Robust stability problem in the mass-spring-damper system with general conformable fractional derivative

Iván Delgado-Carmona; Raúl Temoltzi-Ávila

doi:10.29057/icbi.v13iEspecial.13494

Autores/as

Iván Delgado-Carmona Área Académica de Matemáticas y Física, Universidad Autónoma del Estado de Hidalgo, México. https://orcid.org/0009-0005-6593-6648
Raúl Temoltzi-Ávila Área Académica de Matemáticas y Física, Universidad Autónoma del Estado de Hidalgo, México. https://orcid.org/0000-0003-4462-2197

DOI:

https://doi.org/10.29057/icbi.v13iEspecial.13494

Palabras clave:

sistema masa-resorte-amortiguador, derivada fraccionaria conformable general, conjunto de alcanzabilidad, problema de máxima desviación, estabilidad robusta

Resumen

Este artículo considera un sistema masa-resorte-amortiguador que admite una perturbación externa que se supone pertenece a un conjunto prefijado de funciones continuas a trozos. La derivada fraccionaria conformable general se introduce en la ecuación diferencial de segundo orden que describe la dinámica del sistema masa-resorte-amortiguador mediante el método de fraccionalización. El problema de la máxima desviación se formula y se estudia en el sistema resultante de ecuaciones diferenciales fraccionarias conformables generales. Utilizando la solución del problema de máxima desviación, se obtiene un ciclo límite máximo, y este se utiliza para establecer un criterio de estabilidad robusta para las soluciones de la ecuación diferencial fraccionaria conformable general. El criterio de estabilidad robusta se obtiene considerando una extensión de la definición de estabilidad bajo perturbaciones de acción constante que se usa en sistemas de ecuaciones diferenciales ordinarias. Los resultados obtenidos se ilustran numéricamente.

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Citas

Abdeljawad, T. (2015). On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279:57–66. https://doi.org/10.1016/j.cam.2014.10.016.

Abu-Shady, M. and Kaabar, M. K. A. (2021). A generalized definition of the fractional derivative with applications. Mathematical Problems in Engineering, 2021:9444803. https://doi.org/10.1155/2021/9444803.

Aleksandrov, V. V., Bugrov, D. I., Zhermolenko, V. N., and Konovalenko, I. S. (2021). Attainability set and robust stability of perturbed oscillatory systems. Moscow University Mechanics Bulletin, 70(1):30–34. https://doi.org/10.3103/S0027133021010027.

Aleksandrov, V. V., Reyes-Romero, M., Sidorenko, G. Y., and Temoltzi-Ávila, R. (2010). Stability of controlled inverted pendulum under permanent horizontal perturbations of the supporting point. Mechanics of Solids, 45(2):187–193. https://doi.org/10.3103/S0025654410020044.

Almeida, R., Guzowska, M., and Odzijewicz, T. (2016). A remark on local fractional calculus and ordinary derivatives. Open Mathematics, 14(1):1122–1124. https://doi.org/10.1515/math-2016-0104.

Bayour, B. and Torres, D. F. (2017). Existence of solution to a local fractional nonlinear diferential equation. Journal of Computational and Applied Mathematics, 312(2017):127–133. https://doi.org/10.1016/j.cam.2016.01.014.

Butkovskiy, A. G. (1990). Phase portraits of control dynamical systems, volume 63 of Mathematics and Its Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3258-9.

Caponetto, R., Capelas de Oliveira, E., and Tenreiro Machado, J. A. (2014). A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering, 2014:238459. https://doi.org/10.1155/2014/238459.

Cruz-Duarte, J. M., Rosales-García, J. J., and Correa-Cely, C. R. (2020). Entropy generation in a mass-spring-damper system using a conformable model. Symmetry, 12(3):395. https://doi.org/10.3390/sym12030395.

Ebaid, A., Masaedeh, B., and El-Zahar, E. (2017). A new fractional model for the falling body problem. Chinese Physics Letters, 34(2):020201. https://doi.org/10.1088/0256-307x/34/2/020201.

