Applied Mathematics, 2010, 1, 118-123
doi:10.4236/am.2010.12015 Published Online July 2010 (http://www.SciRP.org/journal/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
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 q RQ, and values of the x(t) belong to com-
pactum m R
X
.
The problem in such statement has been widely cov-
ered in the literature. For its decision we will adhere to
the approach, considered in [1]. 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
)).()())((,(
)(
1
ihihihi
Ttwtxtxtf
h 
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()()
h
Fxh
F
ab
huu 


, where
F[a,b] – some functional space. If 0)(lim
0

h
F
h

, then
algorithm is called F[a,b] – regularizing.
Statement 1: [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

h
hh
h

.
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 ≤)(h F

≤)(
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 )(
1
h
hC

≤)(h
F

≤)(
2
h
hC

.

A. Y. VDOVIN ET AL.
119
The number 0

is called asymptotic order of accu-
racy of the algorithm Dh in F [a,b], if 0
0)(lim



h h 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 [4] 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 [2], 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 2
0 hv h 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(
h
D 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:
.
)(
)()(
))(,(h
t
w
t
x
txtfv ihih
ihi
T
i



Let us fix )( h

. The vector
))()())((,(
)(
1
)( 00 twtxtxtf
h
tv T

and the system - model
a
xaw
h
twtx
txtAtxtgtw 


)(,
)(
)()(
))(,())(,()( 0
0
0

, (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(
h
D.
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
mkt
k,1),( 
 are non-negatives: )(
1t

≥≥,0)( t
r

0)()(
1



tt mr


, and ))(,( txtA≤
)(
1
t
r

.
Statement 4: [5] If matrix mm HR

 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  [6];
4) ))(,())(,()))(,((
1txtAtxtAtxtAP 
 [6].
The solution of the Cauchy problem (3) is of the form



dxxA
h
xgAtxAattw
t
a
a
))())(,(
)(
1
))(,())((;,())(;,()(
0

 
(5)
here

 )(;,

At


— is a solution of the equation
 ))(,())(;,(
)(
1
))(;,( 



xAAt
h
At 

 (6)
with initial condition EAtt ))(;,(

.
Integration by parts from a to t on the right-hand side

A. Y. VDOVIN ET AL.
120
of (5), and taking (2) and (6) into account, we obtain


dxfAt
hh
twtx t
a)(,())(;,(
)(
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


0
1
()() /( )1/()(,;())
((,()))(,())(),
t
a
x
twthhtA
PA xA xFd


 

 (8)
where )()))(,(()(  vxfF T .
Both parts of the previous equality are multiplied on
)))(,((
1txtAP :
0
11
1
()() 1
((,()))(,;())
() ()
((,()))(,())(),
t
a
x
twt
P
At xttA
hh
PA xA xFd


 




 (9)
where ))(;,()))(,(())(;,( 11  AttxtAPAt




is a solution
of the differential equation (6) with initial condition
11
(,;( ))((,()))ttAPAtxt   .
Statement 6: [2] 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( 1 hh  the following estimation holds
)))(,(())(;,( 1  xAPAt  ≤ 2
)(4
)(
1)( KheK h
t






,
here

is positive constant such that, for all ],[ bat

minimal positive eigenvalue )(t
r

of matrix A(t,x(t))
satisfies the inequality )(t
r

≥

> 0.
Corollary 1 According to (8) and boundedness F(t)
such positive constant K0 exists that for all ],[ bat

inequality )(/)()(( 0htwtx   ≤ 0 K is valid.
Corollary 2 For all ],[ bat , ],[ta

, ],0( 1 hh

the following inequality holds:
)))(,(())(;,( 11  xAPAt  ≤ 2
)(4
)(
1)( KheK h
t







Definition. Suppose that ),0( 

h, )(h:

],[ ba
[,]mm abR  , m
Rba  ],[:)(

and )(),( ttt
h
  )(t  .
Consider the representation 
t
ahdssst )(),(

)(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

; ),(st
h
 — is its kernel, and )( h

—
error of the reconstruction.
Consider a function
 )))(,(())(;,(),( 11
)1(


 xAPAtt
h


 .
Let us show that

t
ahdFt
 )(),(
)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)(: [,]mm
abR  , mapping
p)(

