Araştırma Makalesi
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Değişken Gecikmeli Diferansiyel Sistemlerin Niteliksel Davranışları için Geliştirilmiş Yeni Şartlar

Yıl 2024, Cilt: 12 Sayı: 2, 106 - 113

Cemil Tunç

https://doi.org/10.18586/msufbd.1584044

Öz

Bilindiği üzere uygulamalı bilim alanlarında, öğreneğin, mühendislik, tıp, ekonomi, yapay sinir ağları vb. bir çok uygulamalı alanda ortaya çıkan problemlerin matematiksel olarak modellenmesine diferansiyel denklemler karşılık gelmektedir. Bu diferansiyel denklemlerin önemli bir türü is gecikmeli diferansiyel denklemlerdir. Lyapunov anlamında bu tür denklemlerin çözümlerinin karalılık vb. problemleri uygulamalarda önemli bir yere sahiptir. Ancak, genel anlamda gecikmeli diferansiyel denklemleri çözmek pek de kolay değildir, hatta numerik olarak hariç, genel anlamda açıkça çözümleri bulmak imkânsızdır. Bu tür zorluklara rağmen, Lyapunov-Krasovskii fonksiyonel metodu, göz önüne alınan gecikmeli diferansiyel denklemleri çözmeksizin, yani çözümler hakkında herhangi bir ön bilgiye ihtiyaç duymaksızın, çözümlerin kararlılığı vb. davranışları hakkında bilgi edinilmesine olanak tanır.
Bu makalede, birinci mertebeden gecikmeli diferansiyel denklemlerin sürekli zaman sistemlerinin çözümlerinin bazı niteliksel analizleri ele alınmaktadır. Burada belli formda sürekli zaman gecikmeli pertürbe ve pertürbe olmayan diferansiyel denklemler sistemleri sırasıyla göz önüne alınmaktadır. Bu sistemlerin çözümlerinin asimptotik kararlılık, integrallenebilirlik ve sınırlılık davranışları Lyapunov-Krasovskii fonksiyonel metodu yardımıyla incelenmektedir. Elden edilen sonuçların uygulanabilirliğini göstermek için iki örnek verilmiştir. Verilen yeni sonuçlar, geçmiş literatürde elde edilmiş sonuçlardan daha genel niteliktedir.

Anahtar Kelimeler

Sistem Gecikmeli diferansiyel denklemler Asimptotik kararlılık Integrallenebilirlik Sınırlılık Lyapunov-Krasovskii fonksiyonel yaklaşımı

