Motion along a Space Curve with a Quasi-Frame in Euclidean 3-Space: Acceleration and Jerk

The resolution of the acceleration and jerk vectors of a particle moving on a space curve in the Euclidean 3-space is considered. By applying this resolution and Siacci’s theorem, alternative resolutions of acceleration and jerk vectors are derived based on the quasi-frame. In the osculating plane, the acceleration vector is resolved as the sum of its tangential and radial components. In addition, in the osculating and rectifying planes, the jerk vector is resolved along the tangential direction and two special radial directions. The maximum permissible speed on a space curve at all trajectory points is established via the jerk vector formula. Finally, some examples are presented to illustrate how the results work

Finite-Time Stability Analysis of Fractional Delay Systems

Nonhomogeneous systems of fractional differential equations with pure delay are considered. As an application, the representation of solutions of these systems and their delayed Mittag-Leffler matrix functions are used to obtain the finite time stability results. Our results improve and extend the previous related results. Finally, to illustrate our theoretical results, we give an example

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

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 Differential Systems with Pure Delay

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 of Stochastic Delay Systems Driven by the Rosenblatt Process

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

Controllability of Stochastic Delay Systems Driven by the Rosenblatt Process

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

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

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

Finite-Time Stability Analysis of Linear Differential Systems with Pure Delay

Nonhomogeneous systems governed by second-order linear differential equations with pure delay are considered. As an application, the exact solutions of these systems and their delayed matrix functions are used to obtain the finite-time stability results. Our results extend and improve some previous results by removing some restrictive conditions. Finally, an example is provided to illustrate our theoretical results

Finite-Time Stability Analysis of Linear Differential Systems with Pure Delay

Nonhomogeneous systems governed by second-order linear differential equations with pure delay are considered. As an application, the exact solutions of these systems and their delayed matrix functions are used to obtain the finite-time stability results. Our results extend and improve some previous results by removing some restrictive conditions. Finally, an example is provided to illustrate our theoretical results

Existence of Hilfer Fractional Stochastic Differential Equations with Nonlocal Conditions and Delay via Almost Sectorial Operators

In this article, we examine the existence of Hilfer fractional (HF) stochastic differential systems with nonlocal conditions and delay via almost sectorial operators. The major methods depend on the semigroup of operators method and the Mo¨nch fixed-point technique via the measure of noncompactness, and the fundamental theory of fractional calculus. Finally, to clarify our key points, we provide an application