: m Rba ],[ satisfies of the Lipschitz condition
with constant Lp and for all ],[ bat  the representations
)(tH ≤
H
M, ()
t
a
H
d


 ≤

are valid, then (
)
t
a
H



 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



v
vfg
L
MMM  )1( so that )()( 21 tFtF ≤1F Lt

2
t.
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

,
)(
)(
h
h


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(
 t
h
 satisfies the estimation
)(h

≤ 4321 )(
)(
)(4
)()2( Kh
h
h
KhLKK
k
F



 








 .
Proof. Let’s put ),(),( )1()1(att hh 
 when


a[

)),(ah

, and define
() (1)
1() (,)(( )())
th
h
ah
I
tF Ftd






 
,
(1)
2()(,)(( )())
t
h
th
I
tF Ftd




 
,
(1)
3()(, )()()
t
h
ah
I
tFtdFt



 
.
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.
121
1
I ≤

()(())()
F
LbaFthFt   
() (1)
() (,)
th
h
ah td







 ≤ 






 vF MabL

2
)(
)))((),((())();(,( 11 htxhtAPAhtt  
)))((),((())();(,(11 haxhaAPAhat  
≤ )(4
))((
1
)(4
)(
1
(
2
)( h
hat
h
h
v
FeKeK
M
abL






 








))(2 2 Kh

 ≤ ))((2 2
)(4
)(
1KheKh
h






)
2
)((vF MabL

 ;
for I2 : 2 I ≤ (1)
()
() (,)
t
Fh
th
Lh td




 ≤ )2( 21 KK


)( hLF

;
for I3, according to statements 2 and 6:
3
I ≤ 11
[(,;())((,()))tt APAtxt  
11
(,( );())(((),(( )))ta hAPAa hxa h    
1(( ,( )))]( )
P
At xtFt ≤
()
4()
12
(2())
h
hv
M
Ke hK





The error of the operator of reconstruction defined by
(10) satisfies the inequality
(1)
()(,)()()
t
h
a
htFdFt   

≤ () (1)
() (,)(( )())
th
h
ah tFFtd







(1)
() (,)(()())
t
h
th tF Ftd 



 

(1)
() (,)( )()
t
h
ah tFdFt 


 
.
The last, taking into account the estimations for Ij at
)
5
)(2(
13

v
F
M
abLKK  , )
3
)((2 24

v
F
M
abLKK  ,
implies relation )( h

≤ )(4
)(
321 )()2( h
h
FeKhLKK





4
)( Kh

.
Note that, for any k we can indicate such h2(k)
> 0 that, for all )](,0( 2khh  inequality )(4
)(
h
h
e



≤
k
h
h







)(
)(
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(
 t
h
, ],[ bat

, ],[ ta

.
According to (6),


)))(,(()(;, 11




xAPAt  is a solu-
tion the problem of Cauchy

)(
))(;,(
)))(,(())(;,( 1
11 h
At
xAPAt










11
1
((,()))(,())(,; ())
((,()),
P
AxAxt A
dPA x
d







11 1
( ,;())((,( )))((,( )))tt APAtxtPAtxt


,
which implies that (9) takes the form:

(1)
0
1
11
() ()
((,())) (,)()
()
(,;())((,()))()
t
h
a
t
a
xtw t
PAtxtt Fd
h
d
tAPAx Fd
d











In [2] 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(3 hh

)))(,(())(;,( 01  xAPAt  ≤ 4 )( Kh

According to the approach offered in [2], 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
)( 987hK
h
h
KhK
k



 








.
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 iiiii
iii
w tw tgtxtftxtv
tttt 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 [2]. For simplicity we assume in what follows
that ba
h


 .

