Singular Fractional Systems Research Articles

Hamilton-Jacobi Treatment of Singular Systems Using Fractional Calculus

In this paper, the theory of fractional singular systems is investigated with second-order derivatives. The fractional Hamilton–Jacobi treatment of these systems is examined. The fractional Hamilton–Jacobi partial differential equations (FHJPDEs) are constructed. The (FHJPDEs) can be solved to obtain the fractional Hamilton–Jacobi function. By building the fractional Hamilton–Jacobi function, the equations of motion can be obtained.

Multiplicity of solutions for a singular system with sign-changing potential

Abstract This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in ℝ d {\mathbb{R}^{d}} . By imposing appropriate conditions on the weight potential, which is allowed to change sign, we establish the existence of multiple solutions using the shape optimization approach. This study represents one of the earliest endeavors to explore and analyze the occurrence of multiple solutions in fractional singular systems involving sign-changing potentials. By explicitly addressing this particular aspect, our paper contributes significantly to the limited body of literature that exists in this specific field.

Disturbation to Lie symmetry for constrained Hamiltonian system within Agrawal’s operators

Lie theorem for a constrained Hamiltonian system under Agrawal’s operators is studied and proved. First, two fractional singular systems are listed. Second, the Lie symmetry of the constrained Hamiltonian system under Agrawal’s operators is defined. Third, Lie theorems of two fractional constrained Hamiltonian systems are derived and proved, and Lie conserved quantities are obtained. Furthermore, the disturbation to symmetry under Agrawal’s operators is studied. An application is given in the end.

A study on the admissibility of fractional singular systems with variable and constant delays

This paper deals with fractional singular systems with mixed delays and several admissibility criteria are obtained by using Lyapunov-Krasovskii functionals, model transformation, useful lemmas, zero equations and other well-known inequalities. Finally, some numerical examples are given with graps to verify and justify the admissibility of practical systems by using our proposed methods.

Noether Theorem for Fractional Singular Systems

Noether theorems for two fractional singular systems are discussed. One system involves mixed integer and Caputo fractional derivatives, and the other involves only Caputo fractional derivatives. Firstly, the fractional primary constraints and the fractional constrained Hamilton equations are given. Then, the fractional Noether theorems of the two fractional singular systems are established, including the fractional Noether identities, the fractional Noether quasi-identities and the fractional conserved quantities. Finally, the results obtained are illustrated by two examples.

On Stability Analysis of Riemann-Liouville Fractional Singular Systems with Delays

In this study, two lagged fractional order singular neutral differential equations are considered. Using the advantage of the association property of the Riemann -Liouville derivative, the derivative of the appropriate Lyapunov function is calculated. Then, with the help of LMI, sufficient conditions for asymptotic stability of zero solutions are obtained.

Conserved Quantities for Constrained Hamiltonian System within Combined Fractional Derivatives

Singular systems, which can be applied to gauge field theory, condensed matter theory, quantum field theory of anyons, and so on, are important dynamic systems to study. The fractional order model can describe the mechanical and physical behavior of a complex system more accurately than the integer order model. Fractional singular systems within mixed integer and combined fractional derivatives are established in this paper. The fractional Lagrange equations, fractional primary constraints, fractional constrained Hamilton equations, and consistency conditions are analyzed. Then Noether and Lie symmetry methods are studied for finding the integrals of the fractional constrained Hamiltonian systems. Finally, an example is given to illustrate the methods and results.

Conserved Quantity for Fractional Constrained Hamiltonian System

Singular system has great relationship with gauge field theory, condensed matter theory and some other research areas. Based on the mixed integer and Riemann-Liouville fractional derivatives, the fractional singular system is studied. Firstly, the fractional constrained Hamilton equation and the fractional inherent constraint are presented. Secondly, Lie symmetry and conserved quantity are analyzed, including determined equation, limited equation, additional limited equation and structural equation. And finally, an example is given to illustrate the methods and results.

Hamiltonian Formulation of Systems Described Using Fractional Singular Lagrangian

Fractional singular systems defined using mixed integer and Caputo fractional derivative are analyzed. Using these derivatives, fractional primary constraints, fractional constrained Hamilton equations and the corresponding Poisson brackets are established. Several examples are presented to demonstrate applications of the formulations.

On singular systems of nonlinear equations involving 3n-Caputo derivatives

We study singular fractional systems of nonlinear differential equations involving 3n-Caputo derivatives. We investigate existence and uniqueness results using the contraction mapping principle. We also discuss the existence of at least one solution by means of Schauder fixed point theorem. Moreover, we define and discuss the Ulam–Hyers stability and the generalized Ulam–Hyers stability of solutions for such systems. To illustrate the main results, we present some examples.

