 Applied Mathematics, 2010, 1, 118-123 doi:10.4236/am.2010.12015 Published Online July 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM On Stable Reconstruction of the Impact in the System of Ordinary Differential Equations Andrei Y. Vdovin, Svetlana S. Rubleva The Ural State Forest-Engineering University, Ekaterinburg, Russia E-mail: rublevas@mail.ru Received February 18, 2010; revised May 27, 2010; accepted June 8, 2010 Abstract Approach to expansion of an opportunity of the reception the guaranteed estimation for a problem of recon-struction the impact within the limits of the dynamical algorithm is considered in the article. Keywords: Dynamical Algorithm, The Reconstruction of the Impact, The Estimation of Accuracy of the Algorithm 1. Problem Statement Consider the problem of reconstruction of entrance im-pact u)( in the dynamical system ],,[,)(),())(,())(,()( bat xax tutxtftxtptxa (1) according to inexact measurements xh)( of states x)( of the system (1) in knots of splitting [a,b]: 10 tta )()(:iihn txtxbt  ≤ h. Here p)(, )(f are mappings from [a,b] × Rm into Rm (with Euclidean norm ) and into Rm×q — the matrix space of dimension m × q (with spectral norm ), respectively; values of the impact u)( belong to com-pactum qRQ, and values of the x(t) belong to com-pactum mRX. The problem in such statement has been widely cov-ered in the literature. For its decision we will adhere to the approach, considered in . It was offered to restore the impact u∗)( with the minimum norm in L2[a,b] among all impacts u)( generating observable move-ment x)( for stability of algorithm. Essence of method consists in the following: let )(, ∆)(: (0, ∞) → (0, ∞), , – scalar product, comp- actum Q is convex, and index T is denote transposing. In each partition interval [ti,ti+1) are formed: 1) the value at the point ti+1 of the system of the model functioning according to the rule ),)())(,())(,(()()( iiihiihiihhttutxtftxtptwtw 2) the value of ui, being the result of projection on Q of the vector )).()())((,()(1ihihihiTtwtxtxtfh So, the considered algorithm (further Dh) puts in con-formity to measurement xh)( the piecewise constant ap-proximation uh)( of the impact u∗)(, where uh(t) = ui, ),[ 1ii ttt . Suppose that ()* [,]() sup()()hFxhFabhuu, where F[a,b] – some functional space. If 0)(lim0hFh, then algorithm is called F[a,b] – regularizing. Statement 1:  Suppose that x(t) at ],[ bat  be-long to compactum X from Rm; functions p(t,x), f(t, x) at Xbatxt],[))(,( are Lipschitz with respect to all the variables with common constant L; parameters )(h, ∆(h) are vanishing together with h so that 0)()(lim 0hhhh. Then Dh — L2[a,b] – regularizing. The question on estimations of accuracy of algorithm is essential at its use. Let’s enter the following concepts: Deﬁnition. A function ),0[),0[:))()((*21hhvhv is called a lower (upper) estimate of the accuracy of the algorithm Dh in a space F [a,b], if for all ],0( *hh inequalities )(1hv ≤)(hF≤)(2hv hold, and ,0[:)(h ),0[)*h is called the order of the accuracy of the algorithm Dh in F [a,b], if there exist positive constant Ci,i = 1,2 such that )(1hhC≤)(hF≤)(2hhC. A. Y. VDOVIN ET AL. Copyright © 2010 SciRes. AM 119The number 0 is called asymptotic order of accu-racy of the algorithm Dh in F [a,b], if 00)(limhh ex-ists. The estimations of the accuracy for the discontinuous impact in space L1[a,b] for