Relative controllability of the nonlinear fractional dynamical systems with multiple delays in control

Mustafa Aydin

doi:10.5269/bspm.75315

Mustafa Aydin Van Yuzuncu Yil University

Abstract

Relative controllability of linear and nonlinear fractional systems with time-variable delays in control variables for finite-dimensional spaces is considered. Sufficient and necessary circumstances for the controllability of a linear fractional system are offered. Employing Schauder's fixed point theorem, sufficient circumstances for the controllability of nonlinear fractional systems are presented.

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References

A. Fernandez, D. Baleanu, Classes of operators in fractional calculus: A case study, Math. Meth. Appl. Sci. 44(11), 9143–9162, (2021).

A. A. Kilbas, M. Saigo, R. K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integr. Transf. Spec. F. 15(1), 31–49, (2004).

T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19, 7–15, (1971).

R. Garra, R. Gorenflo, F. Polito, Z. Tomovski, Hilfer–Prabhakar derivatives and some applications, Appl. Math. Comput. 242, 576–589, (2014).

A. Giusti, I. Colombaro, R. Garra, R. Garrappa, F. Polito, M. Popolizio, F. Mainardi, A practical guide to Prabhakar fractional calculus, Fract. Calc. Appl. Anal. 23(1), 9–54, (2020).

R. Garrappa, G. Maione, Fractional Prabhakar derivative and applications in anomalous dielectrics: A numerical approach, in: A. Babiarz, A. Czornik, J. Klamka, M. Niezabitowski (eds.), Theory and Applications of Non-integer Order Systems, Springer, Cham, 2017.

Z. Tomovski, J. L. A. Dubbeldam, J. Korbel, Applications of Hilfer–Prabhakar operator to option pricing financial model, Fract. Calc. Appl. Anal. 23(4), 996–1012, (2020).

E. T. Whittaker, G. N. Watson, A course of modern analysis, Cambridge Univ. Press, Cambridge, (1927).

S. Eshaghi, R. K. Ghaziani, A. Ansari, Stability and dynamics of neutral and integro-differential regularized Prabhakar fractional differential systems, Comput. Appl. Math. 39(4), 1–21, (2020).

O. Sebakhy, M. M. Bayoumi, Controllability of linear time-varying systems with delay in control, Int. J. Control 17(1), 127–135, (1973).

J. P. Dauer, Nonlinear perturbations of quasi-linear control systems, J. Math. Anal. Appl. 54, 717–725, (1976).

K. Balachandran, J. Kokila, J. J. Trujillo, Relative controllability of fractional dynamical systems with multiple delays in control, Comput. Math. Appl. 64, 3037–3045, (2012).

M. Aydin, N. I. Mahmudov, Relative controllability of nonlinear delayed multi-agent systems, Int. J. Control 97(2), 348–357, (2024).

J. H. He, Nonlinear oscillation with fractional derivative and its applications, in: Int. Conf. Vibrating Engineering’98, Dalian, China, 1998, 288–291.

M. D. Ortigueira, On the initial conditions in continuous time fractional linear systems, Signal Process. 83, 2301–2309, (2003).

T. S. Chow, Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Phys. Lett. A 342, 148–155, (2005).

R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng. 32, 1–377, (2004).

J. Sabatier, O. P. Agrawal, J. A. Tenreiro-Machado (eds.), Advances in fractional calculus: theoretical developments and applications in physics and engineering, Springer-Verlag, New York, (2007).

M. Aydin, N. I. Mahmudov, Iterative learning control for impulsive fractional order time-delay systems with nonpermutable constant coefficient matrices, Int. J. Adapt. Control Signal Process. 36(6), 1419–1438, (2022).

R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol. 27, 201–210, (1983).

R. L. Bagley, P. J. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J. 23, 918–925, (1985).

F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (eds.), *Fractals and fractional calculus in continuum mechanics*, Springer-Verlag, New York, 1997, 291–348.

I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations to methods of their solution and some of their applications, Academic Press, USA, (1999).

J. L. Adams, T. T. Hartley, Finite time controllability of fractional order systems, J. Comput. Nonlinear Dyn. 3, 021402-1–021402-5, (2008).

M. Aydin, et al., On a study of the representation of solutions of a Ψ-Caputo fractional differential equations with a single delay, Electron. Res. Arch. 30(3), 625–635, (2022).

Y. Q. Chen, H. S. Ahn, D. Xue, Robust controllability of interval fractional order linear time invariant systems, Signal Process. 86, 2794–2802, (2006).

C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Xue, V. Feliu, Fractional-order systems and controls; fundamentals and applications, Springer, London, (2010).

A. B. Shamardan, M. R. A. Moubarak, Controllability and observability for fractional control systems, J. Fract. Calc. 15, 25–34, (1999).

K. Balachandran, J. P. Dauer, Controllability of nonlinear systems via fixed point theorems, J. Optim. Theory Appl. 53, 345–352, (1987).

M. Aydin, N. I. Mahmudov, ψ-Caputo type time-delay Langevin equations with two general fractional orders, Math. Meth. Appl. Sci. 46(8), 9187–9204, (2023).

K. Balachandran, D. Somasundaram, Controllability of nonlinear systems with time varying delays in control, Kybernetika 21, 65–72, (1985).

K. Balachandran, Global relative controllability of nonlinear systems with time varying multiple delays in control, Int. J. Control 46, 193–200, (1987).

J. Klamka, Relative controllability of nonlinear systems with delay in control, Automatica 12, 633–634, (1976).

J. Klamka, Controllability of nonlinear systems with distributed delay in control, Int. J. Control 31, 811–819, (1980).