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Boundary Regional Controllability of Semilinear Systems Involving Caputo Time Fractional Derivatives

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

Abstract

We study boundary regional controllability problems for a class of semilinear fractional systems. Sufficient conditions for regional boundary controllability are proved by assuming that the associated linear system is approximately regionally boundary controllable. The main result is obtained by using fractional powers of an operator and the fixed point technique under the approximate controllability of the corresponding linear system in a suitable subregion of the space domain. An algorithm is also proposed and some numerical simulations performed to illustrate the effectiveness of the obtained theoretical results.

Boundary Regional Controllability of Semilinear Systems Involving Caputo Time Fractional Derivatives

Abstract

We study boundary regional controllability problems for a class of semilinear fractional systems. Sufficient conditions for regional boundary controllability are proved by assuming that the associated linear system is approximately regionally boundary controllable. The main result is obtained by using fractional powers of an operator and the fixed point technique under the approximate controllability of the corresponding linear system in a suitable subregion of the space domain. An algorithm is also proposed and some numerical simulations performed to illustrate the effectiveness of the obtained theoretical results.
Paper Structure (5 sections, 8 theorems, 46 equations, 5 figures, 1 algorithm)

This paper contains 5 sections, 8 theorems, 46 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regional $\mathcal{B}$-Controllability
  4. Numerical Approach and Examples
  5. Conclusion

Key Result

Lemma 2.4

For any $\alpha,q\in ]0,1]$, the following properties hold:

Figures (5)

  • Figure 1: Reached state and the state $d_s$ in $\Omega$ for Example \ref{['ex01']}.
  • Figure 2: Desired state and reached one on $\varGamma$ for Example \ref{['ex01']}.
  • Figure 3: Reached state and the state $d_s$ in $\Omega$ for Example \ref{['ex02']}.
  • Figure 4: Desired state and reached one on $\varGamma$ for Example \ref{['ex02']}.
  • Figure 5: Evolution of the control function of Example \ref{['ex02']}.

Theorems & Definitions (21)

  • Definition 2.1: See 11
  • Definition 2.2: See 11
  • Definition 2.3: See borai
  • Lemma 2.4: See majoritmea
  • Definition 2.5
  • Definition 2.6
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 11 more