Abstract
This manuscript is about infinite-delayed semilinear fractional differential systems. The considered system is of weighted type so that the initial condition may be non null at time zero. The principle contribution is the approximate controllability of the considered system. The state space is not required to be reflexive and the non linear function is not supposed to be Lipschitz. More than that, the nonlinear function in this work can involve spatial derivatives which enlarges the area of application of the obtained result.
Similar content being viewed by others
References
Baleanu, D., Machado, J.A.T., Luo, A.C. (eds.): Fractional Dynamics and Control. Springer Science and Business Media, New York (2011)
Bashirov, A.E., Mahmudov, N.I.: On concepts of controllability for linear deterministic and stochastic systems. SIAM J. Control Optim. 37, 1808–1821 (1999)
Bassanini, P., Elcrat, A.R.: Theory and Applications of Partial Differential Equations. Springer Science and Business Media, New York (2013)
Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008)
Curtain, R., Zwart, H.J.: An Introduction to Infinite Dimensional Linear Systems Theory. Springer, New York (1995)
Dong, Q.: Existence and continuous dependence for weighted fractional differential equations with infinite delay. Adv. Differ. Equ. 190, 1–10 (2014)
Dong, Q., Liu, C., Fan, Z.: Weighted fractional differential equations with infinite delay in Banach spaces. Open Math. 14, 370–383 (2016)
Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Hale, J., Kato, J.: Phase space for retarded equations with infinite delay. Funk. Ekvac. 21, 11–41 (1978)
Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheol. Acta 45, 765–771 (2006)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Hino, Y., Murakami, S., Naito, T.: Functional Differential Equations with Infinite Delay. Springer, New York (1991)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
Lian, T., Fan, Z., Li, G.: Time optimal controls for fractional differential systems with Riemann–Liouville derivatives. Fract. Calculus Appl. Anal. 6, 1524–1541 (2018)
Liu, Z., Li, X.: Approximate controllability of fractional evolution systems with Riemann–Liouville fractional derivatives. SIAM J. Control Optim. 53, 1920–1933 (2015)
Mahmudov, N., Mckibben, M.: On the approximate controllability of fractional evolution equations with generalized Riemann–Liouville fractional derivative. J. Funct. Spaces 2015, 1–9 (2015)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
Podlubny, I.: Fractional Differential Equations, Mathematics in Sciences and Engineering. Academic Press, San Diego (1999)
Wang, J.R., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonl. Anal. Real World Appl. 12, 262–272 (2011)
Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (2012)
Yang, M., Wang, Q.: Approximate controllability of Riemann Liouville fractional differential inclusions. App. Math. Comput. 274, 267–281 (2016)
Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010)
Zhou, Y., Zhang, L., Shen, X.H.: Existence of mild solutions for fractional evolution equations. J. Int. Equ. Appl. 25, 557–586 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mokkedem, F.Z. Approximate Controllability for Weighted Semilinear Riemann–Liouville Fractional Differential Systems with Infinite Delay. Differ Equ Dyn Syst 31, 709–727 (2023). https://doi.org/10.1007/s12591-020-00521-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12591-020-00521-z
Keywords
- Approximate controllability
- Riemann–Liouville fractional derivatives
- \(\alpha \)-Mild solutions
- Weighted fractional equations
- Infinite delay