arXiv subjects

Nazim I. Mahmudov

Publications and source records attributed to Nazim I. Mahmudov.

At least 37 records · Page 2Linked to original sources

arXiv2024

Necessary first and second order optimality conditions for a fractional order differential equation with state delay

In this research paper, we examine an optimal control problem involving a dynamical system governed by a nonlinear Caputo fractional time-delay state equation. The primary objective of this study is to obtain the necessary conditions for optimality, both the first and second order, for the Caputo fractional time-delay optimal control problem. We derive the first-order necessary condition for optimality for the given fractional time-delay optimal control problem. Moreover, we focus on a case where the Pontryagin maximum principle degenerates, meaning that it is satisfied in a tivial manner. Consequently, we proceed to derive the second order optimality conditions specific to the problem under investigation. At the end illustrative examples are provided.

math.OC↗

arXiv2024

Discrete delayed perturbation of Mittag-Leffler function and its application to linear fractional delayed difference system

The linear nonhomogeneous fractional difference system with constant coefficients is introduced. An explicit solution to the system is acquired by proposing a newly discrete retarded perturbation of the nabla Mittag-Leffer-type function containing such matrix equations that provide non-permutability. A couple of special cases obtained from our results are discussed.

math.DS↗

arXiv2023

Pontryagin maximum principle for fractional delay differential equations and controlled weakly singular Volterra delay integral equations

In this article, we explore two distinct issues. Initially, we examine the utilization of the Pontriagin maximum principle in relation to fractional delay differential equations. Additionally, we discuss the optimal approach for solving the control problem for equation (1.1) and its associated payoff function (1.2). Following that, we investigate the application of the Pontryagin Maximum principle in the context of Volterra delay integral equations (1.3). We strengthen the results of our study by providing illustrative examples at the end of the article.

math.OC↗

arXiv2022

Picard approximation of a singular backward stochastic nonlinear Volterra integral equation

Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. In this paper we prove that Picard iterations of BSDEs with globally Lipschitz continuous nonlinearities converge exponentially fast to the solution. Our main result in this paper is to establish a fundamental lemma to prove the global existence and uniqueness of an adapted solution to a singular backward stochastic nonlinear Volterra integral equation (for short singular BSVIE) of order $α\in (\frac{1}{2},1)$ under a weaker condition than Lipschitz one in a Hilbert space.

math.PR↗

arXiv2022

Some Results on Backward Stochastic Differential Equations of Fractional Order

In this article, we deal with fractional stochastic differential equations, so-called Caputo type fractional backward stochastic differential equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to Caputo fBSDEs of order $α\in (\frac{1}{2},1)$ whose coefficients satisfy a Lipschitz condition. A novelty of the article is that we introduce a new weighted norm in the square integrable measurable function space that is useful for proving a fundamental lemma and its well-posedness. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and the mild solution.

math.PR↗

arXiv2022

Maximum principle for discrete time mean-field stochastic optimal control problems

In this paper, we study the optimal control of a discrete-time stochastic differential equation (SDE) of mean-field type, where the coefficients can depend on both a function of the law and the state of the process. We establish a new version of the maximum principle for discrete-time stochastic optimal control problems. Moreover, the cost functional is also of the mean-field type. This maximum principle differs from the classical principle since we introduce new discrete-time backward (matrix) stochastic equations. Based on the discrete-time backward stochastic equations where the adjoint equations turn out to be discrete backward SDEs with mean field, we obtain necessary first-order and sufficient optimality conditions for the stochastic discrete optimal control problem. To verify, we apply the result to production and consumption choice optimization problem.

math.OC↗

arXiv2022

Mean-field type discrete stochastic linear quadratic optimal control problems

In this paper, we consider linear quadratic optimal control with mean-field type for discrete-time stochastic systems with state and control dependent noise. An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional. The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic.

math.OC↗

arXiv2022

Perturbation theory and linear partial differential equations with delay

Functional evolution equations are used in the modeling of numerous physical processes. In this work, our main tool is perturbation theory of strongly continuous semigroups. The advantage of this technique is that one can provide functional evolution equations with the explicit representation formulas of the solution. First, we introduce a closed form of the fundamental solution of the evolution equation with a discrete delay using the delayed Dyson-Phillips series. Then we set up the analytical representation formulas of the classical solutions of linear homogeneous/non-homogeneous evolution equations with a constant delay in a Banach space. In the special case, when a strongly continuous group $\left\lbrace \mathcal{T}(t)\right\rbrace_{t\in \mathbb{R}}$ commutes with a bounded linear operator $A_{1}$, we obtain an elegant formula for the fundamental solution using the powers of the resolvent operator of $A_{0}$. Furthermore, we consider delay evolution equations with permutable/non-permutable linear bounded operators and derive crucial results in terms of non-commutative analysis. Finally, we present an example, in the context of a one-dimensional heat equation with a discrete delay to demonstrate the applicability of our theoretical results and give some comparisons with existing results.

math.FA↗

arXiv2022

The $μ$-neutral fractional multi-delayed differential equations

The $μ$-neutral linear fractional multi-delayed differential nonhomogeneous system with noncommutative coefficient matrices is introduced. The novel $μ$-neutral multi-delayed perturbation of Mittag-Leffler type matrix function is proposed. Based on this, an explicit solution to the system is investigated step by step. The existence uniqueness of solutions to $μ$-neutral nonlinear fractional multi-delayed differential system is obtained with regard to the supremum norm. The notion of stability analysis in the sense of solutions to the described system is discussed on the grounds of the fixed point approach.