the described algorithm are received, for example [2,3]. The purpose of the article is construction of modiﬁcation Dh  and indication the additional assumptions at which receipting of asymptotic order in C[a,b] is possible. For this purpose we will ad-here to the approach offered in , therefore we will omit common proves of lemmas. Note, that on ﬁrst step of work of the algorithm as ap-proximation of u∗)( we select projection of zero on Q. The last make receipt of estimation with condition 0)(lim 20hvh is impossible. If the initial condition u(a) is known (we ﬁx left end of interval), then system (1) can be led to the kind: 0)(,)(),())(,())(,()(av xax tvtxtftxtgtxa (2) where v(t)= u(t) − u(a), g(t,x(t)) = p(t,x(t)) + f(t,x(t))u(a), t ∈ [a, b]. Consider the problem of reconstruction of impact v∗)( in new system (2). In what follows, we assume of performance of the fol- lowing set of conditions: Condition (*). In additional to the assertions of state-ment 1 we suppose that for all ],[bat  1) rank (f(t,x(t))) is equal r, 2) v∗)( is satisﬁed to condition of Lipschitz with constant Lv; 3) the inﬁma with respect to t of dis-tances between the boundaries of compactumes Q and X and v∗(t) and x(t), respectively, are positive; 4) the value v(a) is known with some error: )()(avav≤)(h. Remark. Condition (∗) involves the existence of posi-tive constants Mf, Mg, Mv such that )(f ≤ Mf , )(g ≤ Mg, )(v ≤ Mv. In the modiﬁcation )1(hD of the algorithm Dh besides transformation of the kind of system we refuse from procedure of projection on compactum Q. The last de-crease of arithmetical operations executed at each step, in which case the approximation for vh)(, where ),[ 1ii ttt , is given by the formula: .)()()())(,(htwtxtxtfv ihihihiTi Let us fix )( h. The vector ))()())((,()(1)( 00 twtxtxtfhtv T and the system - model axawhtwtxtxtAtxtgtw )(,)()()())(,())(,()( 000, (3) where ))(,())(,())(,( txtftxtftxtA T (4) we’ll name the ideal. In practice, it is impossible to construct v0)( on the basis of measurement of xh)(, but, in what follows, we shall only need estimates of 0*() ()vt vt and the norms of difference between v0)( and vh)(, which allow us to obtain the asymptotic order of accuracy of )1(hD. 2. Estimation of Norm of a Difference v0)( and v*)( Let’s consider some important statements. Statement 2: [5,6] If H – arbitrary matrix, H+ is its pseudo-inverse, then equalities H = HHT (HT )+ , HH+ = (HH+)T , (HT )+ = (H+)T are valid. Statement 3: [5,7] If ],[ bat , and the matrix A(t, x(t)) is deﬁned by equality (4), then its eigenvalues mktk,1),(  are non-negatives: )(1t ≥≥,0)( tr 0)()(1tt mr, and ))(,( txtA≤)(1tr. Statement 4:  If matrix mmHR is hermitian, R0(H) is its kernel, and R1(H) its image, then Rm = R0(H) ⊕ R1(H), where ⊕ is the sign of the direct sum. Further, for k = 0, 1 the projection operator Pk(H) onto subspace Rk(H) is identiﬁed with the matrix Pk(H), k = 0, 1, corresponding to it in a ﬁxed basis in Rm. Statement 5: If k = 0, 1, and matrix Pk(A(t,x(t))) is projector on Rk(A(t,x(t))), then: 1) EtxtAPtxtAP)))(,(()))(,(( 10 , (E is unity mat- rix); 2) projectors )))(,((0txtAP, )))(,((1txtAP are ortho- gonal; 3) )))(,(()))(,((2txtAPtxtAP kk  ; 4) ))(,())(,()))(,((1txtAtxtAtxtAP  . The solution of the Cauchy problem (3) is of the form dxxAh xgAtxAattwtaa))())(,()(1))(,())((;,())(;,()(0  (5) here )(;,At— is a solution of