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

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

Kaynakça

1. [1] G. M. Bahha, Fractional optimal control problem for differential system with control constraints, Filomat 30 (2016) 2177- 2189.
2. [2] R. F. Curtain and H. Zwart, An introduction to in?nite-dimensional linear systems theory, Springer-Verlag, New York (1995).
3. [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. [4] M. M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Soliton. Fract. 14 (2002) 433-440.
5. [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. [6] G. Fudong, C. YangQuan, K. Chunhai, Regional analysis of time-fractional diffusion processes, Springer International Publishing (2018).
7. [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. [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. [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. [10] A. A. Kilbas, H.M.Srivastava, J.J.Trujillo, Theory and applications of fractional di?erential equations, Elsevier (2006).
11. [11] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris 1 (1968).
12. [12] J. L. Lions, Contrôlabilité exacte perturbation et stabilisation des systèmes distribués, Masson (1988).
13. [13] Y. Louartassi, A new generalization of lemma Gronwall-Bellman, Applied Mathematical Sciences 6 (2012) 621-628.
14. [14] A. Pazy, Semigroups of Linear Operators and Applications to Partial Di?erential Equations, Applied Mathematical Sci- ences. New York 44 (1983).
15. [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. [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. [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. [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. [19] E. Zerrik, A. Kamal, Output controllability for semi-linear distributed systems, J. Dyn. Control Syst. 13 (2007) 289-306.
20. [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. [21] Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. with Appl. 59 (2010) 1063-1077.