arXiv subjects

Elimhan N. Mahmudov

Publications and source records attributed to Elimhan N. Mahmudov.

9 recordsLinked to original sources

arXiv2025

Isoperimetric Problem and Weierstrass Necessary Condition for Fractional Calculus of Variations

Summary]{In this paper, we study problems of minimization of a functional depending on the fractional Caputo derivative of order $0<α\leq 1$ and the fractional Riemann- Liouville integral of order $β> 0$ at fixed endpoints. A fractional analogue of the Du Bois-Reymond lemma is proved, and the Euler-Lagrange conditions are proved for the simplest problem of fractional variational calculus with fixed ends and for the fractional isoperimetric problem. An approach is proposed to obtain the necessary first-order conditions for the strong and weak extrema, and the necessary optimality conditions are obtained. From these necessary conditions, as a consequence, we obtain the Weierstrass condition and its local modification. It should be noted that some papers in the literature claim that the standard proof of the Legendre condition in the classical case $α=1$ cannot be adapted to the fractional case $0< α<1$ with final constraints. Despite this, we prove the Legendre conditions by the standard classical method via the Weierstrass condition. In addition, the necessary Weierstrass-Erdmann conditions at the corner points are obtained. Examples are provided to illustrate the significance of the main results obtained.

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arXiv2025

The Weierstrass necessary condition for fractional calculus of variations

In this paper, we study problems of minimization of a functional depending on the fractional Caputo derivative of order $0<α\leq 1$ and the fractional Riemann- Liouville integral of order $β> 0$ at fixed endpoints. A fractional analogue of the Du Bois-Reymond lemma is proved, and the Euler-Lagrange conditions are proved for the simplest problem of fractional variational calculus with fixed ends and for the fractional isoperimetric problem. An approach is proposed to obtain the necessary first-order conditions for the strong and weak extrema, and the necessary optimality conditions are obtained. From these necessary conditions, as a consequence, we obtain the Weierstrass condition and its local modification. It should be noted that some papers in the literature claim that the standard proof of the Legendre condition in the classical case $α=1$ cannot be adapted to the fractional case $0< α<1$ with final constraints. Despite this, we prove the Legendre conditions by the standard classical method via the Weierstrass condition. In addition, the necessary Weierstrass-Erdmann conditions at the corner points are obtained. Examples are provided to illustrate the significance of the main results obtained.

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arXiv2025

The Euler-Lagrange and Legendre Necessary Conditions for Fractional Calculus of Variations

In this paper, we study the problems of minimizing a functional depending on the Caputo fractional derivative of order $0< α\leq 1$ and the Riemann- Liouville fractional integral of order $β>0$ under certain constraints. A fractional analogue of the Du Bois-Reymond lemma is proved. Using this lemma for various weak local minimum problems, the Euler-Lagrange equation is derived in integral form. Some serious works in the literature claim that the standard proof of the Legendre condition in the classical case $α=1$ cannot be adapted to the fractional case $0<α<1$ with final constraints. In spite of this, we prove the Legendre conditions using the standard classical method. The obtained necessary conditions are illustrated by appropriate examples.

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arXiv2021

Optimization of the first mixed boundary value problem for parabolic differential inclusions in a three spatial dimension

The paper is devoted to the optimization of a first mixed boundary value problem for parabolic differential inclusions (DFIs) with Laplace operator. For this, a problem with a parabolic discrete inclusion is defined, which is the main auxiliary problem. With the help of locally adjoint mappings, necessary and sufficient conditions for the optimality of parabolic discrete inclusions are proved. Then, using the method of discretization of parabolic DFIs and the already obtained optimality conditions for discrete inclusions, the necessary and sufficient conditions for the discrete-approximate problem are formulated in the form of the Euler-Lagrange type inclusion. Thus, using specially proved equivalence theorems, without which it would hardly be possible to obtain the desired result for the problem posed, we establish sufficient optimality conditions for a parabolic DFIs. To demonstrate the above approach, some linear problems and polyhedral optimization with inclusions of parabolic type are investigated.

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arXiv2020

Optimization of System of Nonlinear Second Order Differential Inequalities

This paper deals with the optimization of Bolza problem with a system of convex and nonconvex, discrete and differential state variable inequality constraints of second order by deriving necessary and sufficient conditions for optimality. According to proposed discretization method and equivalence theorems for subdifferential inclusions, the problem with a system of discrete-approximation inequalities are investigated which highly contributes to the derivation of adjoint discrete inclusions generated by given system of nonlinear inequality constraints. Moreover, in the limit case, we obtain sufficient conditions for optimality of the continuous problem. A numerical example is presented to illustrate the theoretical result.

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arXiv2020

Optimal control of differential inclusions with endpoint and state constraints and duality

The paper studies optimal control problem described by higher order evolution differential inclusions (DFIs) with endpoint and state constraints. In the term of Euler-Lagrange type inclusion is derived sufficient condition of optimality for higher order DFIs. It is shown that the adjoint inclusion for the first order DFIs, defined in terms of locally adjoint mapping, coincides with the classical Euler-Lagrange inclusion. Then a duality theorem is proved, which shows that Euler-Lagrange inclusions are "duality relations" for both problems. At the end of the paper duality problems for third order linear and fourth order polyhedral DFIs are considered.

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arXiv2019

On the Duality Theory for Problems with Higher Order Differential Inclusions

This paper on the whole concerns with the duality of Mayer problem for k-th order differential inclusions, where k is an arbitrary natural number. Thus, this work for constructing the dual problems to differential inclusions of any order can make a great contribution to the modern development of optimal control theory. To this end in the form of Euler-Lagrange type inclusions and transversality conditions the sufficient optimality conditions are derived. The principal idea of obtaining optimal conditions is locally adjoint mappings. It appears that the Euler-Lagrange type inclusions for both primary and dual problems are "duality relations". To demonstrate this approach, some semilinear problems with k-th order differential inclusions are considered. Also, the optimality conditions and the duality theorem in problems with second order polyhedral differential inclusions are proved. These problems show that maximization in the dual problems are realized over the set of solutions of the Euler-Lagrange type differential inclusions/equations.

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arXiv2019

Infimal Convolution and Duality in Convex Optimal Control Problems with Second Order Evolution Differential Inclusions

The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying infimal convolution concept of convex functions, step by step we construct the dual problems for discrete, discrete-approximate and differential inclusions and prove duality results. It seems that the Euler-Lagrange type inclusions are "duality relations" for both primary and dual problems and that the dual problem for discrete-approximate problem make a bridge between them. Finally, relying to the method described within the framework of the idea of this paper a dual problem can be obtained for any higher order differential inclusions. In this way relying to the described method for computation of the conjugate and support functions of discrete-approximate problems a Pascal triangle with binomial coefficients, can be successfully used for any "higher order" calculations.

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arXiv2019

Observability and Controllability of Second Order Linear Time Invariant Systems and Kalman Type Conditions

In the present paper we consider controllability and observability of second order linear time invariant systems in matrix form. Without reducing into first order systems we show how the classical conditions for first order linear systems can be generalized to this case. In term of Kalman type criterions these concepts are investigated for second order discrete and continuous time linear systems. It should be pointed out that by repeated differentiation of state and output vector-functions we derive two different systems of linear algebraic equations. Then the initial values x_0, x_1 and input functions can be determined uniquely from these systems if and only if the observability and controllability matrices have full rank, respectively. Also the transfer function of the second order continuous-time linear state-space system is constructed. A numerical example is given to illustrate the feasibility and effectiveness of the theoretic results obtained.

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