Etik Beyan

Uygulanmamaktadır

Destekleyen Kurum

Yok

Proje Numarası

Yok

Teşekkür

Yok

Kaynakça

  • [1] Arino, O., Hbid, M. L., & Ait Dads, E. Delay differential equations and applications, Proceedings of the NATO Advanced Study Institute held at the Cadi Ayyad University, Marrakech, September 9–21, NATO Science Series II: Mathematics, Physics and Chemistry, 205. Springer, Dordrecht, 2006.
  • [2] Azbelev, N., Maksimov, V., & Rakhmatullina, L. Introduction to the theory of linear functional-differential equations, Advanced Series in Mathematical Science and Engineering, 3. World Federation Publishers Company, Atlanta, GA, 1995.
  • [3] Burton, T. A. Stability and periodic solutions of ordinary and functional differential equations, Corrected version of the 1985 original, Dover Publications, Inc., Mineola, NY, 2005.
  • [4] Chuan-Ke, Z., He, Y. , Lin Jiang, M. W., & Qing-Guo, W. An extended reciprocally convex matrix inequality for stability analysis of systems with time-varying delay, Automatica J. IFAC. 85 481–485, 2017.
  • [5] Chuan-Ke, Z., Long, F., He, Y., Yao, W., Jiang, L., & Wu, M. A relaxed quadratic function negative-determination lemma and its application to time-delay systems, Automatica J. IFAC. 113 6 pp, 2020.
  • [6] Xue Tang, D. Some kinds of Liapunov functional in stability theory of RFDE, Acta Math. Appl. Sinica. 11: 2 214–224, 1995.
  • [6] Ge, X., Wang, Z., Lei, S. -L.; Vong, S. Exponential stability of system with time delay based on matrix-refined weighted functions, Eur. J. Control . 79, Paper No. 101076, 7 pp, 2024.
  • [7] Ge, X., Shi, C., Vong, S. Weng Exponential stability of time delay systems based on intermediate polynomial-based weighted functions, Appl. Math. Lett. 116 Paper No. 107055, 7 pp., 2021.
  • [8] Graef, J. R., Tunç, C. Continuability and boundedness of multi-delay functional integro-differential equations of the second order, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109: 1 169–173, 2015.
  • [9] Hale, J. K., Verduyn Lunel, S. M. Introduction to functional-differential equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993.
  • [10] Kolmanovskii, V., Myshkis, A. Applied theory of functional-differential equations, Mathematics and its Applications (Soviet Series), 85. Kluwer Academic Publishers Group, Dordrecht, 1992.
  • [11] Kolmanovskii, V., Myshkis, A. Introduction to the theory and applications of functional-differential equations, Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999.
  • [12] Kolmanovskii, V. B.,Nosov, V. R. Stability of functional-differential equations, Mathematics in Science and Engineering, 180. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986.
  • [13] Krasovskiĭ, N. N. Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay, Translated by J. L. Brenner Stanford University Press, Stanford, Calif, 1963.
  • [14] Kuang, Y. Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993.
  • [15] Lakshmikantham, V., Wen, L. Z., & Zhang, B. G. Theory of differential equations with unbounded delay, Mathematics and its Applications, 298. Kluwer Academic Publishers Group, Dordrecht, 1994.
  • [16] Lee, T.H., Park, J. H., Xu, S. Relaxed conditions for stability of time-varying delay systems, Automatica J. IFAC. 75 11–15, 2017.
  • [17] Li, X., Yang, X., Song, S. Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica J. IFAC. 103 135–140, 2019.
  • [18] Li, Z., Yan, H., Zhang, H., Zhan, X., Huang, C. Improved inequality-based functions approach for stability analysis of time delay system, Automatica J. IFAC 108 8 pp, 2019.
  • [19] Seuret, A., Gouaisbaut, F. Hierarchy of LMI conditions for the stability analysis of time-delay systems, Systems Control Lett. 81 1–7, 2015.
  • [20] Slyn’ko, V.I., Tunç, C. Instability of set differential equations, J. Math. Anal. Appl. 467: 2 935–947, 2018.
  • [21] Smith, H. An introduction to delay differential equations with applications to the life sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011.
  • [22] Tian, J., Ren, Z. Stability analysis of systems with time-varying delays via an improved integral inequality, IEEE Access. 8 90889–90894, 2020.
  • [23] Tunç, C. A note on boundedness of solutions to a class of non-autonomous differential equations of second order, Appl. Anal. Discrete Math. 4: 2 361–372, 2010.
  • [24] Tunç, C. Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations, Appl. Comput. Math. 10: 3 449–462, 2011.
  • [25] Tunç, C. Stability to vector Liénard equation with constant deviating argument, Nonlinear Dynam. 73:3 1245–1251, 2013.
  • [26] Tunç, C., Erdur, S. New qualitative results for solutions of functional differential equations of second order., Discrete Dyn. Nat. Soc, Art. ID 3151742: 13 pp, 2018.
  • [27] Tunç, C., Golmankhaneh, A.K. On stability of a class of second alpha-order fractal differential equations. AIMS Math. 5: 3 2126–2142, 2020.
  • [28] Tunç, C., Tunç, O. A note on certain qualitative properties of a second order linear differential system, Appl. Math. Inf. Sci. 9: 2 953–956, 2015.
  • [29] Tunç, C., Tunç, O. On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order, Journal of Advanced Research. 7:1 165-168, 2016.
  • [30] Tunç, C., Tunç, O. A note on the stability and boundedness of solutions to non-linear differential systems of second order, Journal of the Association of Arab Universities for Basic and Applied Sciences. 24 169-175, 2017.
  • [31] Tunç, C., Tunç, O. Qualitative analysis for a variable delay system of differential equations of second order, Journal of Taibah University for Science. 13: 1 468–477, 2019.
  • [32] Tunç, C., Tunç, O., Wang, Y., & Yao, J.-C. Qualitative analyses of differential systems with time-varying delays via Lyapunov–Krasovskiĭ approach, Mathematics. 9, 1196, 2021.
  • [33] Tunç, O. On the behaviors of solutions of systems of non-linear differential equations with multiple constant delays, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115: 4 22 pp, 2021.
  • [34] Tunç, O., Tunç, C., Wang, Y. Delay-dependent stability, integrability and boundedeness criteria for delay differential systems, Axioms. 10: 3 138, 2021.
  • [35] Wang, Y., Liu, H., & Li, X. A novel method for stability analysis of time-varying delay systems, IEEE Trans. Automat. Control. 66: 3 1422–1428, 2021.
  • [36] Wu, M., He, Y. Parameter-dependent Lyapunov functional for systems with multiple time delays, J. Control Theory Appl. 2:3 239–245, 2004.
  • [37] Zevin, A. A. Maximum Lyapunov exponents and stability criteria of linear systems with variable delay, J. Appl. Math. Mech. 79: 1 1–8, 2015.