the equation ))(,())(;,()(1))(;,(xAAthAt  (6) with initial condition EAtt ))(;,(. Integration by parts from a to t on the right-hand side A. Y. VDOVIN ET AL. Copyright © 2010 SciRes. AM 120 of (5), and taking (2) and (6) into account, we obtain dxfAthhtwtx ta)(,())(;,()(1)()()(0. (7) Note a few properties of the functions in the right part of (7). According to statement 2 and point 4 of a state-ment 5 we have 01()() /( )1/()(,;()) ((,()))(,())(),taxtwthhtAPA xA xFd  (8) where )()))(,(()(vxfF T. Both parts of the previous equality are multiplied on )))(,((1txtAP : 0111()() 1((,()))(,;())() () ((,()))(,())(),taxtwtPAt xttAhhPA xA xFd  (9) where ))(;,()))(,(())(;,( 11  AttxtAPAtis a solution of the differential equation (6) with initial condition 11(,;( ))((,()))ttAPAtxt . Statement 6:  Suppose that assertions of statement 1 hold and for all ],[ bat  1) rank )))(,(( txtf is equal r, 2) the inﬁma with respect to t of distances between the boundaries of compactumes Q and X and v∗(t) and x(t), respectively, are positive. Then there exist positive con-stants h1, K1, K2 such that, for all ],[ bat , ],[ta, ],0( 1hh  the following estimation holds )))(,(())(;,( 1xAPAt  ≤ 2)(4)(1)( KheK ht, here  is positive constant such that, for all ],[ bat minimal positive eigenvalue )(tr of matrix A(t,x(t)) satisﬁes the inequality )(tr ≥  > 0. Corollary 1 According to (8) and boundedness F(t) such positive constant K0 exists that for all ],[ bat inequality )(/)()(( 0htwtx ≤ 0K is valid. Corollary 2 For all ],[ bat , ],[ta, ],0( 1hh the following inequality holds: )))(,(())(;,( 11xAPAt  ≤ 2)(4)(1)( KheK ht Deﬁnition. Suppose that ),0( h, )(h: ],[ ba [,]mmabR , mRba  ],[:)( and )(),( ttth )(t. Consider the representation tahdssst )(),()(t )( h. Let’s name the integral operator on the left-hand side of this equality is operator of reconstruction of the value of )(t; ),(sth — is its kernel, and )( h — error of the reconstruction. Consider a function )))(,(())(;,(),( 11)1(xAPAtth . Let us show that tahdFt)(),()1( (10) is the operator of reconstruction of the value of F(t), and let us estimate the error of the reconstruction. It is not difﬁcult to receive the following results. Lemma 1: If matrix H)(: [,]mmabR , mapping p)(: mRba ],[ satisﬁes of the Lipschitz condition with constant Lp and for all ],[ bat  the representations )(tH ≤ HM, ()taHd ≤  are valid, then ()taH dtpp ))()((  ≤ )( abLp. Lemma 2: If ],[ bat, v)( satisﬁes of the condition Lipschitz, and the rank of the matrix f(t,x(t)) is constant, then for all t1, ],[2bat there is constant vFLML6 vvfgLMMM  )1( so that )()( 21 tFtF ≤1FLt 2t. The formulated lemmas allow to pass to an estimation the error of the reconstruction operator of the function value: Lemma 3: Suppose that condition (∗) hold; )(h, )()(hh tend to zero together with h, 0)(tv for at [ )),(ah and k. Then there exist positive constants h2(k), K3, K4 such that, for )](,0( 2khh  the error of operator of reconstruction of the value of F(t) with kernel ),()1(th satisﬁes the estimation )(h ≤ 4321 )()()(4)()2( KhhhKhLKKkF . Proof. Let’s put ),(),( )1()1(att hh  when a[ )),(ah, and deﬁne () (1)1() (,)(( )())thhahItF Ftd , (1)2()(,)(( )())ththItF Ftd , (1)3()(, )()()thahItFtdFt . Let’s estimate each of these quantities. According to lemmas 1, 2, and statements 3, 6 for I1 are valid: A. Y. VDOVIN ET AL. Copyright © 2010 SciRes. AM 