Regional Controllability of Riemann–Liouville Time-Fractional Semilinear Evolution Equations

Abstract

In this paper, we discuss the exact regional controllability of fractional evolution equations involving Riemann–Liouville fractional derivative of order q ∈ 0,1 . The result is obtained with the help of the theory of fractional calculus, semigroup theory, and Banach fixed-point theorem under several assumptions on the corresponding linear system and the nonlinear term. Finally, some numerical simulations are given to illustrate the obtained result.

Regional boundary controllability of semilinear subdiffusion Caputo fractional systems

“…It should be noted that Theorem 3.3 is not a consequence of known results and that its proof employs a novel method. Here we study the boundary regional controllability for a class of Riemann-Liouville fractional semilinear systems under a condition on the nonlinear part of the system that appears in some real models, for example in the nonlinear growth population model, that is not covered by the results of [26,27]. To do so, we use in the proof of Theorem 3.3 the HUM approach and a generalization of the Gronwall-Bellman inequality, which contrasts with the method used in papers [26,27] that is based on the standard Gronwall's lemma.…”

“…For the regional controllability problem, Ge et al studied the regional controllability of a linear diffusion system with a Caputo derivative of order α ∈ (0, 1] in two cases: when the control operator is bounded, by using the Hilbert Uniqueness Method (HUM) approach; and when one has an unbounded operator, by using a compactness condition [25]. Tajani et al established sufficient conditions for regional controllability of Riemann-Liouville fractional semilinear sub-diffusion systems using two approaches: an analytical approach, based on the contraction mapping theorem; and using the HUM approach, where the regional approximate controllability of the associated linear system is assumed to hold [26,27].…”