# American Institute of Mathematical Sciences

January  2018, 17(1): 267-283. doi: 10.3934/cpaa.2018016

## On stability of functional differential equations with rapidly oscillating coefficients

 1 Department of Nonlinear Oscillations, Voronezh State University, 1, Universitetskaya Square, Voronezh 394018, Russia 2 Department of Applied Mathematics and Mechanics, Voronezh State Technical University, 14, Mos-cow Avenue, Voronezh 394026, Russia

* Corresponding author

Received  January 2017 Revised  July 2017 Published  September 2017

Fund Project: This work was supported by the Ministry of Education and Science of the Russian Federation under state order No. 3.1761.2017.

The paper deals with the functional differential equation
 $\begin{equation*} y'(t)+∈t_0^{∞}μ_0(ds)\, y(t-s)+\sum_{k=1}^∞ e^{iω_kt}∈t_0^{∞}μ_k(ds)\, y(t-s)=f(t), \end{equation*}$
where the functions
 $y$
and
 $f$
take their values in a Hilbert space,
 $ω_k∈\mathbb{R}$
,
 $μ_k$
are bounded operator-valued measures concentrated on
 $[0, +∞)$
, and
 $\sum_{k=1}^∞\Vertμ_k\Vert < ∞$
. It is shown that the equation is stable provided the unperturbed equation
 $y'(t)+\int{{_0^{∞}}}μ_0(ds)\, y(t-s)=f(t)$
is at least strictly passive (and consequently stable) and a special estimate holds; this estimate is certainly true if
 $|ω_k|$
are sufficiently large.
Citation: Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure and Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016
##### References:
 [1] N. Bourbaki, Éléments de mathématique. Fasc. XXXII. Théories spectrales. Chapitre I: Algébres normées. Chapitre II: Groupes localement compacts commutatifs Actualités Scientifiques et Industrielles, No. 1332, Hermann, Paris, 1967 (in French). [2] N. Bourbaki, Éléments de mathématique. Premiere partie. Les structures fondamentales de l’analyse. Livre VI. Intégration., Hermann, Paris, Chapitres 1-4, 1965 (in French); English translation in Springer-Verlag, Berlin, Chapters 1-4, 2004. [3] A. Defant and K. Floret, Tensor Norms and Operator Ideals, vol. 176 of North-Holland Mathematics Studies, vol. 176 of North-Holland Mathematics Studies, North-Holland, Amsterdam-London{New York-Tokyo, (1993). [4] C. Desoer and M. Vidyasagar, Feedback Systems: Input-output Properties, Academic Press, New York{London, (1975). [5] J. J. F. Fournier and J. Stewart, Amalgams of $L^p$ and $l^q$, Bull. Amer. Math. Soc. (N.S.), 13 (1985), 1-21.  doi: 10.1090/S0273-0979-1985-15350-9. [6] M. Gil', Stability of functional differential equations with oscillating coefficients and distributed delays, Differential Equations and Applications, 3 (2011), 11-19.  doi: 10.7153/dea-03-02. [7] M. Gil', Stability of vector functional differential equations with oscillating coefficients, Journal of Advanced Research in Dynamical and Control Systems, 3 (2011), 26-33. [8] R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable vol. 40 of Graduate Studies in Mathematics 3rd edition, Amer. Math. Soc. , Providence, RI, 2006. doi: 10.1090/gsm/040. [9] J. K. Hale and S. M. Verduyn Lunel, Averaging in infinite dimensions, Journal of Integral Equations and Applications, 2 (1990), 463-494.  doi: 10.1216/jiea/1181075583. [10] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations vol. 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [11] N. Higham, Stable iterations for the matrix square root, Numerical Algorithms, 15 (1997), 227-242.  