Improved New Conditions for Qualitative Behaviors of Time-varying Delay Differential Systems

Yıl 2024, Cilt: 12 Sayı: 2, 106 - 113

Cemil Tunç

https://doi.org/10.18586/msufbd.1584044

Öz

Bilindiği üzere uygulamalı bilim alanlarında, öğreneğin, mühendislik, tıp, ekonomi, yapay sinir ağları vb. bir çok uygulamalı alanda ortaya çıkan problemlerin matematiksel olarak modellenmesine diferansiyel denklemler karşılık gelmektedir. Bu diferansiyel denklemlerin önemli bir türü is gecikmeli diferansiyel denklemlerdir. Lyapunov anlamında bu tür denklemlerin çözümlerinin karalılık vb. problemleri uygulamalarda önemli bir yere sahiptir. Ancak, genel anlamda gecikmeli diferansiyel denklemleri çözmek pek de kolay değildir, hatta numerik olarak hariç, genel anlamda açıkça çözümleri bulmak imkânsızdır. Bu tür zorluklara rağmen, Lyapunov-Krasovskii fonksiyonel metodu, göz önüne alınan gecikmeli diferansiyel denklemleri çözmeksizin, yani çözümler hakkında herhangi bir ön bilgiye ihtiyaç duymaksızın, çözümlerin kararlılığı vb. davranışları hakkında bilgi edinilmesine olanak tanır.
Bu makalede, birinci mertebeden gecikmeli diferansiyel denklemlerin sürekli zaman sistemlerinin çözümlerinin bazı niteliksel analizleri ele alınmaktadır. Burada belli formda sürekli zaman gecikmeli pertürbe ve pertürbe olmayan diferansiyel denklemler sistemleri sırasıyla göz önüne alınmaktadır. Bu sistemlerin çözümlerinin asimptotik kararlılık, integrallenebilirlik ve sınırlılık davranışları Lyapunov-Krasovskii fonksiyonel metodu yardımıyla incelenmektedir. Elden edilen sonuçların uygulanabilirliğini göstermek için iki örnek verilmiştir. Verilen yeni sonuçlar, geçmiş literatürde elde edilmiş sonuçlardan daha genel niteliktedir

Anahtar Kelimeler

System Delay differential equations Asymptotic stability Integrability Boundedness Lyapunov-Krasovskii functional approach

