Regional Controllability for Caputo Type Semi-Linear Time-Fractional Systems.

Year 2022, Volume: 6 Issue: 1, 1 - 13, 31.03.2022

Fatima Zahrae El Alaoui , Ali Boutoulout Asmae Tajani

https://doi.org/10.31197/atnaa.799248

Cited By: 1

Abstract

The main purpose of this paper, is to study the regional controllability concept of a semi-linear time-fractional
diffusion systems involving Caputo derivative of order α ∈ (0,1). The main result is obtained by using an
extension of the Hilbert Uniqueness Method (HUM) in addition to a fixed point technique and under several
assumptions on the data of the considered equation. At the end, some numerical simulations are given to
illustrate the efficiently of our result.

Keywords

Regional Controllability, Fractional Calculus, Caputo Time-Fractional Systems, Fixed Point Theorems, HUM Approach, Compact Operators

Supporting Institution

Moualy Ismail University

References

• [1] G. M. Bahha, Fractional optimal control problem for differential system with control constraints, Filomat 30 (2016) 2177- 2189.
• [2] R. F. Curtain and H. Zwart, An introduction to in?nite-dimensional linear systems theory, Springer-Verlag, New York (1995).
• [3] S. R. Duraisamy, P. Sundararajan,& K. Karthikeyan, Controllability problem for fractional impulsive integrodifferential evolution systems of mixed type with the measure of noncompactness, Results in Nonlinear Anal. 3 (2020) 85-99 .
• [4] M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Soliton. Fract. 14 (2002) 433-440.
• [5] A. El Jai and A.J. Pritchard, Sensors and actuators in distributed systems analysis, Ellis Horwood series in Applied mathematics, Wiley, New York (1988).
• [6] G. Fudong, C. YangQuan, K. Chunhai, Regional analysis of time-fractional diffusion processes, Springer International Publishing (2018).
• [7] G. Fudong, C. YangQuan, K. Chunhai, Regional Controllability analysis of fractional diffusion equations with Riemann- Liouville time fractional derivatives, Automatica 76 (2017) 193-199.
• [8] G. Fudong, C. YangQuan, K. Chunhai, On the regional controllability of the sub-diffusion process with Caputo fractional derivative, Fract. Calc. Appl. Anal. 19 (2016) 1261-1281.
• [9] M. Kandasamy, A. Annamalai, Existence results for fractional integrodi?erential systems with interval impulse via sectorial operator, Results in Nonlinear Anal. 2 (2019) 169-181 .
• [10] A. A. Kilbas, H.M.Srivastava, J.J.Trujillo, Theory and applications of fractional di?erential equations, Elsevier (2006).
• [11] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris 1 (1968).
• [12] J. L. Lions, Contrôlabilité exacte perturbation et stabilisation des systèmes distribués, Masson (1988).
• [13] Y. Louartassi, A new generalization of lemma Gronwall-Bellman, Applied Mathematical Sciences 6 (2012) 621-628.
• [14] A. Pazy, Semigroups of Linear Operators and Applications to Partial Di?erential Equations, Applied Mathematical Sci- ences. New York 44 (1983).
• [15] R. Sakthivel, Y. Ren and N. I. Mahmudov, On the approximate controllability of semilinear fractional di?erential systems, Comput. Math. with Appl. 62 (2011) 1451-1459.
• [16] P.J. Torvik, R.L. Bageley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech.-T. ASME 451 (1984) 294-298.
• [17] J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl. 12 (2011) 262-272.
• [18] E. Zerrik, A. El Jai and A. Boutoulout, Actuators and regional boundary controllability of parabolic system, Int. J. Syst. Sci. 31 (2000) 73-82.
• [19] E. Zerrik, A. Kamal, Output controllability for semi-linear distributed systems, J. Dyn. Control Syst. 13 (2007) 289-306.
• [20] Y. Zhou, L. Zhang and X. H. Shen, Existence of mild solutions for fractional evolution equations, J. Integral Equ. Appl. 25 (2013) 557-586.
• [21] Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. with Appl. 59 (2010) 1063-1077.