Advances in Pure Mathematics, 2013, 3, 390-393
http://dx.doi.org/10.4236/apm.2013.34056 Published Online July 2013 (http://www.scirp.org/journal/apm)
Inverse Problems for Dynamic Systems: Classification
and Solution Methods
Menshikov Yu
Differential Equations Department, Dnepropetrovsk University, Gagarina, Ukraine
Received February 15, 2013; revised March 20, 2013; accepted April 25, 2013
Copyright © 2013 Menshikov Yu. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The inverse problems for motions of dynamic systems of which are described by system of the ordinary differential
equations are examined. The classification of such type of inverse problems is given. It was shown that inverse prob-
lems can be divided into two types: synthesis inverse problems and inverse problems of measurement (recognition).
Each type of inverse problems requires separate approach to statements and solution methods. The regularization
method for obtaining of stable solution of inverse problems was suggested. In some cases, instead of recognition of in-
verse problems solution, the estimation of solution can be used. Within the framework of this approach, two practical
inverse problems of measurement are considered.
Keywords: Inverse Problems; Dynamic Systems; Classification; Regularization; Estimation
1. Introduction
Many important practical problems related to mechanical
systems, economical characteristics etc. can be reduced
to investigation of inverse problems for dynamic systems
[1-3].
For the authentic forecast of motion of dynamic sys-
tems it is necessary to use the adequate mathematical de-
scription of physical process as an example [4,5]. One
way to obtain such mathematical description is the solu-
tion of an inverse problems [4,5]. Besides, it is important
that this description was steady against small changes of
initial data of an inverse problem. The decisions can be
accepted on the basis of forecast within conditions of un-
certainty.
Side by side with problems of the motion forecast of
dynamical systems there exist the practical inverse prob-
lems by the purpose of which solutions is the definition
or estimation of real properties of dynamic systems or
real loads on these systems [2,3,6,7]. The problems of te-
chnical diagnostics [3], medical diagnostics [8], diagnos-
tics of the economic characteristics [9], study of the real
external loads on dynamic systems [10,11] may be inclu-
ded into such inverse problems. Some problems related
to obtaining of the decisions are also reduced to the solu-
tion of inverse problems for dynamic systems [7]. The
equations of inverse problems can be linear, nonlinear, as
well as in partial derivatives, with delay etc.
So it is necessary to take classification of inverse pro-
blems to obtain more convenient statements of problems,
more effective methods of study and more right interpre-
tation of approximate solutions.
The given paper is limited to consideration only linear
inverse problems for dynamic systems with the concen-
trated parameters.
Practically all linear inverse problems for dynamic
systems of such type can be reduced to the solution of
Fredgolm (Volltera) integral equation of the first kind
[12]:
 
,d ,
b
a
K
xz uxcxd  
, (1)


, where
K
x

is given kern, ux is given function.

Equation (1) can be represented in the form
A
zu

, (2)
A
where zZ
uU
is continuous operator, is the function
to be found,


,
is given function.
2. Classification of Inverse Problems
Z
Let’s assume, that the functional spaces U
u

u

are met-
ric spaces.
The function in Equation (2) is presented appro-
ximately. In some cases is defined from experiment
[4,5], in other cases this function represents the approxi-
C
opyright © 2013 SciRes. APM

M. YU 391
mation of the given function [12,15].
The error of initial function is given
u


,
Uex
uu

 

u

, (3)
where ex is exact initial function,

,
U
 
,
is distance
between arguments.
In most cases practical problems determine such func-
tional spaces
Z
U that the operator
A
 is compact
operator [13]. Such property of the operator
A

1
repre-
sents significant difficulties in the solution of Equation
(2), as the inverse operator
A

 is not continuous.
The set of the possible solutions of Equation (2) for
this reason is unbounded and is defined as
 
,uAz

:,
U
QzzZ


 


 . (4)
In a number of inverse problems for dynamic systems
an ultimate goal of research is the definition of such so-
lution of Equation (2) which satisfies Equation (2) with
accuracy of the initial data

. In other words, the func-
tion from set of the possible solutions
z
Q

is deter-
mined only: zQ



Q
. Other purposes in the solution of an
inverse problem are not pursued. For example, in inverse
problems [5,14] a ultimate goal is the obtaining of the
adequate mathematical description of motion of dynamic
system for the purposes of reception of the authentic
forecast of motion. Any function from set of the possible
solutions z


 together with the operator
A


,uAz satis- fies the specified requirements U

 

. If instead
operator
A
 is used an exact operator ex
A
with func-
tion , then even the similar result will not be received
with guarantee.
z

The change of size of the initial data error

does
not change a situation. The investigation of problem so-
lution depending on decrease of initial data error

has
no sense as the limiting function does not contain any
additional information.
It is necessary to distinguish two opportunities:
1) function of initial data is obtained as a result of
approximation of experimental measurements [4,
5];
u

exp
u
u
u 
2) function of initial data is obtained as a re-
sult of approximation some dependence which are given
a priori
g
u [15].
In the first case the solution of an inverse problem can
be used at synthesis of the adequate mathematical de-
scription of dynamic systems [4,5].
In the second case the solution of an inverse problem
allows to receive the approximate control of dynamic
system for obtaining of given motion of this system with
accuracy

