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23 pages, 343 KB

Open AccessArticle

Controllability and Minimum-Energy Control of Fractional Differential Systems with Time-Varying State and Control Delays

by Musarrat Nawaz, Naiqing Song and Jahan Zeb Alvi

Fractal Fract. 2026, 10(1), 23; https://doi.org/10.3390/fractalfract10010023 - 29 Dec 2025

Viewed by 513

Abstract

This paper presents a unified framework for controllability and minimum-energy control of linear fractional differential systems with Caputo derivative order

γ \in (0, 1)

and fully time-varying state and control delays. An explicit mild solution representation is derived using the [...] Read more.

This paper presents a unified framework for controllability and minimum-energy control of linear fractional differential systems with Caputo derivative order

γ \in (0, 1)

and fully time-varying state and control delays. An explicit mild solution representation is derived using the fractional fundamental matrix, and a new controllability Gramian is introduced. Using analytic properties of the matrix-valued Mittag-Leffler function, we prove a fractional Kalman-type theorem showing that bounded time-varying delays do not change the algebraic controllability structure determined by

(F, G, K)

. The minimum-energy control problem is solved in closed form through Hilbert space methods. Efficient numerical strategies and several examples—including delayed viscoelastic, neural, and robotic models—demonstrate practical applicability and computational feasibility. Full article

17 pages, 1025 KB

Open AccessArticle

A Vectorization Approach to Solving and Controlling Fractional Delay Differential Sylvester Systems

by Fatemah Mofarreh and Ahmed M. Elshenhab

Mathematics 2025, 13(22), 3631; https://doi.org/10.3390/math13223631 - 12 Nov 2025

Viewed by 304

Abstract

This paper addresses the solvability and controllability of fractional delay differential Sylvester matrix equations with non-permutable coefficient matrices. By applying a vectorization approach and Kronecker product algebra, we transform the matrix-valued problem into an equivalent vector system, enabling the derivation of explicit solution [...] Read more.

This paper addresses the solvability and controllability of fractional delay differential Sylvester matrix equations with non-permutable coefficient matrices. By applying a vectorization approach and Kronecker product algebra, we transform the matrix-valued problem into an equivalent vector system, enabling the derivation of explicit solution representations using a delayed perturbation of two-parameter Mittag-Leffler-type matrix functions. We establish necessary and sufficient conditions for controllability via a fractional delay Gramian matrix, providing a computationally verifiable criterion that requires no commutativity assumptions. The theoretical results are validated through numerical examples, demonstrating effectiveness in noncommutative scenarios where classical methods fail. Full article

(This article belongs to the Special Issue New Trends in Fractional Differential Equations with Applications)

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18 pages, 646 KB

Open AccessArticle

Analytical Solutions and Controllability of Delay Differential Matrix Equations via Kronecker Product and Delayed Matrix Functions

by Fatemah Mofarreh, Faridah Alruwaili, Xingtao Wang and Ahmed M. Elshenhab

Mathematics 2025, 13(22), 3581; https://doi.org/10.3390/math13223581 - 7 Nov 2025

Viewed by 386

Abstract

This work introduces a unified framework for analyzing linear delay differential Sylvester matrix equations with noncommuting coefficients. The methodology employs a Kronecker product-based vectorization to transform the system, yielding explicit closed-form solutions via a novel delayed perturbation matrix function. Additionally, a delay-adapted Gramian [...] Read more.

This work introduces a unified framework for analyzing linear delay differential Sylvester matrix equations with noncommuting coefficients. The methodology employs a Kronecker product-based vectorization to transform the system, yielding explicit closed-form solutions via a novel delayed perturbation matrix function. Additionally, a delay-adapted Gramian matrix is formulated to derive necessary and sufficient controllability criteria. The approach’s efficacy is confirmed through a numerical example, demonstrating its capability in complex, noncommutative scenarios where classical methods are inapplicable. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Practice of Time-Delay Systems Analysis and Control)

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19 pages, 320 KB

Open AccessArticle

Relative Controllability and Ulam–Hyers Stability of the Second-Order Linear Time-Delay Systems

by Kinda Abuasbeh, Nazim I. Mahmudov and Muath Awadalla

Mathematics 2023, 11(4), 806; https://doi.org/10.3390/math11040806 - 5 Feb 2023

Cited by 2 | Viewed by 1832

Abstract

We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and

Ω

. We also introduce a delay Gramian matrix and examine [...] Read more.

We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and

Ω

. We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam–Hyers stability of a second-order semi-linear time-delayed system. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

16 pages, 360 KB

Open AccessArticle

Controllability Criteria for Nonlinear Impulsive Fractional Differential Systems with Distributed Delays in Controls

by Amar Debbouche, Bhaskar Sundara Vadivoo, Vladimir E. Fedorov and Valery Antonov

Math. Comput. Appl. 2023, 28(1), 13; https://doi.org/10.3390/mca28010013 - 15 Jan 2023

Cited by 10 | Viewed by 2149

Abstract

We establish a class of nonlinear fractional differential systems with distributed time delays in the controls and impulse effects. We discuss the controllability criteria for both linear and nonlinear systems. The main results required a suitable Gramian matrix defined by the Mittag–Leffler function, [...] Read more.

