The Existence of Mild Solutions and Approximate Controllability for Nonlinear Fractional Neutral Evolution Systems

Echchaffani, Zoubida; Aberqi, Ahmed; Karite, Touria; Leiva, Hugo

doi:10.61822/amcs-2024-0002

Abstract

The existence of mild solutions and approximate controllability for Riemann–Liouville fractional neutral evolution systems with nonlocal conditions of a fractional order is investigated. The Laplace transform and semigroup theory are the tools used to prove the existence. In turn, approximate controllability is proved on the basis of a Nemytskii operator, a Mittag-Leffler function and certain hypotheses using fixed point theorems, as well as the construction of a Cauchy sequence. An example is provided to highlight the main results.

References

Agarwal, R.P., Leiva, H., Riera, L. and Lalvay, S. (2022). Existence of solutions for impulsive neutral semilinear evolution equations with nonlocal conditions, Discontinuity, Nonlinearity, and Complexity 11(2): 1–18.
Search in Google Scholar Back to article
Almarri, B. and Elshenhab, A.M. (2022). Controllability of fractional stochastic delay systems driven by the Rosenblatt process, Fractal and Fractional 6(11): 664.
Search in Google Scholar Back to article
Babiarz, A., Klamka, J. and Niezabitowski, M. (2016). Schauder’s fixed point theorem in approximate controllability problems, International Journal of Applied Mathematics and Computer Sciences 26(2): 263–275, DOI: 10.1515/amcs-2016-0018.
Search in Google Scholar Back to article
Bárcenas, D., Leiva, H. and Sívoli, Z. (2005). A broad class of evolution equations are approximately controllable but never exactly controllable, IAM Journal of Mathematical Control and Information 22(3): 310–320.
Search in Google Scholar Back to article
Chang, J. and Liu, H. (2009). Existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the α-norm, Nonlinear Analysis: Theory, Methods & Applications 71(9): 3759–3768.
Search in Google Scholar Back to article
Dhayal, R., Malik, M. and Abbas, S. (2019). Approximate and trajectory controllability of fractional neutral differential equation, Advances in Operator Theory 4(4): 802–820.
Search in Google Scholar Back to article
Diethelm, K. and Ford, N.J. (2004). Multi-order fractional differential equations and their numerical solution, Applied Mathematics and Computation 154(3): 621–640.
Search in Google Scholar Back to article
Diethelm, K. and Freed, A.D. (1999). On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity, in F. Keil et al. (Eds), Scientific Computing in Chemical Engineering II—Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, pp. 217–224.
Search in Google Scholar Back to article
Du, J., Cui, D., Sun, Y. and Xu, J. (2020). Approximate controllability for a kind of fractional neutral differential equations with damping, Mathematical Problems in Engineering 2020: 9, Article ID: 7592818.
Search in Google Scholar Back to article
Ech-chaffani, Z., Aberqi, A. and Karite, T. (n.d.). Controllability and optimal control for fractional neutral evolution systems with Caputo derivative, Boletim da Sociedade Paranaense de Matemática, (accepted for publication).
Search in Google Scholar Back to article
Ech-chaffani, Z., Aberqi, A., Karite, T. and Torres, D.F.M. (2022). Minimum energy problem in the sense of Caputo of fractional neutral evolution systems in Banach spaces, Axioms 11(8): 379.
Search in Google Scholar Back to article
Heymans, N. and Podlubny, I. (2006). Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives, Rheologica Acta 45: 765–771.
Search in Google Scholar Back to article
Hu, L., Ren, Y. and Sakthivel, R. (2009). Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum 79: 507–514.
Search in Google Scholar Back to article
Kilbas, A.A., Srivastava, H.M. and Trujillo, J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam.
Search in Google Scholar Back to article
Kiryakova, V. (1994). Generalized Fractional Calculus and Applications, Vol. 301, Longman-Wiley, New York.
Search in Google Scholar Back to article
Leiva, H. and Sundar, P. (2017). Existence of solutions for a class of semilinear evolutions equations with impulses and delays communicated by Toka Diagana, Journal of Nonlinear Evolution Equations and Applications 2017(7): 95–108.
Search in Google Scholar Back to article
Li, K., Peng, J. and Gao, J. (2013). Controllability of nonlocal fractional differential systems of order 1 <q < 2 in Banach spaces, Reports on Mathematical Physics 71(1): 33–43.
Search in Google Scholar Back to article
Li, X., Liu, X. and Tang, M. (2021). Approximate controllability of fractional evolution inclusions with damping, Chaos, Solitons & Fractals 148: 111073.
Search in Google Scholar Back to article
Li, Y. (2015). Regularity of mild solutions for fractional abstract Cauchy problem with order 1 < q < 2, Zeitschrift für angewandte Mathematik und Physik 66: 3283–3298.
Search in Google Scholar Back to article
