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Damping Problem for Multidimensional Control System with Delays

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Distributed Computer and Communication Networks (DCCN 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 678))

Abstract

We consider damping problem for control system with delay described by the system of differential-difference equations of neutral type, and establish the relationship of the variational problem for the nonlocal functionals and the corresponding boundary value problem for differential-difference equations. We prove the existence and uniqueness of generalized solution to the boundary value problem for this system of differential-difference equations.

A.S. Adkhamova and A.L. Skubachevskii—This research was carried out with the support of the State’s Programme of the Russian Ministry for Education and Science (No. 1.1974.2014/K).

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References

  1. Krasovskii, N.N.: Control Theory of Motion. Nauka, Moscow (1968)

    Google Scholar 

  2. Halanay, A.: Optimal controls for systems with time lag. SIAM J. Control 6, 213–234 (1968)

    MathSciNet  MATH  Google Scholar 

  3. Gabasov, R., Kirillova, F.M.: Qualitative Theory of Optimal Processes. Nauka, Moscow (1971)

    MATH  Google Scholar 

  4. Weiss, L.: On the controllability of delay-differential systems. SIAM J. Control 5, 575–587 (1967)

    Article  MathSciNet  Google Scholar 

  5. Kent, G.A.: A maximum principle for optimal control problems with neutral functional differential systems. Bull. Am. Math. Soc. 77, 565–570 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  6. Banks, H.T., Kent, G.A.: Control of functional differential equations of retarded and neutral type to target sets in function space. SIAM J. Control 10, 567–593 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kharatishvili, G.L., Tadumadze, T.A.: Nonlinear optimal control systems with variable delays. Mat. Sb. 107(149), 613–628 (1978). English transl. in Mathem. USSR Sb. 35 (1979)

    MathSciNet  Google Scholar 

  8. Onanov, G.G., Skubachevskii, A.L.: Differential equations with displaced arguments in stationary problems in the mechanics of a deformed body. Prikladnaya Mekh. 15, 39–47 (1979). English transl. in Soviet Applied Mech. 15

    MathSciNet  Google Scholar 

  9. Onanov, G.G., Tsvetkov, E.L.: On the minimum of the energy with respect to the functions with deviating argument in a stationary problem of elasticity theory. Russ. J. Math. Phys. 3(4), 491–500 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Skubachevskii, A.L.: Elliptic functional differential equations and applications. Birkhauser, Basel, Boston, Berlin (1997)

    Google Scholar 

  11. Feller, W.: Diffusion processes in one dimension. Trans. Am. Math. Soc. 77, 1–30 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ventsel’, A.D.: On boundary conditions for multidimensional diffusion processes. Theoriya Veroyatn. i ee Primen. 4, 172–185 (1959). English transl. in Theory Prob. and its Appl. 4

    MathSciNet  Google Scholar 

  13. Skubachevskii, A.L.: On some problems for multidimensional diffusion processes. Dokl. Akad. Nauk SSSR 307, 287–292 (1989)

    Google Scholar 

  14. Bitsadze, A.V., Samarskii, A.A.: On some simple generalizations of linear elliptic boundary value problems. Dokl. Akad. Nauk SSSR 185, 739–740 (1969). English transl. in Soviet Math. Dokl. 10

    MathSciNet  MATH  Google Scholar 

  15. Samarskii, A.A.: Some problems of the theory of differential equations. Differentsial’nye Uravneniya 16, 1925–1935 (1980). English transl. in Differential Equations 16

    MathSciNet  Google Scholar 

  16. Vorontsov, M.A., Ricklin, J.C., Carhart, G.W.: Optical simulation of phase-distorted imaging systems: nonlinear and adaptive optics approach. Opt. Eng. 34, 3229–3238 (1995)

    Article  Google Scholar 

  17. Razgulin, A.V.: Rotational multi-petal waves in optical system with 2D feedback, In: Roy, R. (ed.) Proceedings SPIE 2039, Chaos in Optics, pp. 342–352 (1993)

    Google Scholar 

  18. Skubachevskii, A.L.: Bifurcation of periodic solutions for nonlinear parabolic functional differential equations arising in optoelectronics. Nonlinear Anal. 32(2), 267–278 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kamenskii, G.A., Myshkis, A.D.: Formulation of boundary-value problems for differential equations with deviating arguments containing highest-order terms. Differentsial’nye Uravneniya 10, 409–418 (1974). English transl. in Differential Equations 10 (1975)

    MathSciNet  MATH  Google Scholar 

  20. Kamenskii, A.G.: Boundary value problems for equations with formally symmetric differential-difference operators. Differentsial’nye Uravneniya 12, 815–824 (1976). English transl. in Differential Equations 12 (1977)

    MathSciNet  Google Scholar 

  21. Skubachevskii, A.L.: On the damping problem for control system with delay. Dokl. Akad. Nauk Ross. 335, 157–160 (1994)

    MATH  Google Scholar 

  22. Leonov, D.D.: On damping problem for control system with delay. Contemp. Math. Fundam. Dir. 37, 28–37 (2010)

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, People’s Friendship University of Russia, Miklukho-Maklaya Street 6, Moscow, 117198, Russia

    A. S. Adkhamova & A. L. Skubachevskii

Authors
  1. A. S. Adkhamova
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  2. A. L. Skubachevskii
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Corresponding author

Correspondence to A. S. Adkhamova .

Editor information

Editors and Affiliations

  1. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    Vladimir M. Vishnevskiy

  2. RUDN University , Moscow, Russia

    Konstantin E. Samouylov

  3. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    Dmitry V. Kozyrev

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Adkhamova, A.S., Skubachevskii, A.L. (2016). Damping Problem for Multidimensional Control System with Delays. In: Vishnevskiy, V., Samouylov, K., Kozyrev, D. (eds) Distributed Computer and Communication Networks. DCCN 2016. Communications in Computer and Information Science, vol 678. Springer, Cham. https://doi.org/10.1007/978-3-319-51917-3_52

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  • DOI: https://doi.org/10.1007/978-3-319-51917-3_52

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-51916-6

  • Online ISBN: 978-3-319-51917-3

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