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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
| 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 |
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
| $y$ |
| $f$ |
| $ω_k∈\mathbb{R}$ |
| $μ_k$ |
| $[0, +∞)$ |
| $\sum_{k=1}^∞\Vertμ_k\Vert < ∞$ |
| $y'(t)+\int{{_0^{∞}}}μ_0(ds)\, y(t-s)=f(t)$ |
| $|ω_k|$ |
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). |
| [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. |
| [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. |
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). |
| [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. |
| [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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