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Boundary Controllability of Riemann-Liouville Fractional Semilinear Equations

Asmae Tajani, Fatima-Zahrae El Alaoui, Delfim F. M. Torres

Abstract

We study the boundary regional controllability of a class of Riemann-Liouville fractional semilinear sub-diffusion systems with boundary Neumann conditions. The result is obtained by using semi-group theory, the fractional Hilbert uniqueness method, and Schauder's fixed point theorem. Conditions on the order of the derivative, internal region, and on the nonlinear part are obtained. Furthermore, we present appropriate sufficient conditions for the considered fractional system to be regionally controllable and, therefore, boundary regionally controllable. An example of a population density system with diffusion is given to illustrate the obtained theoretical results. Numerical simulations show that the proposed method provides satisfying results regarding two cases of the control operator.

Boundary Controllability of Riemann-Liouville Fractional Semilinear Equations

Abstract

We study the boundary regional controllability of a class of Riemann-Liouville fractional semilinear sub-diffusion systems with boundary Neumann conditions. The result is obtained by using semi-group theory, the fractional Hilbert uniqueness method, and Schauder's fixed point theorem. Conditions on the order of the derivative, internal region, and on the nonlinear part are obtained. Furthermore, we present appropriate sufficient conditions for the considered fractional system to be regionally controllable and, therefore, boundary regionally controllable. An example of a population density system with diffusion is given to illustrate the obtained theoretical results. Numerical simulations show that the proposed method provides satisfying results regarding two cases of the control operator.
Paper Structure (7 sections, 8 theorems, 40 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 7 sections, 8 theorems, 40 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setting and Preliminaries
  3. The Boundary Regional Controllability Problem
  4. Algorithm and Simulations
  5. Example 1: Zonal Actuator
  6. Example 2: Pointwise Actuator
  7. Conclusion

Key Result

Lemma 2.6

For all $\beta\geq 0$, the probability density function $\mathrm{\phi}_\alpha$ satisfies

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (17)

  • Definition 2.1: See 11
  • Definition 2.2: See 11
  • Definition 2.3: See 11
  • Definition 2.4: See pazy2012semigroups
  • Definition 2.5: See pazy2012semigroups
  • Lemma 2.6: See 11
  • Lemma 2.7: See exis
  • Definition 2.8: See mee
  • Lemma 2.9: See majo
  • Lemma 2.10: See ren
  • ...and 7 more