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

On stability of functional differential equations with rapidly oscillating coefficients

Vitalii G. Kurbatov 1,, and  Valentina I. Kuznetsova 2,

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.
Keywords: Stability, linear functional differential equation, averaging principle, causal invertibility, passivity.
Mathematics Subject Classification: Primary:34K20;Secondary:34K06, 26A33, 34K33.
Citation: Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & 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).  Google Scholar

[2]

N. Bourbaki, Éléments de mathématique. Premi`ere 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.  Google Scholar

[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).   Google Scholar

[4]

C. Desoer and M. Vidyasagar, Feedback Systems: Input-output Properties, Academic Press, New York{London, (1975).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

N. Higham, Stable iterations for the matrix square root, Numerical Algorithms, 15 (1997), 227-242.  doi: 10.1023/A:1019150005407.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[16]

N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics Princeton University Press, Princeton, 1949. Google Scholar

[17]

V. G. Kurbatov, Functional Differential Operators and Equations vol. 473 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

V. G. Kurbatov and I. S. Frolov, An operator variant of the maximum principle, Mat. Zametki, 36 (1984), 531-535.   Google Scholar

[21]

S. Kwapień, Isomorphic characterization of inner product spaces by orthogonal series with vector valued coefficients, Studia Math., 44 (1972), 583-595.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[26]

W. Rudin, Functional Analysis 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill Inc. , New York, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

L. Schwartz, Théorie des distributions á valeurs vectorielles. I, Ann. Inst. Fourier. Grenoble, 7 (1957), 1-141.   Google Scholar

[30]

L. Schwartz, Théorie des distributions á valeurs vectorielles. II, Ann. Inst. Fourier. Grenoble, 8 (1958), 1-209.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

V. S. Vladimirov, Generalized Functions in Mathematical Physics Mir Publishers, Moscow, 1979; Translated from the second Russian edition.  Google Scholar

[35]

J. C. Willems, The Analysis of Feedback Systems The MIT Press, Cambridge, 1971. Google Scholar

[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.  Google Scholar

show all references

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).  Google Scholar

[2]

N. Bourbaki, Éléments de mathématique. Premi`ere 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.  Google Scholar

[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).   Google Scholar

[4]

C. Desoer and M. Vidyasagar, Feedback Systems: Input-output Properties, Academic Press, New York{London, (1975).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

N. Higham, Stable iterations for the matrix square root, Numerical Algorithms, 15 (1997), 227-242.  doi: 10.1023/A:1019150005407.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[16]

N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics Princeton University Press, Princeton, 1949. Google Scholar

[17]

V. G. Kurbatov, Functional Differential Operators and Equations vol. 473 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

V. G. Kurbatov and I. S. Frolov, An operator variant of the maximum principle, Mat. Zametki, 36 (1984), 531-535.   Google Scholar

[21]

S. Kwapień, Isomorphic characterization of inner product spaces by orthogonal series with vector valued coefficients, Studia Math., 44 (1972), 583-595.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[26]

W. Rudin, Functional Analysis 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill Inc. , New York, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

L. Schwartz, Théorie des distributions á valeurs vectorielles. I, Ann. Inst. Fourier. Grenoble, 7 (1957), 1-141.   Google Scholar

[30]

L. Schwartz, Théorie des distributions á valeurs vectorielles. II, Ann. Inst. Fourier. Grenoble, 8 (1958), 1-209.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

V. S. Vladimirov, Generalized Functions in Mathematical Physics Mir Publishers, Moscow, 1979; Translated from the second Russian edition.  Google Scholar

[35]

J. C. Willems, The Analysis of Feedback Systems The MIT Press, Cambridge, 1971. Google Scholar

[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.  Google Scholar

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