doi: 10.1023/A:1019150005407. [12] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups vol. 31 of American Mathematical Society Colloquium Publications, Amer. Math. Soc. , Providence, Rhode Island, 1957. [13] Y. Hino, S. Murakami and T. Naito, Functional-differential Equations with Infinite Delay vol. 1473 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432. [14] T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, vol. 13 of Studies in Systems, Decision and Control, vol. 13 of Studies in Systems, Decision and Control, Springer, Heidelberg-New York-Dordrecht- London, (2015). [15] V. Kolmanovskii and A. Myshkis, Introduction to The Theory and Applications of Functional Differential Equations vol. 463 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. [16] N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics Princeton University Press, Princeton, 1949. [17] V. G. Kurbatov, Functional Differential Operators and Equations vol. 473 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1999. [18] V. G. Kurbatov, Efficient estimation in the averaging principle, Avtomatika i Telemekhanika, 8, 50-55 (in Russian); English translation in Automation and Remote Control, 50 (1989), 1040{1045. [19] V. G. Kurbatov, The effect of rapidly oscillating feedbacks on stability, Doklady Akademii Nauk SSSR, 316 (1991), 558-561 (in Russian); English translation in Doklady Mathematics, 36 (1991), 20-22. [20] V. G. Kurbatov and I. S. Frolov, An operator variant of the maximum principle, Mat. Zametki, 36 (1984), 531-535. [21] S. Kwapień, Isomorphic characterization of inner product spaces by orthogonal series with vector valued coefficients, Studia Math., 44 (1972), 583-595. [22] M. Lakrib and T. Sari, Time averaging for ordinary differential equations and retarded functional differential equations, Electronic Journal of Differential Equations, 2010 (2010), 1-24. [23] B. Lehman, The influence of delays when averaging slow and fast oscillating systems: overview, IMA Journal of Mathematical Control and Information, 19 (2002), 201-215. [24] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, vol. Band 181 of Die Grundlehren der mathematischen Wissenschaften, Springer- Verlag, Berlin-Heidelberg-New York, 1972, Translated from French. [25] R. Ortega, J. Perez, P. Nicklasson and H. Sira-Ramirez, Passivity-based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications Communications and Control Engineering, Springer-Verlag, London, 1998. [26] W. Rudin, Functional Analysis 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill Inc. , New York, 1991. [27] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives Nauka i Tekhnika, Minsk, 1987 (in Russian); English translation in Gordon and Breach Science Publishers, Yverdon, 1993. [28] J. A. Sanders, F. Verhulst and J. A. Murdock, Averaging Methods in Nonlinear Dynamical Systems vol. 59 of Applied Mathematical Sciences, 2nd edition, Springer, New York, 2007. [29] L. Schwartz, Théorie des distributions á valeurs vectorielles. I, Ann. Inst. Fourier. Grenoble, 7 (1957), 1-141. [30] L. Schwartz, Théorie des distributions á valeurs vectorielles. II, Ann. Inst. Fourier. Grenoble, 8 (1958), 1-209. [31] L. Schwartz, Théorie des distributions Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966. [32] R. L. Stratonovich, Conditional Markov Processes and Their Application to The Theory of Optimal Control Moscow State University, Moscow, 1966 (in Russian); English translation in American Elsevier, New York, 1968. [33] A. van der Schaft, $L_2$-gain and Passivity Techniques in Nonlinear Control, vol. 218 of Lecture Notes in Control and Information Sciences, Springer{Verlag, Berlin{Heidelberg{New Yor, (1996).  doi: 10.1007/3-540-76074-1. [34] V. S. Vladimirov, Generalized Functions in Mathematical Physics Mir Publishers, Moscow, 1979; Translated from the second Russian edition. [35] J. C. Willems, The Analysis of Feedback Systems The MIT Press, Cambridge, 1971. [36] P. P. Zabreiko and O. M. Petrova, Theorem on the continuation of bounded solutions for differential equations and the averaging principle, Sibirskii Matematicheskii Zhurnal, 21 (1980), 50-61 (in Russian); English translation in Siberian Mathematical Journal, 21 (1980), 517-526.