Study on Asymptotic Stability of Fractional Singular Systems with Time Delay

The stability of fractional singular systems with time delay is discussed. Considering the singularity of the system, a system is decomposed into two subsystems. Through fractional Laplacian transformation and inverse Laplacian transformation on the subsystems, the expression of the state variables in time domain is obtained. According to the characteristics of Mittag-Leffler function, some inequalities that have important influence on stability are derived. Finally, a new sufficient condition is found to make the fractional singular systems with time delay asymptotically stable when the fractional order belongs to1 < alpha < 2. Meanwhile, the sufficient condition is also obtained to make the system stable under the nonlinear disturbance. All processes are proved and numerical examples are provided to show the validity and feasibility of the proposed method.

Stability of Singular Fractional Systems of Nonlinear Integro-Differential Equations

In this paper, we study singular fractional systems of nonlinear integro-differential equations. We investigate the existence and uniqueness of solutions by means of Schauder fixed point theorem and using the contraction mapping principle. Moreover, we define and study the Ulam-Hyers stability and the generalized Ulam-Hyers stability of solutions. Some applications are presented to illustrate the main results.

Quantization of fractional singular Lagrangian systems using WKB approximation

In this paper, the fractional singular Lagrangian system is studied. The Hamilton–Jacobi treatment is developed to be applicable for fractional singular Lagrangian system. The equations of motion are obtained for the fractional singular systems and the Hamilton–Jacobi partial differential equations are obtained and solved to determine the action integral. Then the wave function for fractional singular system is obtained. Besides, to demonstrate this theory, the fractional Christ-Lee model is discussed and quantized using the WKB approximation.

Several results for high dimensional singular fractional systems involving $n^2$ Caputo derivatives

Several results for high dimensional singular fractional systems involving $n^2$ Caputo derivatives

Realisation of Linear Continuous-Time Fractional Singular Systems Using Digraph-Based Method. First approach.

In this paper, a new method for computation of a minimal realisation of a given proper transfer function of fractional continuous-time singular one-dimensional linear systems using one-dimensional digraphs theory D has been presented. For the proposed method, an algorithm was constructed. The proposed solution allows minimal digraphs construction for any one-dimensional system. The proposed method was discussed and illustrated with numerical examples.

Asymptotical stability of Riemann–Liouville fractional singular systems with multiple time-varying delays

Stability of Riemann–Liouville fractional singular systems remains an open problem. This paper deals with Riemann–Liouville fractional singular systems with multiple time-varying delays, and two asymptotic stability criteria are derived. The criteria are described as matrix equations or matrix inequalities, which are computationally flexible and efficient. The advantage of our employed method is that one may directly calculate integer-order derivatives of the Lyapunov functions. Finally, a simple example is given to illustrate the effectiveness of our main results.

Stability and Robustness of Singular Systems of Fractional Nabla Difference Equations

In this article, we study the stability and robustness of a class of singular linear systems of fractional nabla difference equations whose coefficients are constant matrices. Firstly, by assuming that the singular fractional system has a unique solution for given initial conditions, we study the asymptotic stability of the equilibria of the homogeneous system. We also prove conditions on the input vector under which the solution of the non-homogeneous system converges. Next, since it is known that existence and uniqueness of solutions depend on the invariants of the pencil of the system, by taking into consideration the fact that small perturbations can change the invariants, we perturb the singular fractional system and obtain bounds on the perturbation effect of the invariants of the pencil. In addition, by using this result, we study the robustness of solutions of the system. Finally, we give numerical examples based on a real singular fractional nabla dynamical system to illustrate our theory.

SINGULAR FRACTIONAL CONTINUOUS-TIME AND DISCRETE-TIME LINEAR SYSTEMS

Abstract New classes of singular fractional continuous-time and discrete-time linear systems are introduced. Electrical circuits are example of singular fractional continuous-time systems. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and Laplace transformation the solution to the state equation of singular fractional linear systems is derived. It is shown that every electrical circuit is a singular fractional systems if it contains at least one mesh consisting of branches with only ideal supercondensators and voltage sources or at least one node with branches with supercoils. Using the Weierstrass regular pencil decomposition the solution to the state equation of singular fractional discrete-time linear systems is derived. The considerations are illustrated by numerical examples.

Singular fractional linear systems and electrical circuits

Singular fractional linear systems and electrical circuitsA new class of singular fractional linear systems and electrical circuits is introduced. Using the Caputo definition of the fractional derivative, the Weierstrass regular pencil decomposition and the Laplace transformation, the solution to the state equation of singular fractional linear systems is derived. It is shown that every electrical circuit is a singular fractional system if it contains at least one mesh consisting of branches only with an ideal supercapacitor and voltage sources or at least one node with branches with supercoils.