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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 Email: Menshikov2003@list.ru 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 Copyright © 2013 SciRes. APM 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 z1 , 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 zz ,h QZz 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 Copyright © 2013 SciRes. APM M. YU Copyright © 2013 SciRes. APM 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. REFERENCES [1] L. P. Lebedev, I. I. Vorovich and G. M. Gladwell, “Func- tional Analysis: Applications in Mechanics and Inverse Problems (Solid Mechanics and Its Applications),” Sprin- ger, Berlin, 2002. [17] Yu. L. Menshikov, “Uncontrollable Distortions of the So- lutions Inverse Problems of Unbalance Identification,” Proceedings of ICSV12 XXII International Congress on Sound and Vibration, Lisbon, 11-14 July 2005, pp. 204- 212. [2] G. M. L. G. Gladwell, “Inverse Problems in Vibration,” PDF Kluwer Academic Publishers, New York, 2005. [18] C. W. Groetsch, “Inverse Problems in the Mathematical Sciences,” Vieweg, 1993. [3] K. P. Gaikovich, “Inverse Problems in Physical Diagnos- tics,” Longman Scientific & Technical, Harlow, 1988. [19] A. V. Goncharskij, A. C. Leonov and A. G. Yagola, “About One Regularized Algorithm for Ill-Posed Problems with Approximate Given Operator,” Journal of Computational Mathematics and Mathematical Physics, Vol. 12, No. 6, 1972, pp. 1592-1594. [4] Yu. L. Menshikov, “Adequate Mathematical Description of Dynamic System: Statement Problem, Synthesis Meth- ods,” Proceedings of the 7th EUROSIM Congress on Mo- delling and Simulation, Vol. 2, 2010, 7 p. [20] Yu. L. Menshikov and N. V. Polyakov, “The Models of Ex- ternal Action for Mathematical Simulation,” Proceedings of 4th International Symposium on Systems Analysis and Simulation, Berlin, 1992, pp. 393-398. [5] Yu. L. Menshikov, “Synthesis of Adequate Mathematical Description as Inverse Problem,” Proceedings of 5th In- ternational Conference on Inverse Problems: Modeling & Simulation, Antalya, 24-29 May 2010, pp. 185-186. [21] Yu. L. Menshikov and A. G. Nakonechnij, “Principle of Maximum Stability in Inverse Problems under Minimum a Priori Initial Information,” Proceedings of Internatio- nal Conference PDMU-2003, Kiev-Alushta, 8-12 Sep- tember 2003, pp. 80-82. [6] Proceedings of International Conference, “Identification of Dynamical Systems and Inverse Problems,” MAI, Moscow, 1994. [7] Yu. L. Menshikov, “The Inverse Krylov Problem,” Com- putational Mathematics and Mathematical Physics, Vol. 43, No. 5, 2003, pp. 633-640. |