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##### References:
 [1] N. Bourbaki, Éléments de mathématique. Fasc. XXXII. Théories spectrales. Chapitre I: Algébres normées. Chapitre II: Groupes localement compacts commutatifs Actualités Scientifiques et Industrielles, No. 1332, Hermann, Paris, 1967 (in French). [2] N. Bourbaki, Éléments de mathématique. Premiere partie. Les structures fondamentales de l’analyse. Livre VI. Intégration., Hermann, Paris, Chapitres 1-4, 1965 (in French); English translation in Springer-Verlag, Berlin, Chapters 1-4, 2004. [3] A. Defant and K. Floret, Tensor Norms and Operator Ideals, vol. 176 of North-Holland Mathematics Studies, vol. 176 of North-Holland Mathematics Studies, North-Holland, Amsterdam-London{New York-Tokyo, (1993). [4] C. Desoer and M. Vidyasagar, Feedback Systems: Input-output Properties, Academic Press, New York{London, (1975). [5] J. J. F. Fournier and J. Stewart, Amalgams of $L^p$ and $l^q$, Bull. Amer. Math. Soc. (N.S.), 13 (1985), 1-21.  doi: 10.1090/S0273-0979-1985-15350-9. [6] M. Gil', Stability of functional differential equations with oscillating coefficients and distributed delays, Differential Equations and Applications, 3 (2011), 11-19.  doi: 10.7153/dea-03-02. [7] M. Gil', Stability of vector functional differential equations with oscillating coefficients, Journal of Advanced Research in Dynamical and Control Systems, 3 (2011), 26-33. [8] R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable vol. 40 of Graduate Studies in Mathematics 3rd edition, Amer. Math. Soc. , Providence, RI, 2006. doi: 10.1090/gsm/040. [9] J. K. Hale and S. M. Verduyn Lunel, Averaging in infinite dimensions, Journal of Integral Equations and Applications, 2 (1990), 463-494.  doi: 10.1216/jiea/1181075583. [10] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations vol. 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [11] N. Higham, Stable iterations for the matrix square root, Numerical Algorithms, 15 (1997), 227-242.  doi: 10.1023/A:1019150005407. [12] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups vol. 31 of American Mathematical Society Colloquium Publications, Amer. Math. Soc. , Providence, Rhode Island, 1957. [13] Y. Hino, S. Murakami and T. Naito, Functional-differential Equations with Infinite Delay vol. 1473 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432. [14] T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits, vol. 13 of Studies in Systems, Decision and Control, vol. 13 of Studies in Systems, Decision and Control, Springer, Heidelberg-New York-Dordrecht- London, (2015). [15] V. Kolmanovskii and A. Myshkis, Introduction to The Theory and Applications of Functional Differential Equations vol. 463 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. [16] N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics Princeton University Press, Princeton, 1949. [17] V. G. Kurbatov, Functional Differential Operators and Equations vol. 473 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1999. [18] V. G. Kurbatov, Efficient estimation in the averaging principle, Avtomatika i Telemekhanika, 8, 50-55 (in Russian); English translation in Automation and Remote Control, 50 (1989), 1040{1045. [19] V. G. Kurbatov, The effect of rapidly oscillating feedbacks on stability, Doklady Akademii Nauk SSSR, 316 (1991), 558-561 (in Russian); English translation in Doklady Mathematics, 36 (1991), 20-22. [20] V. G. Kurbatov and I. S. Frolov, An operator variant of the maximum principle, Mat. Zametki, 36 (1984), 531-535. [21] S. Kwapień, Isomorphic characterization of inner product spaces by orthogonal series with vector valued coefficients, Studia Math., 44 (1972), 583-595. [22] M. Lakrib and T. Sari, Time averaging for ordinary differential equations and retarded functional differential equations, Electronic Journal of Differential Equations, 2010 (2010), 1-24. [23] B. Lehman, The influence of delays when averaging slow and fast oscillating systems: overview, IMA Journal of Mathematical Control and Information, 19 (2002), 201-215. [24] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, vol. Band 181 of Die Grundlehren der mathematischen Wissenschaften, Springer- Verlag, Berlin-Heidelberg-New York, 1972, Translated from French. [25] R. Ortega, J. Perez, P. Nicklasson and H. Sira-Ramirez, Passivity-based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications Communications and Control Engineering, Springer-Verlag, London, 1998. [26] W. Rudin, Functional Analysis 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill Inc. , New York, 1991. [27] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives Nauka i Tekhnika, Minsk, 1987 (in Russian); English translation in Gordon and Breach Science Publishers, Yverdon, 1993. [28] J. A. Sanders, F. Verhulst and J. A. Murdock, Averaging Methods in Nonlinear Dynamical Systems vol. 59 of Applied Mathematical Sciences, 2nd edition, Springer, New York, 2007. [29] L. Schwartz, Théorie des distributions á valeurs vectorielles. I, Ann. Inst. Fourier. Grenoble, 7 (1957), 1-141. [30] L. Schwartz, Théorie des distributions á valeurs vectorielles. II, Ann. Inst. Fourier. Grenoble, 8 (1958), 1-209. [31] L. Schwartz, Théorie des distributions Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966. [32] R. L. Stratonovich, Conditional Markov Processes and Their Application to The Theory of Optimal Control Moscow State University, Moscow, 1966 (in Russian); English translation in American Elsevier, New York, 1968. [33] A. van der Schaft, $L_2$-gain and Passivity Techniques in Nonlinear Control, vol. 218 of Lecture Notes in Control and Information Sciences, Springer{Verlag, Berlin{Heidelberg{New Yor, (1996).  doi: 10.1007/3-540-76074-1. [34] V. S. Vladimirov, Generalized Functions in Mathematical Physics Mir Publishers, Moscow, 1979; Translated from the second Russian edition. [35] J. C. Willems, The Analysis of Feedback Systems The MIT Press, Cambridge, 1971. [36] P. P. Zabreiko and O. M. Petrova, Theorem on the continuation of bounded solutions for differential equations and the averaging principle, Sibirskii Matematicheskii Zhurnal, 21 (1980), 50-61 (in Russian); English translation in Siberian Mathematical Journal, 21 (1980), 517-526.
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