (3.3) by formula (3.4). I s case, e corol e model s foud by e feedback prcple. Tus, equao (3.3) as e form I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23.

6 58 V. I. Maksmov w () = f(, ξ, ξ k ) BB [ w ( ) ] foraa ξ... (3.5) Lemma. Le e followg codos ( ), ( ), ( ) ( ), ( ) as (3.6) be vald. Te, uformly w respec o (, ) ad ξ ( Ξ ) ( x(, ) ), e equaly + w () s ds C (3.7) s fulflled. Here, C = cos >, = ( ), =,. Proof. Usg (3.5), we oba e followg equales ad Here, d [ w ( ) x ( )] = f (, ξ, ξ k ) BB [ w ( ) ] f ( x ( ) x ( )) Bu ( ) ξ,, = d () = BB [ w ( ) x( )] +Ψ ( s) fora. a. w () x() =. () s s BB w s w Ψ () =Ψ () + [ () ( )], Ψ () s = BB [() xs ξ ] + [ f(, ξ, ξ k ) f( sxs, (), xs ( ))] Bus (), s. Noe a, due o (3.), (3.2) ad (3.6), e famly of fucos Ψ ( ) s bouded: Ψ () ( ) for almos all ad all (, ) (3.8) uformly w respec o. Te, we ave s M T Deoe BB ( s) () (3.9) w () x() = e Ψ ( s) ds, T. μ() = max w ( ) x( ), f () = f(, ξ, ξ ) for. Te followg esmaes are rue: k I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23.

7 O a algorm of e dyamc recosruco of pus sysems w me-delay K BB w () s ds f() s BB[ w ( ) ξ ] ds (3.) K ( ( ) ) ( ) ( ) + K2 2 μ +, μ μ +. Moreover, we ave + () () () BB w ( s) ds при Ψ Ψ +. (3.) Terefore, from (3.9) (3.), we deduce a Usg (3.8), we oba BB ( s) μ() K3( + μ( ) ) e 2 + ds+ 2 (3.2) BB ( s) + e Ψ ( s) ds,. BB ( s) BB ( s) e ( s) 4 e Ψ ds K ds. (3.3) I vrue of codo, e marx BB s posve defe. I s case, all e egevalues of s marx s real ad e leas value (deoe by ν ) s posve. Te, e followg equaly ν BB ( s ) ( s) 5 e ds K e ds = (3.4) ν ( s) ν = K5 e = K5 ( e ) K6 ν ν s fulflled. From (3.3) ad (3.4), follows a BB ( s) e Ψ ( s) ds K7. (3.5) I ur, assumg = ad akg o accou (3.5), from (3.2) we derve KK μ K ( ) ( ) 8( ) KK 3 6 Terefore, for suffcely small (for example, suc a ), we ave 2 I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23.

8 6 V. I. Maksmov μ( ) K9 + + K( + ) (3.6) (see (3.6)). By aalogy w (3.), we oba + w () s ds K + ( μ( ) + ). I addo, usg (3.6) aga, we ave + ( μ( ) + ) + K ( + ) K2. Hece, + w () s ds K3. Iequaly (3.7) s esablsed. Te lemma s proved. Lemma mples e followg eorem., Teorem 2. If u() W ( T; R ) ad agreeme relaos (3.6) for e parameers are fulflled, e, waever ε > maybe, e covergece v ( ) u( ) C([ εϑ, ]; R ) akes place as. If, addo, u () =, e v () u() CT ( ; R). Te followg esmae for e covergece rae s vald: v () u() BB c ( ) + c 2( + ( )) ( ) + c 3ω( ( )) + c 4 e Bu(). Proof. I s easly see a e equaly olds. Here, d BB ( s) () e ds (3.7) BB[ w ( ) x( )] ( s) ds = Ψ = 3 d BB ( s) d BB ( s) ( j) e Bu sds e γ ds ds j= = () + () sds γ () s = BB [ w () s w ( )], () γ () = BB [() x s ξ ], (3) γ () s = f(, ξ, ξ ) f( sxs, (), xs ( )) foraa.. s. (2) s k I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23.

9 O a algorm of e dyamc recosruco of pus sysems w me-delay 6 From (3.7), we ave γ () Takg o accou (2.2) ad (3.), we coclude a Moreover (see (3.2)), γ (2) () s C, s T. (3.8) + () s C2, s T. (3.9) γ (3) () s M( + + ω( )), s T. (3.2) I s case, from (3.8) (3.2), akg o accou (3.4) ad (3.5), we derve 3 d BB ( s) ( ) ( e j ) γ ( s ) ds (3.2) j= ds + ρ(,, ) = C3( + + ω( ) + ). Iegrag e frs erm e rg-ad sde of equaly (3.7) by pars, we oba d BB ( s) ( e ) Bu ( s ) ds (3.22) ds = BB BB ( s) = e Bu() Bu( ) + e Bu ( s) ds. Moreover, from (2.2), (3.), ad (3.7), for all, we derve e esmae I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23. BB {[ w ( ) x( )] [ w ( ) ξ ]} (3.23) w () s ds C5 C + +. From (3.7), (3.2), (3.23), ad (3.6), due o e boudedess of u () ( u () L ( T; R ) follows a e equaly ), BB [ w ( ) x( )] Bu( ) (3.24) + ( ) BB ρ,, + C 5 + C6 + e Bu() s rue. Te cocluso of e eorem follows from (3.24). Te eorem s proved. Teorem 2 mples e ex oe.