1211I ≤ ()(())() FLbaFthFt  () (1)() (,)thhah td ≤  vF MabL2)( )))((),((())();(,( 11 htxhtAPAhtt )))((),((())();(,(11 haxhaAPAhat ≤ )(4))((1)(4)(1(2)( hhathhvFeKeKMabL  ))(2 2Kh ≤ ))((2 2)(4)(1KheKhh )2)((vF MabL ; for I2 : 2I ≤ (1)()() (,)tFhthLh td ≤ )2( 21 KK )( hLF; for I3, according to statements 2 and 6: 3I ≤ 11[(,;())((,()))tt APAtxt 11(,( );())(((),(( )))ta hAPAa hxa h   1(( ,( )))]( )PAt xtFt ≤ ()4()12(2())hhvMKe hK The error of the operator of reconstruction defined by (10) satisfies the inequality (1)()(,)()()thahtFdFt  ≤ () (1)() (,)(( )())thhah tFFtd (1)() (,)(()())thth tF Ftd  (1)() (,)( )()thah tFdFt . The last, taking into account the estimations for Ij at )5)(2(13vFMabLKK  , )3)((2 24vFMabLKK  , implies relation )( h ≤ )(4)(321 )()2( hhFeKhLKK 4)( Kh. Note that, for any k we can indicate such h2(k) > 0 that, for all )](,0( 2khh  inequality )(4)(hhe ≤ khh)()(4 is valid. This fact implies the assertion of the lemma. Let’s pass from the integral on the right-hand side of (9) to the operator of reconstruction of value F(t) with kernel ),()1(th, ],[ bat, ],[ ta. According to (6), )))(,(()(;, 11xAPAt  is a solu-tion the problem of Cauchy )())(;,()))(,(())(;,( 111 hAtxAPAt 111((,()))(,())(,; ()) ((,()),PAxAxt AdPA xd 11 1( ,;())((,( )))((,( )))tt APAtxtPAtxt, which implies that (9) takes the form: (1)0111() ()((,())) (,)()() (,;())((,()))()thataxtw tPAtxtt FdhdtAPAx Fdd In  the following result has been received: Lemma 4: If conditions of statement 6 are satisﬁed, then there are such positive constant K4 and h3 that for all ],[ bat, ],[ ta, ],0(3hh )))(,(())(;,( 01xAPAt  ≤ 4)( Kh According to the approach offered in , it is not difﬁcult to receive the following result. Lemma 5: Suppose that the assumptions of lemma 3 hold. Then there exist positive constants K5, K6, K7 such that, for all ],[ bat the following estimate is valid. )()( *0 tvtv  ≤ )()()(4)( 987hKhhKhKk. 3. Estimate of the Norm of Difference between v0)( and vh)( To derive this estimate, we need, ﬁrst, to estimate )()( 0twtwh. Note that the rule 10())((,())( ,()))( (-), [,), ()hhi iiiiiiiiw tw tgtxtftxtvtttt tvva  (11) can be regarded as the implementation of the Euler method for solving problem (3) with an inexact calculated right- hand side. In view of the speciﬁc character of our equa-tion, we cannot use familiar results. For obtaining of a required estimation we will adhere to the approach of-fered in . For simplicity we assume in what follows that bah . A. Y. VDOVIN ET AL. Copyright © 2010 SciRes. AM 122 Consider the Euler method for the differential equation (3) with exactly calculated right-hand side: for ,[ itt )1it ).()(),())()(())(,()(1))(,()()(0awaw t-ttwtx txtAhtxtgtwtweiieiiiiiiee (12) In  the following result has been received: Lemma 6: Let condition (∗) hold. Then there exist positive constants h4, K10 such that for all ],0(4hh and ],[ bat  the following estimate holds: )()(0twtw e ≤ 10)()( Khh Lemma 7: Let condition (∗) hold. Then there exist positive constants h5, K12, K13 such that for all ],0(5hh  and ],[ bat  the following estimate holds: )()(twtw eh  ≤ 1312 )()( KhKhh. Proof. According to (11) and (12), the following rela-tion holds ),()))(,())(,(()()()()()()()())(,())(,()()()()())(,())()(()()())(,(()()(0011htxtgtxtg hhtwtwhtwtx txtAtxtAh htxtxtxtAtwtwhhtxtAEtwtwiiihiieiiiiiihiiihihiieihihiieih  Taking into account (∗), corollary 