We establish a class of nonlinear fractional differential systems with distributed time delays in the controls and impulse effects. We discuss the controllability criteria for both linear and nonlinear systems. The main results required a suitable Gramian matrix defined by the Mittag–Leffler function, using the standard Laplace transform and Schauder fixed-point techniques. Further, we provide an illustrative example supported by graphical representations to show the validity of the obtained abstract results. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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20 pages, 340 KB

Open AccessArticle

Controllability of Stochastic Delay Systems Driven by the Rosenblatt Process

by Barakah Almarri, Xingtao Wang and Ahmed M. Elshenhab

Mathematics 2022, 10(22), 4223; https://doi.org/10.3390/math10224223 - 11 Nov 2022

Cited by 1 | Viewed by 1635

Abstract

In this work, we consider dynamical systems of linear and nonlinear stochastic delay-differential equations driven by the Rosenblatt process. With the aid of the delayed matrix functions of these systems, we derive the controllability results as an application. By using a delay Gramian [...] Read more.

In this work, we consider dynamical systems of linear and nonlinear stochastic delay-differential equations driven by the Rosenblatt process. With the aid of the delayed matrix functions of these systems, we derive the controllability results as an application. By using a delay Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear stochastic delay systems. In addition, by employing Krasnoselskii’s fixed point theorem, we present some necessary criteria for the controllability of nonlinear stochastic delay systems. Our results improve and extend some existing ones. Finally, an example is given to illustrate the main results. Full article

(This article belongs to the Special Issue Dynamics and Control Theory with Applications)

18 pages, 350 KB

Open AccessArticle

Controllability of Fractional Stochastic Delay Systems Driven by the Rosenblatt Process

by Barakah Almarri and Ahmed M. Elshenhab

Fractal Fract. 2022, 6(11), 664; https://doi.org/10.3390/fractalfract6110664 - 10 Nov 2022

Cited by 5 | Viewed by 1861

Abstract

In this work, we consider linear and nonlinear fractional stochastic delay systems driven by the Rosenblatt process. With the aid of the delayed Mittag-Leffler matrix functions and the representation of solutions of these systems, we derive the controllability results as an application. By [...] Read more.

In this work, we consider linear and nonlinear fractional stochastic delay systems driven by the Rosenblatt process. With the aid of the delayed Mittag-Leffler matrix functions and the representation of solutions of these systems, we derive the controllability results as an application. By introducing a fractional delayed Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear fractional stochastic delay systems. Furthermore, by employing Krasnoselskii’s fixed point theorem, we establish sufficient conditions for the controllability of nonlinear fractional stochastic delay systems. Finally, an example is given to illustrate the main results. Full article

(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)

17 pages, 344 KB

Open AccessArticle

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

by Barakah Almarri, Xingtao Wang and Ahmed M. Elshenhab

Fractal Fract. 2022, 6(10), 611; https://doi.org/10.3390/fractalfract6100611 - 20 Oct 2022

Cited by 13 | Viewed by 1991

Abstract

Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers–Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions. We provide some sufficient and necessary conditions for [...] Read more.

Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers–Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions. We provide some sufficient and necessary conditions for the controllability of linear fractional-delay systems by introducing a fractional delay Gramian matrix. Furthermore, we establish some sufficient conditions of controllability and Hyers–Ulam stability of nonlinear fractional-delay systems by applying Krasnoselskii’s fixed-point theorem. Our results improve, extend, and complement some existing ones. Finally, numerical examples of linear and nonlinear fractional-delay systems are presented to demonstrate the theoretical results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

18 pages, 359 KB

Open AccessArticle

Existence of Solutions and Relative Controllability of a Stochastic System with Nonpermutable Matrix Coefficients

by Kinda Abuasbeh, Nazim I. Mahmudov and Muath Awadalla

Fractal Fract. 2022, 6(6), 307; https://doi.org/10.3390/fractalfract6060307 - 31 May 2022

Cited by 6 | Viewed by 2133

Abstract

In this study, time-delayed stochastic dynamical systems of linear and nonlinear equations are discussed. The existence and uniqueness of the stochastic semilinear time-delay system in finite dimensional space is investigated. Introducing the delay Gramian matrix, we establish some sufficient and necessary conditions for [...] Read more.

In this study, time-delayed stochastic dynamical systems of linear and nonlinear equations are discussed. The existence and uniqueness of the stochastic semilinear time-delay system in finite dimensional space is investigated. Introducing the delay Gramian matrix, we establish some sufficient and necessary conditions for the relative approximate controllability of time-delayed linear stochastic dynamical systems. In addition, by applying the Banach fixed point theorem, we establish some sufficient relative approximate controllability conditions for semilinear time-delayed stochastic differential systems. Finally, concrete examples are given to illustrate the main results. Full article

18 pages, 318 KB

Open AccessArticle

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

by Ahmed M. Elshenhab and Xingtao Wang

Mathematics 2022, 10(8), 1248; https://doi.org/10.3390/math10081248 - 11 Apr 2022

Cited by 16 | Viewed by 2268

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 [...] Read more.

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. Full article

(This article belongs to the Topic Dynamical Systems: Theory and Applications)

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