Controllability of nonlinear fractional integrodifferential systems involving multiple delays in control
This work studies the existence of solutions and approximate controllability of fractional integrodifferential systems with Riemann-Liouville derivatives and with multiple delays in control. We establish suitable assumptions to prove the existence of solutions. Controllability of the system is shown by assuming a range condition on control operators and Lipschitz condition on non-linear functions. We use the concepts of strongly continuous semigroup rather than resolvent operators. Finally, an example is give to illustrate the theory.
[1] Kilbas, A.A., Srivastava, H.M. & Trujillo J.J.(2006). Theory and Applications of Fractional Dif- ferential Equations. North-Holland Math. Stud., 204, Elsevier Science, Amsterdam.
[2] Heymans, N. & Podlubny, I. (2006). Physical in- terpretation of initial conditions for fractional dif- ferential equations with Riemann-Liouville frac- tional derivatives. Rheologica Acta, 45, 765-771.
[3] Galucio, A.C., Deu, J.F. & Ohayon, R. (2005). A fractional derivative viscoelastic model for hy- brid active-passive damping treatments in time domain-application to sandwich beams. Journal of Intelligent Material Systems and Structures, 16(1), 33-45.
[4] Baleanu, D. & Golmankhaneh, A.K. (2010). On electromagnetic field in fractional space. Nonlin- ear Analysis: Real World Applications, 11(1), 288-292.
[5] Hilfer, R. (2000). Applications of Fractional Cal- culus in Physics. Singapore: World Scientfic Publ Co.
[6] Jia, J.H., Shen, X.Y. & Hua, H.X. (2007). Vis- coelastic behavior analysis and application of the fractional derivative Maxwell model. Journal of Vibration and Control, 13(4), 385-401.
[7] Koeller, R.C. (1984). Applications of fractional calculus to the theory of viscoelasticity. Journal of Applied Mechanics, 51(2), 299-307.
[8] Li, J., Liu, F., Feng, L. & Turner I. (2017). A novel finite volume method for the Riesz space distributed-order diffusion equation. Computers & Mathematics with Applications, 74, 772-783.
[9] Liu, X.Y., Liu, Z.H. & Fu, X. (2014). Relax- ation in nonconvex optimal control problems de- scribed by fractional differential equations. Jour- nal of Mathematical Analysis and Applications, 409(1), 446-458.
[10] Liu, Z.H., Zeng, S.D. & Bai, Y.R. (2016). Maximum principles for multi-term space-time variable-order fractional diffusion equations and their applications.Fractional Calculus and Applied Analysis, 19(1), 188-211.
[11] Liu, Z.H. & Zeng, S.D. (2017). Differential varia- tional inequalities in infinite Banach spaces. Acta Mathematica Scientia, 37B(1), 26-32.
[12] Samko, S.G., Kilbas, A.A. & Marichev, O.I.(1993). Fractional Integral and Derivatives: The- ory and Applications, Gordon and Breach, New York.
[13] Podlubny, I. (1999). Fractional Differential Equa- tions, Academic Press, San Diego, CA.
[14] Balachandran, K., Govindaraj, V. , Rivero, M. & Trujillo J.J. (2015). Controllability of fractional damped dynamical systems. Applied Mathematics and Computation, 257, 66-73.
[15] Liu, Z.H., Sun, J.H. & Szanto, I. (2013). Mono- tone iterative technique for Riemann-Liouville fractional integrodifferential equations with ad- vanced arguments. Results in Mathematics, 63, 1277–1287.
[16] Hosseini, S.M. & Shahmorad, S. (2003). Nu- merical solution of a class of integrodifferential equations by the tau method with an error es- timation.Applied Mathematics and Computation, 136(2-3), 559-570.
[17] Shakeri, F. & Dehghan, M. (2013). A high order finite volume element method for solving elliptic partial integrodifferential equations. Applied Nu- merical Mathematics, 65, 105-118.
