Controllability and Hyers–Ulam Stability of Differential Systems with Pure Delay

Abstract

Dynamic systems of linear and nonlinear differential equations with pure delay are considered in this study. As an application, the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions is used to obtain the controllability and Hyers–Ulam stability results. By introducing a delay Gramian matrix, we establish some sufficient and necessary conditions for the controllability of linear delay differential systems. In addition, by applying Krasnoselskii’s fixed point theorem, we establish some sufficient conditions of controllability and Hyers–Ulam stability of nonlinear delay differential systems. Our results improve, extend, and complement some existing ones. Finally, two examples are given to illustrate the main results.

Controllability and Hyers–Ulam Stability of Fractional Systems with Pure Delay

“…Remark 7. We note that Theorems 1-3 improve, extend, and complement some existing results in [16,19,21,35].…”

“…Remark 4. We note in the case of α = 2 in (2) that Theorem 2 coincides with the conclusion of Corollary 3 in [16].…”

“…Remark 3. Under condition A, a nonsingular n × n matrix, we note in the case of α = 2, A = A 2 in (1) that Theorem 1 coincides with the conclusion of Theorem 3.1 in [21] and Corollary 2 in [16].…”

“…Recently, the representation of solutions of time-delay systems has been considered. In particular, the pioneering study [9,10] produced several innovative findings on the representations of solutions of time-delay systems, which were used in the control problems and stability analysis; see, for instance, [11][12][13][14][15][16][17][18][19][20][21] and the references therein.…”

“…Motivated by [11,16], the explicit solutions Formula (8) of (3) combined with the delayed Mittag-Leffler matrix functions are employed as an application to derive controllability results on Ω = [0, x 1 ].…”