[15]. Sometimes such inverse problems are
named as problems of output restoration of dynamic sys-
tems [11].
The important characteristic in the solution of inverse
problems is the size of an error of the solution depending
on size of an error of the initial data. However, in this
case the function in relation of which the error must be
calculated is not defined. If such function represents the
exact solution ex of the equation ex
z

A
zu

z

, then in this
case the size of an error of inverse problem solution (in
relation to function ex ) does not have any meaning.
Any function from unbounded set of the possible solu-
tions together with the operator
A
 gives the necessary
result at further use. The size of the specified error can
reach unnatural sizes for engineering calculations (100%
- 10000%).
The inverse problems of such type in some works are
named as inverse problems of synthesis [11,12].
Some expansions of a class of inverse problems of
such type on a case when the operator
A
 in the equa-
tion can be selected from the beforehand given class of
the operators A
A
K
 is examined in works [11,12]. Va-
rious non-standard statements of inverse problems in this
case are possible [16]. The basic purpose of research of
inverse problems in the specified statements is the re-
search of the solutions at the various additional require-
ments: the unitary solution, steadiest solution, most con-
venient solution, optimum solution for the purposes of
the forecast etc. [16]. The inverse problems of such type
are named as inverse problems of synthesis for a class of
models [11,16].
The qualitative distortion of the solution of an inverse
problem of synthesis can be caused by an uncontrollable
error in the initial data [17].
Essentially, other situation arises in research of inverse
problems, when result of the solution of an inverse prob-
lem is the obtaining of the information about the exact
solution of an inverse problem (2) [7,18]. The problems
of technical diagnostics, medical diagnostics, acceptance
of the decisions in conditions under uncertainties may be
included into such problems [1,2,3,7,10].
At the solution of inverse problems of such type it is
necessary to have the information about the error of the
initial data
A
u
 , in relation to the exact initial data
ex
A
, ex . For a case of metric spaces u,, ,
Z
UzZu U



 

of such information will be the size of an error:
 
sup ,
,,
,0
,0 0,,.
Uex
zZ
Aex Z
ZUex
Az Az
A
Ah
z
zuu









(5)
The set of possible solutions such inverse problem
(with account of operator
A
 error) is defined as:





,:, ,,0 hUU
QzzZAzuhz






Q
. (6)
The set ,h

is unbounded for the same reason and
obviously includes the set Q

.
It is impossible to use a priori given function as initial
function here. u 

M. YU 392
The any element ,h
zQ

 does not represent interest
from the point of view of obtaining of the information
about the exact solution of Equation (2). Besides, here is
not represented the opportunity to study the solution be-
havior at reduction of error of the initial data size. The
size

is determined by quality of the measuring equip-
ments and cannot be changed.
The basic difficulty at study of the solutions of such
inverse problem consists in absence of the information
about properties of the exact operator ex
A
. Therefore,
the error of the operator
h
A
 in relation to the exact
operator ex
A
can be defined rather approximately with
big overestimate. On the basis of this it is problematic the
consideration of limiting transitions at 0, 0h


,
. It
is necessary to interpret the approximate solutions of in-
verse problems at the fixed sizes of initial data error.
It is clearly evident that the error of the solution of an
inverse problem of such type has decisive meaning.
The inverse problems of the specified type are named
as inverse proble ms of measurements (interpretation).
Such a classification of inverse problems is not com-
mon and can be replaced by a different classification.
However, this classification is useful because it avoids
some methodological errors.
3. Methods of Solution
It is well known that the inverse problems are unstable
with respect to small changes of the initial data and for
their solution are used special regularized algorithms [13,
19].
Let us assume that functional spaces
Z
U
u

are Ba-
nach spaces. The error of function from the function
g
u has the size:
gU
uu



ms for fixeoperator
. (7)
The set of possible solutions of synthesis inverse
probled
A
 is oted by Qden

:

QzzZAzu:, U


 
.
The solution of following extreme problem can be ac-
cepted as stable solution of synthesis inverse problem:




1
inf ,
zQ Z
zz

 


 (8)
where

z is stabilizing functional which is defined
on 1
Z
1
(set
Z
is everywhere dense into
Z
) [13].
The obtaining of function in a synthesis inverse
problem is important. The operator
z

A

z

together with so-
lution provides the stable adequate mathematical des-
cription of process [4,5].
In some cases the “simplest” solution can be chosen as
solution of synthesis inverse problem [20].
In inverse problems of interpretation it is necessary
additionally to take into account the inaccuracy of op-
erator
A
ex
with respect to the exact operator
A
[11,
13].
Let us suppose that the characteristic of an error of the
operator
A
ex
is given if the operator
A
is linear op-
erator:
A
ex ZU
Ah




,
Q
. (9)
The set of possible solution of Equation (1) is neces-
sary to extend to set h

taking into account the inac-
curacy of the operator
A
:

 ,:, hZ
U
QzzZAzuhz




zQ
. (10)

The algorithm of the solution of the incorrect problem
with approximate operator was proposed in work [19]
which is based on Tikhonov regularization method [13].
The statement of such interpretation inverse problem
can be formulated for obtaining of the stable solution as
follows: it is necessary to find an element ,est h


on
which the greatest lower bound of some stabilizing func-
tional


z is reached




,1
inf
hest
zQ Zzz

 , (11)
where 1
Z
is subset of
Z
, on subset 1
Z
has been de-
fined stabilizing functional


z1
, the set
Z
is every-
where dense in
Z
[13].
Sometimes in inverse problems of measurement it is
enough to find the value


est
One of the important characteristics for the specified
algorithm is the size of an error. The obtaining of
represents significant difficulties, as the exact operator
z
h h
ex
only.
A
is unknown.
As a result of the solution of interpretation inverse
problem it is necessary to accept some approximation
to the exact solution ex of Equation (1) or its estima-
tion in the beforehand certain sense [7,10].
z

z
zest
The functional


z can characterize the chosen
property of the exact solution (for example, smoothness).
The approximate solution will give the estimation from
below of exact solution on a degree of smoothness. If a
functional


zz
,h
QZz
z,1h QZ


characterizes a deviation of the appro-
ximate solution from the given function ap , then the
solution of an extreme problem (11) will give function
from set 1

closest to function ap . Thus, it is
obvious that should not belong to the set .
ap
The estimation of a deviation of the operator
A
 from
exact operator ex
A
cannot be done effectively during a
consideration of interpretation problems.
For overcoming the specified difficulties it is offered
to accept the following hypothesis: for the exact solution
of the equation
ex
zex ex
A
zu

the inequality is valid




ex
zz

z 
, (12)
where is regularized solution of Equation (1) with
approximate operator
A
 and approximate initial data

M. YU 393
u



[8] W. Q. Yang and L. H. Peng, “Image Reconstruction Algo-
rithms for Electrical Capacitance Tomography,” Journal
of Measurement Science and Technology, Vol. 14, No. 1,
pp. 123-134.
z

is stabilizing functional [13]. ,

Theorem. If z

is stabilizing functional [13] then
estimation

z
h
of exact solution exists and is stable
with respect to small change of initial data. [9] Fr. Zirilli, “Inverse Problems in Mathematical Finance,”
Proceedings of 5th International Conference on Inverse
Problems: Modeling & Simulation, Turkey, 24-29 May
2009, pp. 185-186.
The offered hypothesis is not supposed to use the size
of inaccuracy of the operator
A
 from the exact ope-
rator ex
A
in the solution of inverse problems of inter-
pretation. [10] Yu. L. Menshikov and G. I. Yach, “Identification of Mo-
ment of Technological Resistance on Rolling Mill of
Sheets,” Proceedings of Higher Institutes. Ferrous Metal-
lurgy, Moscow, No. 9, 1977, pp. 69-73.
The satisfaction of an inequality (12) is obvious if the
operators ex
A
,A
 are linear. For the nonlinear operator
ex
A
(that in the greater degree corresponds to a reality)
the inequality (12) can be proved by properties of the
approximate operators which are used in calculations
[21].
[11] Yu. L. Menshikov, “Identification of External Loads un-
der Minimum of a Priori Information: Statements, Clas-
sification and Interpretation,” Bulletin of Kiev National
University, Mathematic, No. 2, 2004, Kiev, pp. 310-315.
Use of the offered hypothesis allows to receive various
objective estimations of the exact solution ex of in-
verse problems such as (1) that is important in recogni-
tion problems [7,10]. Moreover, the size is not used
in calculations. At ex
[12] Yu. L. Menshikov, “Identification of Models of External
Loads,” In: Book of Robotics, Automation and Control,
Vienna, 2008.
z
h
A
A

z the estimation of function
ex will be more exact. For definition of parameter re-
gularization it is possible to use the usual discrepancy
method [13] where the value is absent.
h
[13] А. N. Tikhonov and V. Yu. Arsenin, “Methods of Incor-
rectly Problems Solution,” Science, Moscow, 1979.
[14] Yu. L. Menshikov, “Algorithms of Construction of Ade-
quate Mathematical Description of Dynamic System,” Pro-
ceedings of MATHMOD 09 Vienna—Full Papers CD Vo-
lume, Vienna University of Technology, Vienna, Februa-
ry 2009, pp. 2482-2485.
The proposed approach was used to solve two practi-
cal inverse problems of measurement [7,10].
Offered algorithm can be used also for estimation of
real unknown parameters of physical processes by identi-
fication method.
[15] Yu. S. Osipov, A. V. Krajgimsky and V. I. Maksimov, “Me-
thods of Dynamical Restoration of Inputs of Controlled
Systems,” Ekaterinburg, 2011.
[16] Yu. L. Menshikov, “Inverse Problems in Non-Classical
Statements,” International Journal of Pure and Applied
Mathematics, Vol. 67, No. 1, 2011, pp. 79-96.
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