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VARIATION FORMULAS OF SOLUTION FOR A CONTROLLED FUNCTIONAL-DIFFERENTIAL EQUATION CONSIDERING DELAY PERTURBATION

Year 2011, Volume 01, Issue 1, 58 - 68, 01.06.2011

Abstract

Variation formulas of solution are proved for a controlled non-linear functionaldifferential equation with constant delay and the continuous initial condition. In this paper, the essential novelty is the effect of delay perturbation in the variation formulas.The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.

References

  • Gamkrelidze,R. V., (1978), Principles of optimal control theory, Plenum Press-New York and London. [2] Kharatishvili, G. L., Machaidze, Z. A., Markozashvili, N. I. and Tadumadze, T. A., (1973), Abstract Variational Theory and Its Application to Optimal Problems with Delays, Metsniereba, Tbilisi, (in Russian).
  • Kharatishvili, G. L. and Tadumadze, T. A., (2007), Variation formulas of solutions and optimal control problems for differential equations with retarded argument, J. Math. Sci.(N.Y.), 104(1), 1-175.
  • Kharatishvili, G. and Tadumadze, T., (2008), Variation formulas for solution of a nonlinear differential equation with time delay and mixed initial condition, J. Math. Sci. (N.Y.), 148 (3), 302-330.
  • Kharatishvili, G., Tadumadze, T. and Gorgodze, N., (2000), Continuous dependence and differen- tiability of solution with respect to initial data and right-hand side for differential equations with deviating argument, Mem. Differential Equations Math. Phys., 19, 3-105.
  • Neustadt, L. W., (1976), Optimization: A theory of necessary conditions, Princeton Univ.Press, Princeton, New York.
  • Ogustoreli, N. M., (1966), Time-Delay Control Systems, Academic Press, New York-London.
  • Ramishvili, I. and Tadumadze, T., (2004), Formulas of variation for a solution of neutral differential equations with continuous initial condition, Georgian Math. J., 11:1, 155-175.
  • Tadumadze, T. A., (1983), Some topics of qualitative theory of optimal control, Tbilisi State University Press, Tbilisi, (in Russian).
  • Tadumadze, T., (2009), Formulas of variation for solutions for some classes of functional-differential equations and their applications, Nonlinear Analysis, 71(12), e706-e710.
  • Tadumadze, T., and Alkhazishvili, L., (2004), The formulas of variation of solution for non-linear controlled delay differential equation with continuous initial condition, Mem. Diff. Eq. Math. Phys., 31, 83-97.
  • Tadumadze, T., (2010), Variation formulas of solution for nonlinear delay differential equation with taking into account delay perturbation and discontinuous initial condition, Georgian International Journal of Sciences and Technology, v. 3, no. 1, 53-71.
  • Tamaz Tadumadze was born in the village Khunjulauri, Samtredya re- gion, Georgia, 1947. Graduated with Honors from Tbilisi State University

Year 2011, Volume 01, Issue 1, 58 - 68, 01.06.2011

Abstract

References

  • Gamkrelidze,R. V., (1978), Principles of optimal control theory, Plenum Press-New York and London. [2] Kharatishvili, G. L., Machaidze, Z. A., Markozashvili, N. I. and Tadumadze, T. A., (1973), Abstract Variational Theory and Its Application to Optimal Problems with Delays, Metsniereba, Tbilisi, (in Russian).
  • Kharatishvili, G. L. and Tadumadze, T. A., (2007), Variation formulas of solutions and optimal control problems for differential equations with retarded argument, J. Math. Sci.(N.Y.), 104(1), 1-175.
  • Kharatishvili, G. and Tadumadze, T., (2008), Variation formulas for solution of a nonlinear differential equation with time delay and mixed initial condition, J. Math. Sci. (N.Y.), 148 (3), 302-330.
  • Kharatishvili, G., Tadumadze, T. and Gorgodze, N., (2000), Continuous dependence and differen- tiability of solution with respect to initial data and right-hand side for differential equations with deviating argument, Mem. Differential Equations Math. Phys., 19, 3-105.
  • Neustadt, L. W., (1976), Optimization: A theory of necessary conditions, Princeton Univ.Press, Princeton, New York.
  • Ogustoreli, N. M., (1966), Time-Delay Control Systems, Academic Press, New York-London.
  • Ramishvili, I. and Tadumadze, T., (2004), Formulas of variation for a solution of neutral differential equations with continuous initial condition, Georgian Math. J., 11:1, 155-175.
  • Tadumadze, T. A., (1983), Some topics of qualitative theory of optimal control, Tbilisi State University Press, Tbilisi, (in Russian).
  • Tadumadze, T., (2009), Formulas of variation for solutions for some classes of functional-differential equations and their applications, Nonlinear Analysis, 71(12), e706-e710.
  • Tadumadze, T., and Alkhazishvili, L., (2004), The formulas of variation of solution for non-linear controlled delay differential equation with continuous initial condition, Mem. Diff. Eq. Math. Phys., 31, 83-97.
  • Tadumadze, T., (2010), Variation formulas of solution for nonlinear delay differential equation with taking into account delay perturbation and discontinuous initial condition, Georgian International Journal of Sciences and Technology, v. 3, no. 1, 53-71.
  • Tamaz Tadumadze was born in the village Khunjulauri, Samtredya re- gion, Georgia, 1947. Graduated with Honors from Tbilisi State University

Details

Primary Language English
Journal Section Research Article
Authors

Tamaz TADUMADZE This is me
Tbilisi State University, Department of Mathematics, 2, University St., 0186, Tbilisi, Georgia


Abdeljalil NACHAOUİ This is me
University of Nantes/CNRS UMR 6629, Jean Leray Laboratory of Mathematics , 2 rue de Houssiniere, B.P. 92208, 44322 Nantes, France

Publication Date June 1, 2011
Published in Issue Year 2011, Volume 01, Issue 1

Cite

Bibtex @ { twmsjaem761809, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2011}, volume = {01}, number = {1}, pages = {58 - 68}, title = {VARIATION FORMULAS OF SOLUTION FOR A CONTROLLED FUNCTIONAL-DIFFERENTIAL EQUATION CONSIDERING DELAY PERTURBATION}, key = {cite}, author = {Tadumadze, Tamaz and Nachaoui, Abdeljalil} }