10 62 V. I. Maksmov, Teorem 3. Le u() W ( T; R ), agreeme relaos (3.6) for e parameers be fulflled, ad u () =. Te, e famly of algorms D = ( Δ, M, U) of form (2.3), (3.3), (2.7), ad (3.4) s recosrucg. Alog w measurg e pase saes a dscree me sas (see (2.2)), oe ca cosder e case we measurg e pase saes x() s performed couously. Namely, s assumed a a every me sa T we measure e pase saes of sysem (2.); as a resul, we oba vecors ξ ( ) R sasfyg e equaly ξ () x(), T, were e fucos ξ ( ), T, are Lebesgue measurable. I s case, s possble also o cosder e problem of recosrucg e pu u() ad o apply e sceme used above for s solvg. Ideed, as a model, we ake e -dmesoal sysem w e al codo w ( ) = f (, ξ ( ), ξ ( )) + Bv ( ) for a. a. T (3.25) w () = x (). Tus, e model s descrbed by e lear ordary dffereal equao. Te corol v= v ( ) model (3.25) s defed by e rule () ( () ξ ()) ( ) v = B w, =. (3.26) I s easly see a e fuco v ( ) of form (3.26) s calculaed as follows: 2 { ξ } v () = argm v + 2( () w ()), Bv) : v R. For suc coce of e corol v (, ) sysem (3.25) akes e form () f ( ()) BB ( w () ()) T w =, ξ ξ,. Te soluo of s sysem s deoed by w (. ) Teorem 4. Le ( ) ad, ( ) as. If u() W ( T; R ), e, waever ε > maybe, e covergece v ( ) u( ) C([ εϑ, ]; R ) akes place as. If, addo, u () =, e v () u() CT ( ; R). Te followg esmae for e covergece rae s vald: 2 3 BB v ( ) u( ) c( ) + c ( ) + c e Bu() I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23.

11 O a algorm of e dyamc recosruco of pus sysems w me-delay 63 Proof. For almos all T, we ave d [ w ( ) x ( )] = f (, ξ ( ), ξ ( )) f ( x, ( ), x ( )) Bu ( ) BB [ w ( ) ξ ( )], d.e. d [ w ( ) x ( )] = BB [ w ( ) ξ ( )] + ρ( ). d Here, ρ ( ) = f (, ξ ( ), ξ ( )) f (, x ( ), x ( )) Bu ( ). Cosequely, Here, d BB ( s) ρ (3.27) ds BB [ w ( ) x( )] = ( e ) ( s) ds = 2 BB ( s) d BB ( s) ( j) d = ( e ) Bu( s) ds + ( e ) ρ ( s) ds. ds ds j= ρ () s = BB [() x s ξ ()] s, (2) ρ ( s) = f( s, ξ ( s), ξ (s )) f( s, x( s), x(s )) for a. a. s T. () Usg (3.) ad (3.2), we derve () (2) ρ () s c, ρ () s c2 foraa.. s T. I s case, e laer equales mply 2 d BB ( s) ( j) ( e ) ρ ( s) ds c3. (3.28) j= ds Iegrag e frs erm e rg-ad sde of equaly (3.27)) by pars, we oba (3.22). From (3.27), (3.28), ad e boudedess of u (), follows a BB BB [ w ( ) x( )] Bu( ) c4 + c5 + e Bu(). (3.29) Esmae (3.29) complees e proof of e eorem. I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23.

12 64 V. I. Maksmov 4 Ackowledgemes Ts work was suppored by e Russa Foudao for Basc Researc (projecs 3-- a ad of-M2) ad by e Russa Fud for Humaes (projec ). Refereces Krasovsk, N. N. ad Subbo, A. I. (988): Game-Teorecal Corol Problems. Sprger Verlag, New York Berl. Kryazmsk, A. V. ad Ospov, Yu. S. (983): O modellg of corol a dyamc sysem. Izv. Akad. Nauk USSR, Tec. Cyber. 2, pp. 5 6 ( Russa). Ospov, Yu. S. ad Kryazmsk, A. V. (995): Iverse problems of ordary dffereal equaos: dyamcal soluos. Gordo ad Breac. Maksmov, V. I. (22): Dyamcal Iverse Problems of Dsrbued Sysems. VSP, Boso. Ospov, Yu. S., Kryazmsk, A. V., ad Maksmov, V. I. (2): Meods of Damcal Recosruco of Ipus of Corolled Sysems. UB RAS, Ekaerburg ( Russa). Blzorukova, M., Maksmov, V., ad Padolf, L. (22): Dyamc pu recosruco for a olear me-delay sysem. Auomao ad remoe corol. 63(2), pp Maksmov, V. I. (24): Te meod of smoog fucoal verse corol problems for delay sysems. Fuc. Dff. Eqs., ( 2), pp. 93. Maksmov, V. I. (27): Lyapuov fuco meod pu recosruco problems of sysems w afereffec. J. Ma. Sc., 4(6), pp I. J. of Adv. Appl. Ma ad Mec. (2): 53-64, 23.

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