[18] Dehghan, M. & Salehi, R. (2012). The numerical solution of the non-linear integrodifferential equa- tions based on the meshless method. Journal of Computational and Applied Mathematics, 236(9), 2367-2377.
[19] Dehghan, M. (2006). Solution of a partial inte- grodifferential equation arising from viscoelastic- ity. International Journal of Computer Mathemat- ics, 83(1), 123-129.
[20] Wang, L. (2009). Approximate controllability of integrodifferential equations with multiple delays. Journal of Optimization Theory and Applications, 143, 185-206.
[21] Ji, S. & Yang, D. (2019). Solution to Riemann- Liouville fractional integrodifferential equations via fractional resolvents. Advances in Difference Equations, 524, 1-17.
[22] Sheng, J. & Jiang, W. (2017). Existence and uniqueness of the solution of fractional damped dynamical systems. Advances in Continuous and Discrete Models, 16, 1-14.
[23] Davies, I. & Jackreece, P. (2005). Controllability and null controllability of linear systems. Journal of Applied Sciences and Environmental Manage- ment, 9, 31-36.
[24] Haq, A. & Sukavanam, N. (2020). Controllability of second-order nonlocal retarded semilinear sys- tems with delay in control. Applicable Analysis, 99(16), 2741-2754.
[25] Klamka, J. (2009). Constrained controllability of semilinear systems with delays. Nonlinear Dy- namics, 56, 169-177.
[26] Liu, S., Debbouche, A. & Wang, J. (2018). ILC method for solving approximate controllability of fractional differential equations with noninstan- taneous impulses. Journal of Computational and Applied Mathematics, 339, 343-355.
[27] Kumar, S. & Sukavanam, N. (2012). Approximate controllability of fractional order semilinear sys- tems with bounded delay. Journal of Differential Equations, 252, 6163-6174.
[28] Rykaczewski, K. (2012). Approximate controlla- bility of differential inclutions in Hilbert spaces. Nonlinear Analysis, 75, 2701-2702.
[29] Wang, J.R. & Zhou, Y. (2011). A class of frac- tional evolution equations and optimal controls. Nonlinear Analysis: Real World Application, 12, 262-272.
[30] Yang, M. & Wang, Q. (2016). Approximate con- trollability of Riemann-Liouville fractional differ- ential inclusions. Applied Mathematics and Com- putation, 274, 267-281.
[31] Mahmudov, N.I. & McKibben, M.A. (2015). On the Approximate controllability of fractional evolution equations with generalized Riemann- Liouville fractional derivative. Jouranl of Function Spaces, 2015, 1-9.
[32] Li, K., Peng, J. & Jia, J. (2012). Cauchy problems for fractional differential equations with Riemann- Liouville fractional derivatives. Journal of Func- tional Analysis, 263, 476-510.
[33] Ibrahim, BHE., Fan, Z. & Li, G. (2017). Approxi- mate controllability for functional equations with Riemann-Liouville derivative by iterative and ap- proximate method. Journal of Function Spaces, 2017, 1-7.
[34] Haq, A. (2022). Partial-approximate con- trollability of semi-linear systems involving two Riemann-Liouville fractional derivatives. Chaos, Solitons & Fractals, 157, 111923. https://doi.org/10.1016/j.chaos.2022.111923
[35] Haq, A. & Sukavanam, N. (2022). Existence and controllability of higher-order nonlinear frac- tional integrodifferential systems via fractional re- solvent, Mathematical Methods in the Applied Sci- ences, 45(16), 9034-9048.
[36] Zhu, S., Fan, Z. & Li, G. (2018). Approxi- mate controllability of Riemann-Liouville frac- tional evolution equations with integral contractor assumption. Journal of Applied Analysis & Com- putation, 8, 532-548.
[37] Chang, Y.K., Pereira, A. & Ponce, R. (2017). Ap- proximate controllability for fractional differential equations of sobolev type via properties on resol- vent operators. Fractional Calculus and Applied Analysis, 20(4), 963-987.