Li, Y., Sun, H. and Feng, Z. (2016). Fractional abstract Cauchy problem with order 1 < q < 2, Dynamics of PDE 13(2): 155–177.
Search in Google Scholar Back to article
Liang, Y. (2022). Existence and approximate controllability of mild solutions for fractional evolution systems of Sobolev-type, Fractal and Fractional 6(2): 56.
Search in Google Scholar Back to article
Liu, Z. and Li, X. (2015). Approximate controllability of fractional evolution systems with Riemann–Liouville fractional derivatives, SIAM Journal on Control and Optimization 53(4): 1920–1933.
Search in Google Scholar Back to article
Mabel Lizzy, R. and Balachandran, K. (2018). Boundary controllability of nonlinear stochastic fractional systems in Hilbert spaces, International Journal of Applied Mathematics and Computer Science 28(1): 123–133, DOI: 10.2478/amcs-2018-0009.
Search in Google Scholar Back to article
Mahmudov, N.I. (2003). Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM Journal on Control and Optimization 42(5): 1604–1622.
Search in Google Scholar Back to article
Mainardi, F. (1997). Fractional calculus: Some basic problems in continuum and statistical mechanics, in A. Carpinteri and F. Mainardi (Eds), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Vienna, pp. 291–348.
Search in Google Scholar Back to article
Mingyuan, S., Chunhai, K. and Shaojun, G. (2016). Approximate controllability of fractional neutral evolution equations with nonlocal conditions, Journal of Shanghai Normal University (Natural Sciences) 45(3): 253–264.
Search in Google Scholar Back to article
Mokkedem, F.Z. and Fu, X.L. (2014). Approximate controllability of semi-linear neutral integrodifferential systems with finite delay, Applied Mathematics and Computation 242: 202–215.
Search in Google Scholar Back to article
Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York.
Search in Google Scholar Back to article
Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego.
Search in Google Scholar Back to article
Sakthivel, R., Mahmudov, N.I. and Nieto, J.J. (2012). Controllability for a class of fractional-order neutral evolution control systems, Applied Mathematics and Computation 218(20): 10334–10340.
Search in Google Scholar Back to article
Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993). Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon.
Search in Google Scholar Back to article
Triggiani, R. (1977). A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM Journal on Control and Optimization 15(3): 407–411.
Search in Google Scholar Back to article
Vijayakumar, V., Udhayakumar, R. and Kavitha, K. (2021). On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay, Evolution Equations and Control Theory 10(2): 271–296.
Search in Google Scholar Back to article
Xi, X.X., Hou, M., Zhou, X.F. and Wen, Y. (2022a). Approximate controllability of fractional neutral evolution systems of hyperbolic type, Evolution Equations and Control Theory 11(4): 1037–1069.
Search in Google Scholar Back to article
Xi, X.X., Zhou, X.F. and Wen, Y. (2022b). Approximate controllability of fractional neutral evolution systems of hyperbolic type, Evolution Equations and Control Theory 11(4): 1037–1069.
Search in Google Scholar Back to article
Yan, Z. (2012). Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces, IMA Journal of Mathematical Control and Information 30(4): 443–462.
Search in Google Scholar Back to article
Ye, H.P., Gao, J.M. and Ding, Y.S. (2007). A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematical Analysis and Applications 328(2): 1075–1081.
Search in Google Scholar Back to article
Zhang, S. (2006). Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electronic Journal of Differential Equations 2006(36): 1–12.
Search in Google Scholar Back to article
Zhao, D. and Liu, Y. (2022). New discussion on approximate controllability for semilinear fractional evolution systems with finite delay effects in Banach spaces via differentiable resolvent operators, Fractal and Fractional 6(8): 424.
Search in Google Scholar Back to article
Zhou, Y. and He, J.W. (2021). New results on controllability of fractional evolution systems with order 1 < q < 2, Evolution Equations and Control Theory 10(3): 491–509.
Search in Google Scholar Back to article
Zhou, Y. and Jiao, F. (2010a). Existence of mild solutions for fractional neutral evolution equations, Computers & Mathematics with Applications 59(3): 1063–1077.
Search in Google Scholar Back to article
Zhou, Y. and Jiao, F. (2010b). Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Analysis: Real World Applications 11(5): 4465–4475.
Search in Google Scholar Back to article
Zhu, B., Han, B.Y. and Yu, W.G. (2020). Existence of mild solutions for a class of fractional non-autonomous evolution equations with delay, Acta Mathematicae Applicatae Sinica 36(4): 870–878.
Search in Google Scholar Back to article

The Existence of Mild Solutions and Approximate Controllability for Nonlinear Fractional Neutral Evolution Systems

Abstract

Paradigm

My account