Proje Numarası

Yok

Kaynakça

  • [1] Arino, O., Hbid, M. L., & Ait Dads, E. Delay differential equations and applications, Proceedings of the NATO Advanced Study Institute held at the Cadi Ayyad University, Marrakech, September 9–21, NATO Science Series II: Mathematics, Physics and Chemistry, 205. Springer, Dordrecht, 2006.
  • [2] Azbelev, N., Maksimov, V., & Rakhmatullina, L. Introduction to the theory of linear functional-differential equations, Advanced Series in Mathematical Science and Engineering, 3. World Federation Publishers Company, Atlanta, GA, 1995.
  • [3] Burton, T. A. Stability and periodic solutions of ordinary and functional differential equations, Corrected version of the 1985 original, Dover Publications, Inc., Mineola, NY, 2005.
  • [4] Chuan-Ke, Z., He, Y. , Lin Jiang, M. W., & Qing-Guo, W. An extended reciprocally convex matrix inequality for stability analysis of systems with time-varying delay, Automatica J. IFAC. 85 481–485, 2017.
  • [5] Chuan-Ke, Z., Long, F., He, Y., Yao, W., Jiang, L., & Wu, M. A relaxed quadratic function negative-determination lemma and its application to time-delay systems, Automatica J. IFAC. 113 6 pp, 2020.
  • [6] Xue Tang, D. Some kinds of Liapunov functional in stability theory of RFDE, Acta Math. Appl. Sinica. 11: 2 214–224, 1995.
  • [6] Ge, X., Wang, Z., Lei, S. -L.; Vong, S. Exponential stability of system with time delay based on matrix-refined weighted functions, Eur. J. Control . 79, Paper No. 101076, 7 pp, 2024.
  • [7] Ge, X., Shi, C., Vong, S. Weng Exponential stability of time delay systems based on intermediate polynomial-based weighted functions, Appl. Math. Lett. 116 Paper No. 107055, 7 pp., 2021.
  • [8] Graef, J. R., Tunç, C. Continuability and boundedness of multi-delay functional integro-differential equations of the second order, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109: 1 169–173, 2015.
  • [9] Hale, J. K., Verduyn Lunel, S. M. Introduction to functional-differential equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993.
  • [10] Kolmanovskii, V., Myshkis, A. Applied theory of functional-differential equations, Mathematics and its Applications (Soviet Series), 85. Kluwer Academic Publishers Group, Dordrecht, 1992.
  • [11] Kolmanovskii, V., Myshkis, A. Introduction to the theory and applications of functional-differential equations, Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999.
  • [12] Kolmanovskii, V. B.,Nosov, V. R. Stability of functional-differential equations, Mathematics in Science and Engineering, 180. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986.
  • [13] Krasovskiĭ, N. N. Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay, Translated by J. L. Brenner Stanford University Press, Stanford, Calif, 1963.
  • [14] Kuang, Y. Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993.
  • [15] Lakshmikantham, V., Wen, L. Z., & Zhang, B. G. Theory of differential equations with unbounded delay, Mathematics and its Applications, 298. Kluwer Academic Publishers Group, Dordrecht, 1994.
  • [16] Lee, T.H., Park, J. H., Xu, S. Relaxed conditions for stability of time-varying delay systems, Automatica J. IFAC. 75 11–15, 2017.
  • [17] Li, X., Yang, X., Song, S. Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica J. IFAC. 103 135–140, 2019.
  • [18] Li, Z., Yan, H., Zhang, H., Zhan, X., Huang, C. Improved inequality-based functions approach for stability analysis of time delay system, Automatica J. IFAC 108 8 pp, 2019.
  • [19] Seuret, A., Gouaisbaut, F. Hierarchy of LMI conditions for the stability analysis of time-delay systems, Systems Control Lett. 81 1–7, 2015.
  • [20] Slyn’ko, V.I., Tunç, C. Instability of set differential equations, J. Math. Anal. Appl. 467: 2 935–947, 2018.
  • [21] Smith, H. An introduction to delay differential equations with applications to the life sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011.
  • [22] Tian, J., Ren, Z. Stability analysis of systems with time-varying delays via an improved integral inequality, IEEE Access. 8 90889–90894, 2020.
  • [23] Tunç, C. A note on boundedness of solutions to a class of non-autonomous differential equations of second order, Appl. Anal. Discrete Math. 4: 2 361–372, 2010.
  • [24] Tunç, C. Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations, Appl. Comput. Math. 10: 3 449–462, 2011.
  • [25] Tunç, C. Stability to vector Liénard equation with constant deviating argument, Nonlinear Dynam. 73:3 1245–1251, 2013.
  • [26] Tunç, C., Erdur, S. New qualitative results for solutions of functional differential equations of second order., Discrete Dyn. Nat. Soc, Art. ID 3151742: 13 pp, 2018.
  • [27] Tunç, C., Golmankhaneh, A.K. On stability of a class of second alpha-order fractal differential equations. AIMS Math. 5: 3 2126–2142, 2020.
  • [28] Tunç, C., Tunç, O. A note on certain qualitative properties of a second order linear differential system, Appl. Math. Inf. Sci. 9: 2 953–956, 2015.
  • [29] Tunç, C., Tunç, O. On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order, Journal of Advanced Research. 7:1 165-168, 2016.
  • [30] Tunç, C., Tunç, O. A note on the stability and boundedness of solutions to non-linear differential systems of second order, Journal of the Association of Arab Universities for Basic and Applied Sciences. 24 169-175, 2017.
  • [31] Tunç, C., Tunç, O. Qualitative analysis for a variable delay system of differential equations of second order, Journal of Taibah University for Science. 13: 1 468–477, 2019.
  • [32] Tunç, C., Tunç, O., Wang, Y., & Yao, J.-C. Qualitative analyses of differential systems with time-varying delays via Lyapunov–Krasovskiĭ approach, Mathematics. 9, 1196, 2021.
  • [33] Tunç, O. On the behaviors of solutions of systems of non-linear differential equations with multiple constant delays, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115: 4 22 pp, 2021.
  • [34] Tunç, O., Tunç, C., Wang, Y. Delay-dependent stability, integrability and boundedeness criteria for delay differential systems, Axioms. 10: 3 138, 2021.
  • [35] Wang, Y., Liu, H., & Li, X. A novel method for stability analysis of time-varying delay systems, IEEE Trans. Automat. Control. 66: 3 1422–1428, 2021.
  • [36] Wu, M., He, Y. Parameter-dependent Lyapunov functional for systems with multiple time delays, J. Control Theory Appl. 2:3 239–245, 2004.
  • [37] Zevin, A. A. Maximum Lyapunov exponents and stability criteria of linear systems with variable delay, J. Appl. Math. Mech. 79: 1 1–8, 2015.
Toplam 38 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematiksel Yöntemler ve Özel Fonksiyonlar
Bölüm Araştırma Makalesi
Yazarlar

