Abstract
In this paper, the controllability of differential systems with the general conformable derivative is studied. By elaborating the rank criterion and the conformable Gram criterion, sufficient and necessary conditions to investigate that a linear general conformable system is null completely controllable are given. We obtain a full generalization to the general conformable fractional-order system case. In addition, Krasnoselskii’s fixed point theorem to obtain a complete controllability result for a semilinear general conformable system is applied.
Highlights
PreliminariesWe start this section by recalling some definitions, some lemmas, and theorems [6, 13, 14, 22]
In recent years, in [9], the conformable fractional derivative has been defined
By elaborating the rank criterion and the conformable Gram criterion, sufficient and necessary conditions to investigate that a linear general conformable system is null completely controllable are given
Summary
We start this section by recalling some definitions, some lemmas, and theorems [6, 13, 14, 22]. Assume a function φ which is defined on [a, b); the general conformable derivative (GCD) starting from real a of φ is defined by. E conformable fractional integral (CFI) of a function φ is defined by. Assume a continuous function φ defined on [a, b]. Assume that φ is an absolutely continuous function on [a, b]. Let ρ1, ρ2, ρ3 ∈ R and the functions φ1, φ2: [a, b) ⟶ R such that Tθa,ψa φ1(t) and Tθa,ψa φ2(t) exist on (a, b).
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