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Vibrosolutions to differential equations in distributions with discontinuous regular functions on the right-hand side

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Abstract

We studyn-dimensional differential equations in distributions of the form\(\dot x\)(t)=f(x, u, t)+ b(x, u, t) \(\dot u\)(t), wheref(x, u, t) andb(x, u, t) are piecewise continuous functions andu(t) is anm-dimensional function of bounded variation with nondecreasing components. The notion of vibrosolution is introduced for equations of this type, and necessary and sufficient conditions for the existence of vibrosolutions are derived. The transition to an equivalent equation with measure is carried out, thus making it possible to explicitly calculate the jumps of the vibrosolutions at the points of discontinuity ofu(t).

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References

  1. M. A. Krasnosel'skii and A. V. Pokrovskii,Systems with Hysteresis [in Russian], Nauka, Moscow (1991).

    Google Scholar 

  2. Yu. V. Orlov, “Vibrocorrect differential equations with measures,”Mat. Zametki [Math. Notes],38, No. 1, 110–119 (1985).

    MATH  MathSciNet  Google Scholar 

  3. Yu. V. Orlov,Theory of Optimal Systems with Generalized Controls [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  4. M. V. Basin and Yu. V. Orlov, “Guaranteed estimation of the state of linear dynamical systems by discrete-continuous observations,”Avtomat. i Telemekh. [Automat. Remote Control], No. 3, 36–45 (1992).

    MathSciNet  Google Scholar 

  5. A. F. Filippov,Differential Equations with Discontinuous Right-Hand Side [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  6. B. M. Miller, “Optimization of dynamical systems with generalized control,”Avtomat. i Telemekh. [Automat. Remote Control], No. 6, 23–34 (1989).

    MATH  Google Scholar 

  7. S. T. Zavalishchin and A. N. Sesekin,Sampled-Data Processes: Models and Applications [in Russian], Nauka, Moscow (1991).

    Google Scholar 

  8. A. A. Kirillov and A. D. Gvishiani,Theorems and Problems of Functional Analysis [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  9. J. Kurzweil, “Generalized ordinary differential equations,”Czechoslovak Math. J.,8, No. 3, 360–388 (1958).

    MATH  MathSciNet  Google Scholar 

  10. V. A. Dykhta and G. A. Kolokol'nikova, “Minimum conditions on a set of sequences in a degenerate variational problem,”Mat. Zametki [Math. Notes],34, No. 5, 735–744 (1983).

    MathSciNet  Google Scholar 

  11. L. Schwartz,Analyse Mathématique, Hermann, Paris (1967).

    Google Scholar 

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Authors and Affiliations

  1. Institute for Problems of Control, Russian Academy of Sciences, USSR

    M. V. Basin

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  1. M. V. Basin
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Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 12–21, July, 1995.

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Basin, M.V. Vibrosolutions to differential equations in distributions with discontinuous regular functions on the right-hand side. Math Notes 58, 685–691 (1995). https://doi.org/10.1007/BF02306177

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  • DOI: https://doi.org/10.1007/BF02306177

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