Elaydi, S. (2005). An introduction to diference equations. Undergraduate Texts in Mathematics. Springer, New York, 3 edition. https://doi.org/10.1007/0-387-27602-5.

Elishakoff, I. and Ohsaki, M. (2010). Optimization and anti-optimization of structures under uncertainty. Imperial College Press, Singapore. https://doi.org/10.1142/p678.

Elsgoltz, L. (1970). Diferential equations and the calculus of variations. Mir, Moscow.

Formalskii, A. M. (2015). Stabilisation and motion control of unstable objects, volume 23 of Studies in Mathematical Physics. De Gruyter, Berlin. https://doi.org/10.1515/9783110375893.

Gómez-Aguilar, J. F., Rosales-García, J. J., Bernal-Alvarado, J. J., Córdova-Fraga, T., and Guzmán-Cabrera, R. (2012). Fractional mechanical oscillators. Revista Mexicana de Física, 58(4):348–352. https://rmf.smf.mx/ojs/index.php/rmf/article/view/3934.

Kajouni, A., Chafiki, A., Hilal, K., and Oukessou, M. (2021). A new conformable fractional derivative and applications. International Journal of Differential Equations, 2021:6245435. https://doi.org/10.1155/2021/6245435.

Khalil, R., Al Horani, M., Yousef, A., and Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264:65–70. https://doi.org/10.1016/j.cam.2014.01.002.

Kurzhanski, A. B. and Varaiya, P. (2014). Dynamics and control of trajectory tubes. Theory and computation. Systems & Control: Foundations & Applications. Springer, Switzerland. https://doi.org/10.1007/978-3-319-10277-1.

Ogrekci, Y., Basci, Y., and Misir, A. (2021). Ulam type stability for conformable fractional diferential equations. Rendiconti del Circolo Matematico di Palermo Series 2, 70(2):807–817. https://doi.org/10.1007/s12215-020-00532-3.

Oldham, K. B. and Spanier, J. (1974). The fractional calculus: Theory and applications of diferentiation and integration to arbitrary order, volume 111 of Mathematics in Science and Engineering. Academic Press, Inc., New York.

Ortega, A. and Rosales, J. J. (2018). Newton’s law of cooling with fractional conformable derivative. Revista Mexicana de Física, 64(2):172–175. https://doi.org/10.31349/RevMexFis.64.172.

Rosales, J., Guía, M., Gómez, F., Aguilar, F., and Martínez, J. (2014). Two dimensional fractional projectile motion in a resisting medium. Open Physics, 12(7):517–520. https://doi.org/10.2478/s11534-014-0473-8.

Rosales, J. J., Gómez, J. F., Guía, M., and Tkach, V. I. (2011). Fractional electromagnetic waves. In International Conference on Laser and Fiber–Optical Networks Modeling (LFNM), pages 1–3. https://doi.org/10.1109/LFNM.2011.6144969.

Sales Teodoro, G., Tenreiro Machado, J. A., and Capelas de Oliveira, E. (2019). A review of definitions of fractional derivatives and other operators. Journal of Computational Physics, 388:195–208. https://doi.org/10.1016/j.jcp.2019.03.008.

Souahi, A., Makhlouf, A. B., and Hammami, M. A. (2017). Stability analysis of conformable fractional-order nonlinear systems. Indagationes Mathematicae, 28(6):1265–1274. https://doi.org/10.1016/j.indag.2017.09.009.

Younas, A., Abdeljawad, T., Batool, R., Zehra, A., and Alqudah, M. A. (2020). Linear conformable diferential system and its controllability. Advances in Diference Equations, 2020(1):449. https://doi.org/10.1186/s13662-020-02899-0.

Zhao, D. and Luo, M. (2017). General conformable fractional derivative and its physical interpretation. Calcolo, 54(3):903–917. https://doi.org/10.1007/s10092-017-0213-8.

Zhong, W. and Wang, L. (2018). Basic theory of initial value problems of conformable fractional diferential equations. Advances in Diference Equations, 2018(1):321. https://doi.org/10.1186/s13662-018-1778-5.