Inverse Problems for Dynamic Systems: Classification and Solution Methods

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 problems 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 inverse problems solution, the estimation of solution can be used. Within the framework of this approach, two practical inverse problems of measurement are considered.

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 problems 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 inverse problems solution, the estimation of solution can be used. Within the framework of this approach, two practical inverse problems of measurement are considered.

Cite this paper

M. Yu, "Inverse Problems for Dynamic Systems: Classification and Solution Methods,"*Advances in Pure Mathematics*, Vol. 3 No. 4, 2013, pp. 390-393. doi: 10.4236/apm.2013.34056.

M. Yu, "Inverse Problems for Dynamic Systems: Classification and Solution Methods,"

References

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[2] G. M. L. G. Gladwell, “Inverse Problems in Vibration,” PDF Kluwer Academic Publishers, New York, 2005.

[3] K. P. Gaikovich, “Inverse Problems in Physical Diagnostics,” Longman Scientific & Technical, Harlow, 1988.

[4] Yu. L. Menshikov, “Adequate Mathematical Description of Dynamic System: Statement Problem, Synthesis Methods,” Proceedings of the 7th EUROSIM Congress on Modelling and Simulation, Vol. 2, 2010, 7 p.

[5] Yu. L. Menshikov, “Synthesis of Adequate Mathematical Description as Inverse Problem,” Proceedings of 5th International Conference on Inverse Problems: Modeling & Simulation, Antalya, 24-29 May 2010, pp. 185-186.

[6] Proceedings of International Conference, “Identification of Dynamical Systems and Inverse Problems,” MAI, Moscow, 1994.

[7] Yu. L. Menshikov, “The Inverse Krylov Problem,” Computational Mathematics and Mathematical Physics, Vol. 43, No. 5, 2003, pp. 633-640.

[8] W. Q. Yang and L. H. Peng, “Image Reconstruction Algorithms for Electrical Capacitance Tomography,” Journal of Measurement Science and Technology, Vol. 14, No. 1, pp. 123-134.

[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.

[10] Yu. L. Menshikov and G. I. Yach, “Identification of Moment of Technological Resistance on Rolling Mill of Sheets,” Proceedings of Higher Institutes. Ferrous Metallurgy, Moscow, No. 9, 1977, pp. 69-73.

[11] Yu. L. Menshikov, “Identification of External Loads under Minimum of a Priori Information: Statements, Classification and Interpretation,” Bulletin of Kiev National University, Mathematic, No. 2, 2004, Kiev, pp. 310-315.

[12] Yu. L. Menshikov, “Identification of Models of External Loads,” In: Book of Robotics, Automation and Control, Vienna, 2008.

[13] А. N. Tikhonov and V. Yu. Arsenin, “Methods of Incorrectly Problems Solution,” Science, Moscow, 1979.

[14] Yu. L. Menshikov, “Algorithms of Construction of Adequate Mathematical Description of Dynamic System,” Proceedings of MATHMOD 09 Vienna—Full Papers CD Volume, Vienna University of Technology, Vienna, February 2009, pp. 2482-2485.

[15] Yu. S. Osipov, A. V. Krajgimsky and V. I. Maksimov, “Methods 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.

[17] Yu. L. Menshikov, “Uncontrollable Distortions of the Solutions Inverse Problems of Unbalance Identification,” Proceedings of ICSV12 XXII International Congress on Sound and Vibration, Lisbon, 11-14 July 2005, pp. 204-212.

[18] C. W. Groetsch, “Inverse Problems in the Mathematical Sciences,” Vieweg, 1993.

[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.

[20] Yu. L. Menshikov and N. V. Polyakov, “The Models of External Action for Mathematical Simulation,” Proceedings of 4th International Symposium on Systems Analysis and Simulation, Berlin, 1992, pp. 393-398.

[21] Yu. L. Menshikov and A. G. Nakonechnij, “Principle of Maximum Stability in Inverse Problems under Minimum a Priori Initial Information,” Proceedings of International Conference PDMU-2003, Kiev-Alushta, 8-12 September 2003, pp. 80-82.

[1] L. P. Lebedev, I. I. Vorovich and G. M. Gladwell, “Functional Analysis: Applications in Mechanics and Inverse Problems (Solid Mechanics and Its Applications),” Springer, Berlin, 2002.

[2] G. M. L. G. Gladwell, “Inverse Problems in Vibration,” PDF Kluwer Academic Publishers, New York, 2005.

[3] K. P. Gaikovich, “Inverse Problems in Physical Diagnostics,” Longman Scientific & Technical, Harlow, 1988.

[4] Yu. L. Menshikov, “Adequate Mathematical Description of Dynamic System: Statement Problem, Synthesis Methods,” Proceedings of the 7th EUROSIM Congress on Modelling and Simulation, Vol. 2, 2010, 7 p.

[5] Yu. L. Menshikov, “Synthesis of Adequate Mathematical Description as Inverse Problem,” Proceedings of 5th International Conference on Inverse Problems: Modeling & Simulation, Antalya, 24-29 May 2010, pp. 185-186.

[6] Proceedings of International Conference, “Identification of Dynamical Systems and Inverse Problems,” MAI, Moscow, 1994.

[7] Yu. L. Menshikov, “The Inverse Krylov Problem,” Computational Mathematics and Mathematical Physics, Vol. 43, No. 5, 2003, pp. 633-640.

[8] W. Q. Yang and L. H. Peng, “Image Reconstruction Algorithms for Electrical Capacitance Tomography,” Journal of Measurement Science and Technology, Vol. 14, No. 1, pp. 123-134.

[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.

[10] Yu. L. Menshikov and G. I. Yach, “Identification of Moment of Technological Resistance on Rolling Mill of Sheets,” Proceedings of Higher Institutes. Ferrous Metallurgy, Moscow, No. 9, 1977, pp. 69-73.

[11] Yu. L. Menshikov, “Identification of External Loads under Minimum of a Priori Information: Statements, Classification and Interpretation,” Bulletin of Kiev National University, Mathematic, No. 2, 2004, Kiev, pp. 310-315.

[12] Yu. L. Menshikov, “Identification of Models of External Loads,” In: Book of Robotics, Automation and Control, Vienna, 2008.

[13] А. N. Tikhonov and V. Yu. Arsenin, “Methods of Incorrectly Problems Solution,” Science, Moscow, 1979.

[14] Yu. L. Menshikov, “Algorithms of Construction of Adequate Mathematical Description of Dynamic System,” Proceedings of MATHMOD 09 Vienna—Full Papers CD Volume, Vienna University of Technology, Vienna, February 2009, pp. 2482-2485.

[15] Yu. S. Osipov, A. V. Krajgimsky and V. I. Maksimov, “Methods 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.

[17] Yu. L. Menshikov, “Uncontrollable Distortions of the Solutions Inverse Problems of Unbalance Identification,” Proceedings of ICSV12 XXII International Congress on Sound and Vibration, Lisbon, 11-14 July 2005, pp. 204-212.

[18] C. W. Groetsch, “Inverse Problems in the Mathematical Sciences,” Vieweg, 1993.

[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.

[20] Yu. L. Menshikov and N. V. Polyakov, “The Models of External Action for Mathematical Simulation,” Proceedings of 4th International Symposium on Systems Analysis and Simulation, Berlin, 1992, pp. 393-398.

[21] Yu. L. Menshikov and A. G. Nakonechnij, “Principle of Maximum Stability in Inverse Problems under Minimum a Priori Initial Information,” Proceedings of International Conference PDMU-2003, Kiev-Alushta, 8-12 September 2003, pp. 80-82.