1 from statement 6 and lemma 6 the following estimations hold: )()( ihtxtx  ≤ h; )()()( 0htwtx ii ≤ K0; ))(,())(,( iiihitxtAtxtA  ≤ hLMf2; )()(0iei twtw  ≤ 11)()( Khh; ))(,())(,( iiihi txtgtxtg  ≤ fvMLMLh )( , hence there exist positive constant h5, K11 = 022KLMM ff 104(1)fvLM MLM such that, for all ],0(5hh  )()( 11 ieih twtw ≤ )()( ieih twtw  fMhhKhhh )()()()(11  (13) Since, in this case, )()( 00twtw eh  ≤ h, it follows from the (13) that, for any i =1,n: )()( ieihtwtw  ≤ ))()()()((11 fMhhKhhhnh  11() () ()() ()fba hhhKhhMhh  ≤ 1312 )()( KhKhh, 12 12() 1KbaK, fMabK )(13  . The lemma is proved. Using lemmas 6, 7, we obtain the following result. Lemma 8: Let the assumption of lemma 6 hold, ∆(h) = h. Then there exist positive constant K14, K15 such that, for all ],[ bat, the following estimate hold: )()( 0twtwh ≤ 1514 )()( KhKhh. Note that the difference between v0(t) and vh(t) for 1[, )iittt can be expressed as: 00()()()()(, ())()() () (,())()() () (,())()() ()(( ,( ))(,()))()ThihThi hThTT hi hiihixtxtvtvtf txthwt wtftxt hwt wtftxt hxtwtftxt ftxth (14) In view of (14) and lemma8 the following result hold: Lemma 9: Suppose that the assumptions of lemma 8 hold, quantities 2h, )()(hh are bounded. Then there exist positive constants Kv, h6 such that |vh(t)| ≤ Kv for all ],0( 6hh and ],[ bat. According to approach, considered in , we can ob-tain the next result. It is required to obtain a sharper es-timate. Lemma 10: Suppose that assumptions of lemma 9 hold. Then there exist constants K16, K17 such that for all ],[ bat the following inequality holds: )()(0twtw h ≤ 1716 )()( KhhhK. Let’s now reﬁne the norm of the difference between v0)( and vh)(. From the (14), the condition (∗), lemmas 5 and 10 the next result hold. Lemma 11: Suppose that the assumptions the lemma 8 hold, then there exist constants K18, K19 > 0 that for all ],[ bat the following inequality holds A. Y. VDOVIN ET AL. Copyright © 2010 SciRes. AM 123)()(0tvtvh ≤ 1918 )()( KhKhh. 4. The Upper Estimation, the Asymptotic Order of Accuracy It is known that there exist constant K20 > 0 such that the lower estimation of accuracy Dh in C[a,b] is of the form )(1hv ≥ hK20 In view of lemmas 5, 11, the following estimate hold: Theorem 1: (upper bound for the accuracy). Let con-dition (∗) hold and )(4)(1hh. Then the upper bound for the accuracy in C[a,b] is of the form: )(2hv ≤ 91817)()()()(4KhhhKhKk 1918 )()( KhKhh . Remark 1: Optimal on h the order of upper estimation of accuracy may be realized, at choice 121)( kkhh, 121)( kkhh, )()( hh, hence 210. Remark 2: In our case the unknowing impact u∗)( can be deﬁned as u∗)( = v∗)( + P1(A(t,x(t)))u(a). 5. References  Y. S. Osipov and A. V. Kryazhimskii, “Inverse Problems for Ordinary Differential Equations: Dynamical Solu-tions,” Gordon and Breach Science Publishers, London, 1995.  A. Y. Vdovin and S. S. Rubleva, “On the Guaranteed Accuracy of a Dynamical Recovery Procedure for Con-trols with Bounded Variation in Systems Depending Linearly on the Control,” Mathematical Notes, Vol. 87, No. 3, 2010, pp. 316-335.  A. Y. Vdovin, A. V. Kim and S. S. Rubleva, “On As-ymptotic Accuracy in L1 of One Dynamical Algorithm for Reconstructing a Disturbance,” Proceedings of the Stek-lov Institute of Mathematics, Vol. 255, 2006, pp. 216-224.  Y. S. Osipov, F. P. Vasilyev and M. M. 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