math.DS↗

arXiv2021

Asymptotic behavior of solutions to fractional stochastic multi-term differential equation systems involving non-permutable matrices

In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs for short). Our goal in this paper is to establish results on the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order $α\in (\frac{1}{2},1)$ and $β\in (0,1)$ whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on $λ$. Also, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general.

math.PR↗

arXiv2021

Perturbation of fractional strongly continuous cosine family operators

Perturbation theory has long been a very useful tool in the hands of mathematicians and physicists. The purpose of this paper is to prove some perturbation results for infinitesimal generators of fractional strongly continuous cosine families. That is, we impose sufficient conditions such that $\mathscr{A}$ is the infinitesimal generator of a fractional strongly continuous cosine family in a Banach space $\mathcal{X}$, and $\mathscr{B}$ is a bounded linear operator in $\mathcal{X}$, then $\mathscr{A}+\mathscr{B}$ is also the infinitesimal generator of a fractional strongly continuous cosine family in $\mathcal{X}$. Our results coincide with the classical ones when $α=2$.

math.AP↗

arXiv2021

Perturbation theory for fractional evolution equations in a Banach space

A strong inspiration for studying perturbation theory for fractional evolution equations comes from the fact that they have proven to be useful tools in modeling many physical processes. In this paper, we study fractional evolution equations of order $α\in (1,2]$ associated with the infinitesimal generator of an operator fractional cosine function generated by bounded time-dependent perturbations in a Banach space. We show that the abstract fractional Cauchy problem associated with the infinitesimal generator $A$ of a strongly continuous fractional cosine function remains uniformly well-posed under bounded time-dependent perturbation of $A$. We also provide some necessary special cases.

math.AP↗

arXiv2021

A new technique for solving Sobolev type fractional multi-order evolution equations

A strong inspiration for studying Sobolev type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev type fractional evolution equations with multi-orders in a Banach space. We propose a new Mittag-Leffler type function which is generated by linear bounded operators and investigate their properties which are productive for checking the candidate solutions for multi-term fractional differential equations. Furthermore, we propose an exact analytical representation of solutions for multi-dimensional fractional-order dynamical systems with nonpermutable and permutable matrices.

math.AP↗

arXiv2020

Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications

In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type differentiation operators with general fractional-order of $n-1 <α\leq n$, $n \in \mathbb{N}$ . A rule of fractional differentiation under integral sign with general order is necessary and applicable tool for verification by substitution for candidate solutions of inhomogeneous multi-term fractional differential equations. We derive explicit analytical solutions of generalized Bagley-Torvik equations in terms of recently defined bivariate Mittag-Leffler type functions that based on fractional Green's function method and verified solutions by substitution in accordance by applying the fractional Leibniz integral rule. Furthermore, we study an oscillator equation as a special case of differential equations with multi-orders via the Leibniz integral rule.

math.CA↗

arXiv2020

A natural extension of Mittag-Leffler function associated with a triple infinite series

We establish a new natural extension of Mittag-Leffler function with three variables which is so called "trivariate Mittag-Leffler function". The trivariate Mittag-Leffler function can be expressed via complex integral representation by putting to use of the eminent Hankel's integral. We also investigate Laplace integral relation and convolution result for a univariate version of this function. Moreover, we present fractional derivative of trivariate Mittag-Leffler function in Caputo type and we also discuss Riemann- Liouville type fractional integral and derivative of this function. The link of trivariate Mittag-Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics. Thus, we provide an exact analytic solutions of homogeneous and inhomogeneous multi-term fractional differential equations by means of a newly defined trivariate Mittag-Leffler functions.

math.CA↗

arXiv2020

Controllability of fractional stochastic delay dynamical systems

In this paper, we consider Caputo type fractional stochastic time-delay system with permutable matrices. We derive stochastic analogue of variation of constants formula via a newly defined delayed Mittag-Leffer type matrix function. Thus, we investigate new results on existence and uniqueness of mild solutions with the help of weighted maximum norm to fractional stochastic time-delay differential equations whose coefficients satisfy standard Lipschitz conditions. The main points in the proof are to apply Ito's isometry and martingale representation theorem, and to show the notion of a coincidence between the integral equation and the mild solution. Finally, we study complete controllability results for linear and nonlinear fractional stochastic delay dynamical systems with Wiener noise.

math.DS↗

arXiv2020

Hadamard type fractional time-delay semilinear differential equations: Delayed Mittag-Leffler function approach

We propose a delayed Mittag-Leffler type matrix function with logarithm, which is an extension of the classical Mittag-Leffler type matrix function with logarithm and delayed Mittag-Leffler type matrix function. With the help of the delayed Mittag-Leffler type matrix function with logarithm, we give an explicit form a of solutions to nonhomogeneous Hadamard type fractional time-delay linear differential equations. Moreover, we study existence uniqueness and stability in Ulam-Hyers sense of the Hadamard type fractional time-delay nonlinear equations.

math.DS↗

arXiv2015

On The Class Of $2D$ $q$-Appell Polynomials

In this research, as the new results of our previously proposed definition for the new class of $2D$ $q$-Appell polynomials, we derive some interesting relations including the recurrence relation and partial $q$-difference equation of the aforementioned family of $q$-polynomials. Next, as some famous examples of this new defined class of $q$-polynomials, we obtain the corresponding relations to the $2D$ $q$-Bernoulli polynomials, $2D$ $q$-Euler polynomials as well as $2D$ $q$-Genocchi polynomials.

math.NT↗