Studies on convergence and stability of iterative learning control in impulsive fractional systems with Hilfer fractional derivative

In this paper, the stability of iterative learning control with data dropouts is discussed. By the super vector formulation, an iterative learning control (ILC) system with data dropouts can be modeled as an asynchronous dynamical system with rate constraints on events in the iteration domain. The stability condition is provided in the form of linear matrix inequalities (LMIS) depending on the stability of asynchronous dynamical systems. The analysis is supported by simulations.

A novel iterative learning control (ILC) algorithm for a two‐wheeled self‐balancing mobile robot with time‐varying, nonlinear, and strong‐coupling dynamics properties is presented to resolve the trajectory tracking problem in this research. A kinematics model and dynamic model of a two‐wheeled self‐balancing mobile robot are deduced in this paper, and the combination of an open‐closed‐loop PD‐ILC law and a variable forgetting factor is presented. The open‐closed‐loop PD‐ILC algorithm adopts current and past learning items to drive the state variables and input variables, and the output variables converge to the bounded scope of their desired values. In addition, introducing a variable forgetting factor can enhance the robustness and stability of ILC. Numerous simulation and experimental data demonstrate that the proposed control scheme has better feasibility and effectiveness than the traditional control algorithm.

In this paper, we initially derive the equivalent fractional integral equation to $$\Psi $$ -Hilfer hybrid fractional differential equations and through it, we prove the existence of a solution in the weighted space. The paper’s primary objective is to obtain estimates on $$\Psi $$ -Hilfer fractional derivative and utilize it to derive the hybrid fractional differential inequalities involving $$\Psi $$ -Hilfer fractional derivative. With the assistance of these fractional differential inequalities, we determine the existence of extremal solutions and comparison theorems.

This paper is devoted to the study of a coupled system of fractional Langevin equations involving the ‐Hilfer fractional derivative and nonlocal Riemann–Stieltjes integral boundary conditions. By transforming the problem into an equivalent system of integral equations, sufficient conditions for the existence of solutions are established via Schauder's fixed point theorem and the Leray–Schauder nonlinear alternative. In addition, uniqueness results are obtained by applying the Banach contraction principle. Several illustrative examples are provided to demonstrate the applicability of the theoretical results and to validate the proposed approach.

In the present paper, we investigate the Hardy–Littlewood type and the integration by parts result for –Riemann–Liouville fractional integrals. Also, we attack the integration by parts for the –Riemann–Liouville and –Hilfer fractional derivatives. To finish, we investigated Sobolev‐type inequalities involving the –Riemann–Liouville and the –Hilfer fractional derivatives in weighted space.

Hilfer fractional derivative is an important and interesting operator in fractional calculus, and it can be applicable in pure theories and other fields. It yields to other notable definitions, Ψ-Hilfer, (k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(k,\\Psi )$\\end{document}-Hilfer derivatives, etc. Motivated by the concepts of the proportional fractional derivative and (k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(k,\\Psi )$\\end{document}-Hilfer fractional derivative, we first introduce new definitions of integral and derivative, termed the (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional integral and (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional Hilfer fractional derivative. This type of fractional derivative is advantageous as it aligns with earlier studies on fractional differential equations. Additionally, we present a more generalized version of the (ρ,α,β,k,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,\\alpha ,\\beta ,k,r)$\\end{document}-resolvent family, followed by an exploration of its properties. By analyzing the generalized resolvent family, we examine the existence of mild solutions to the (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional Hilfer fractional Cauchy problem, supported by an illustrative example to show the main result.

In this paper, we study the forced oscillatory theory for higher order fractional differential equations with damping term via $\Psi$-Hilfer fractional derivative. We get sufficient conditions which ensure the oscillation of all solutions and give an illustrative example for our results. The $\Psi$-Hilfer fractional derivative according to the choice of the $\Psi$ function is a generalization of the different fractional derivatives defined earlier. The results obtained in this paper are a generalization of the known results in the literature, and present new results for some fractional derivatives.

In this paper, we introduce an extension of the Hilfer fractional derivative, the “Hilfer fractional quantum derivative”, and establish some of its properties. Then, we introduce and discuss initial and boundary value problems involving the Hilfer fractional quantum derivative. The existence of a unique solution of the considered problems is established via Banach’s contraction mapping principle. Examples illustrating the obtained results are also presented.

The determination of a space‐dependent source term along with the solution for a 1‐dimensional time fractional diffusion equation with nonlocal boundary conditions involving a parameter β>0 is considered. The fractional derivative is generalization of the Riemann‐Liouville and Caputo fractional derivatives usually known as Hilfer fractional derivative. We proved existence and uniqueness results for the solution of the inverse problem while over‐specified datum at 2 different time is given. The over‐specified datum at 2 time allows us to avoid initial condition in terms of fractional integral associated with Hilfer fractional derivative.

Because of the prevalent time-delay characteristics in real-world phenomena, this paper investigates the existence of mild solutions for diffusion equations with time delays and the Hilfer fractional derivative. This derivative extends the traditional Caputo and Riemann–Liouville fractional derivatives, offering broader practical applications. Initially, we constructed Banach spaces required to handle the time-delay terms. To address the challenge of the unbounded nature of the solution operator at the initial moment, we developed an equivalent continuous operator. Subsequently, within the contexts of both compact and non-compact analytic semigroups, we explored the existence and uniqueness of mild solutions, considering various growth conditions of nonlinear terms. Finally, we presented an example to illustrate our main conclusions.

Existence of mild solution for evolution equation with Hilfer fractional derivative

On the nonlinear [formula omitted]-Hilfer fractional differential equations

In this paper we deals with the existence of solution for a new kind of Langevin inclusion involving -Hilfer fractional derivative. The suggested study is based on some basic definitions of fractional calculus and multivalued analysis. The existence result is obtained by making use of the nonlinear alternative of Leray-Schauder type. In the end, we are giving an example to illustrate our results.

ABSTRACTThis paper investigates fractional differential equations and inclusions involving two distinct fractional derivatives, specifically the ‐Hilfer and ‐Hilfer fractional derivatives, within the framework of the Navier boundary value problem. For the single‐valued case, we establish the existence and uniqueness of solutions using classical iterative methods, including the Leray–Schauder, Banach, and Krasnoselskii fixed‐point theorems. In the multi‐valued case, we apply the Leray–Schauder nonlinear alternative when the right‐hand side (RHS) of the inclusion has convex values, while the Covitz–Nadler fixed‐point theorem is employed for non‐convex RHS mappings. To illustrate the practical applicability of our results, we present numerical examples that validate our theoretical findings.

Considerable attention has been recently given to the existence of solutions of initial or boundary value problems for fractional differential equations and inclusions with Hilfer fractional derivative. Motivated by these results, in this paper we will present existence, data dependence and Ulam stability results for some differential inclusions with Hilfer fractional derivative. The results follow as applications of the multi-valued weakly Picard operator theory. An example illustrates the main result of the paper.