A. Y. VDOVIN ET AL.
122
Consider the Euler method for the differential equation
(3) with exactly calculated right-hand side: for ,[ i
tt

)
1i
t
 ). ( ) ( ),())()((
))(,(
)(
1
))(,()
(
)(
0awaw t-ttwtx
txtA
h
txtgtwtw
eiiei
iiiiiee






(12)
In [2] the following result has been received:
Lemma 6: Let condition (∗) hold. Then there exist
positive constants h4, K10 such that for all ],0(4 hh

and ],[ bat  the following estimate holds:
)()(
0twtw e
 ≤ 10
)(
)( K
h
h


Lemma 7: Let condition (∗) hold. Then there exist
positive constants h5, K12, K13 such that for all
],0(5 hh  and ],[ bat  the following estimate holds:
)()(twtw eh  ≤ 1312 )(
)( KhK h
h


.
Proof. According to (11) and (12), the following rela-
tion holds

),()))(,())(,((
)(
)(
)()(
)(
)()(
))(,())(,()(
)(
)()(
))(,())()((
)(
)(
))(,(()()(
00
11
htxtgtxtg
h
h
twtw
h
twtx
txtAtxtAh
h
txtx
txtAtwtw
h
h
txtAEtwtw
iiihi
ieiii
iiihi
iih
ihiieih
ihiieih

















 



Taking into account (∗), corollary 1 from statement 6
and lemma 6 the following estimations hold:
)()( ihtxtx  ≤ h; )(
)()( 0
h
twtx ii

 ≤ K0;
))(,())(,( iiihitxtAtxtA  ≤ hLMf 2;
)()(
0iei twtw  ≤ 11
)(
)( K
h
h

;
))(,())(,( iiihi txtgtxtg  ≤ fvMLMLh

 )( ,
hence there exist positive constant h5, K11 = 0
22KLMM ff
10
4(1)
fv
LM MLM
such that, for all ],0(5 hh 
)
(
)( 11 ieih twtw ≤ )
(
)( ieih twtw 
f
MhhK
h
hh )()(
)(
)(
11 

 

(13)
Since, in this case, )()( 00twtw eh  ≤ h, it follows
from the (13) that, for any i =1,n:
)
(
)( ieihtwtw  ≤ ))()(
)(
)(
(11 f MhhK
h
hh
nh 

 

11
() () ()
() ()
f
ba hh
hKhhM
hh


 
 


≤ 1312 )(
)( KhK h
h


,
12 12
() 1KbaK


, f MabK )(
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 )(
)( KhK h
h


.
Note that the difference between v0(t) and vh(t) for
1
[, )
ii
ttt


can be expressed as:
0
0
()()
()()(, ())()
() ()
(,())()
() ()
(,())()
() ()
(( ,( ))(,()))()
Thi
h
Thi h
Th
TT hi hi
ihi
x
txt
vtvtf txth
wt wt
ftxt h
wt wt
ftxt h
x
twt
ftxt 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, )(
)(
h
h


are bounded. Then there
exist positive constants Kv, h6 such that |vh(t)| ≤ Kv for
all ],0( 6 hh

and ],[ bat

.
According to approach, considered in [2], 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.
123
)()(
0tvtvh
 ≤ 1918 )(
)( KhK h
h


.
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 ≤ 9
1
81
7)(
)(
)(
)(
4Kh
h
h
Kh
Kk















1918 )(
)( KhK h
h


 .
Remark 1: Optimal on h the order of upper estimation
of accuracy may be realized, at choice 12
1)( 
k
k
hh 
,
12
1
)( 

k
k
hh  , )()( hh


, hence 2
1
0

.
Remark 2: In our case the unknowing impact u∗)(

can be defined as u∗)( = v∗)( + P1(A(t,x(t)))u(a).
5. References
[1] Y. S. Osipov and A. V. Kryazhimskii, “Inverse Problems
for Ordinary Differential Equations: Dynamical Solu-
tions,” Gordon and Breach Science Publishers, London,
1995.
[2] 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.
[3] 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.
[4] Y. S. Osipov, F. P. Vasilyev and M. M. Potapov, “Bases
of the Method Dynamic Regulation,” in Russian, Mos-
cow State University Press, Moscow, 1999.
[5] V. V. Voevodin and Y. A. Kuznetzov, “Matrixes and
Calculations,” in Russian, Publishing House “Nauka”, Mos-
cow, 1984.
[6] A. Albert, “Regression and the Moor–Penrose Pseudoin-
verse,” Academic Press, New York, 1972.
[7] P. A. Wedin, “Pertubation Theory for Pseudoinverces,”
BIT Numerical Mathematics, Vol. 13, No. 2, 1973, pp.
217-232.