Cemil Tunç 0000-0003-2909-8753

Proje Numarası Yok
Erken Görünüm Tarihi 21 Aralık 2024
Yayımlanma Tarihi
Gönderilme Tarihi 13 Kasım 2024
Kabul Tarihi 4 Aralık 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 12 Sayı: 2

Kaynak Göster

APA Tunç, C. (2024). Improved New Conditions for Qualitative Behaviors of Time-varying Delay Differential Systems. Mus Alparslan University Journal of Science, 12(2), 106-113. https://doi.org/10.18586/msufbd.1584044
AMA Tunç C. Improved New Conditions for Qualitative Behaviors of Time-varying Delay Differential Systems. MAUN Fen Bil. Dergi. Aralık 2024;12(2):106-113. doi:10.18586/msufbd.1584044
Chicago Tunç, Cemil. “Improved New Conditions for Qualitative Behaviors of Time-Varying Delay Differential Systems”. Mus Alparslan University Journal of Science 12, sy. 2 (Aralık 2024): 106-13. https://doi.org/10.18586/msufbd.1584044.
EndNote Tunç C (01 Aralık 2024) Improved New Conditions for Qualitative Behaviors of Time-varying Delay Differential Systems. Mus Alparslan University Journal of Science 12 2 106–113.
IEEE C. Tunç, “Improved New Conditions for Qualitative Behaviors of Time-varying Delay Differential Systems”, MAUN Fen Bil. Dergi., c. 12, sy. 2, ss. 106–113, 2024, doi: 10.18586/msufbd.1584044.
ISNAD Tunç, Cemil. “Improved New Conditions for Qualitative Behaviors of Time-Varying Delay Differential Systems”. Mus Alparslan University Journal of Science 12/2 (Aralık 2024), 106-113. https://doi.org/10.18586/msufbd.1584044.
JAMA Tunç C. Improved New Conditions for Qualitative Behaviors of Time-varying Delay Differential Systems. MAUN Fen Bil. Dergi. 2024;12:106–113.
MLA Tunç, Cemil. “Improved New Conditions for Qualitative Behaviors of Time-Varying Delay Differential Systems”. Mus Alparslan University Journal of Science, c. 12, sy. 2, 2024, ss. 106-13, doi:10.18586/msufbd.1584044.
Vancouver Tunç C. Improved New Conditions for Qualitative Behaviors of Time-varying Delay Differential Systems. MAUN Fen Bil. Dergi. 2024;12(2):106-13.
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