[38] Liu, Z. & Li, X. (2015). Approximate controllabil- ity of fractional evolution systems with Riemann– Liouville fractional derivatives. SIAM Journal on Control Optimization, 53(1), 1920-1933.
[39] He, B., Zhou, H. & Kou, C. (2016). The control- lability of fractional damped dynamical systems with control delay. Communications in Nonlinear Science and Numerical Simulation, 32, 190-198.
[40] Debbouche, A. & Antonov, V. (2017). Approxi- mate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclu- sion conditions in Banach spaces. Chaos, Solitons & Fractals, 102, 140-148.
[41] Li, X., Liu, Z., Li, J. & Tisdell, C. (2019). Exis- tence and controllability for non-linear fractional control systems with damping in Hilbert spaces. Acta Matematica Scientia, 39B(1), 229-242.
[42] Aimene, D., Baleanu, D. & Seba, D. (2019). Controllability of semilinear impulsive Atangana- Baleanu fractional differential equations with de- lay.Chaos, Solitons & Fractals, 128, 51-57.
[43] Ye, H.P., Gao, J.M. & Ding, Y.S. (2007). A gener- alized Gronwall inequality and its application to a fractional differential equation. Journal of Math- ematical Analysis and Applications, 328, 1075- 1081.
[44] Haq, A. & Sukavanam, N. (2022). Mild solution and approximate controllability of second-order retarded systems with control delays and nonlo- cal conditions. Bulletin of the Iranian Mathemat- ical Society, 48(2), 447-464.
[45] Haq, A. & Sukavanam, N. (2021). Mild solu- tion and approximate controllability of retarded semilinear systems with control delays and nonlo- cal conditions. Numerical Functional Analysis and Optimization, 42(6), 721-737.
[46] Sharma, M. (2021). Solvability and optimal con- trol of nonautonomous fractional dynamical sys- tems of neutral-type with nonlocal conditions. Ira- nian Journal of Science and Technology, Transac- tion A: Science, 45, 2121-2133.https://doi.org/10.1007/s40995-021-01215-z
[47] Patel, R., Shukla, A. & Jadon, S.S. (2020). Ex- istence and optimal control problem for semi- linear fractional order (1, 2] control system.Mathematical Methods in the Applied Sciences, https://doi.org/10.1002/mma.6662.
[48] Shukla, A., Vijayakumar, V. & Nisar, K.S. (2021). A new exploration on the existence and approx- imate controllability for fractional semilinear im- pulsive control systems of order r ∈ (1, 2). Chaos, Solitons & Fractals, 1-20.
[49] Dineshkumar, C., Udhayakumar, R., Vijayaku- mar, V., Shukla, A. & Nisar, K.S. (2021). A note on approximate controllability for nonlocal frac- tional evolution stochastic integrodifferential in- clusions of order r ∈ (1, 2) with delay. Chaos, Soli- tons & Fractals, 153, 111565.
[50] Shukla, A., Sukavanam, N. & Pandey, D.N.(2015). Complete controllability of semi-linear stochastic system with delay. Rendiconti del Cir- colo Matematico di Palermo, 64, 209-220.
[51] Sahijwani, L. & Sukavanam, N. (2023). Approx- imate controllability for Riemann-Liouville frac- tional differential equations. International Journal of Optimization & Control: Theories & Applica- tions, 13, 59-67.
[52] Raja, M.M., Vijayakumar, V., Shukla, A., Nisar, K.S. & Baskonus, H.M. (2022). On the approx- imate controllability results for fractional inte- grodifferential systems of order 1 < r < 2 with sectorial operators. Journal of Computa- tional and Applied Mathematics, 415, 114492. https://doi.org/10.1016/j.cam.2022.114492
[53] Shukla, A., Sukavanam, N. & Pandey, D.N.(2015). Approximate controllability of semi- linear fractional control systems of order α ∈ (1, 2]. SIAM Proceedings of the Con- ference on Control and its Applications. https://doi.org/10.1137/1.9781611974072.2