Problems In Calculus Of Variations Research Articles

Necessary conditions for the extremum in non-smooth problems of variational calculus with delay

In this article, with the help of a special variation of the Weierstrass type, the necessary condition for a minimum is obtained. As consequences, the analogue of the Weierstrass condition and its local modification, as well as the analogue of the Legendre condition are obtained.

Differential Transform Method and Neural Network for Solving Variational Calculus Problems

Differential Transform Method and Neural Network for Solving Variational Calculus Problems

On parametric representation of the Newton's aerodynamic problem

Newton's problem of finding the surface shape of a rotation body based on the condition of minimal resistance of the body when it moves in a rarefied medium is discussed. The problem is formulated in the form of a classical isoperimetric problem in calculus of variations. The exact solution is given in the class of piecewise differentiable functions. The numerical results of specific calculations of the functional for cone and hemisphere are presented. We prove that the optimization effect is significant by comparison of the results for cone and hemisphere with the value of the optimized functional for the optimal contour.

A Poincaré type inequality with three constraints

In this paper, we consider a problem in calculus of variations motivated by a quantitative isoperimetric inequality in the plane. More precisely, the aim of this article is the computation of the minimum of the variational problem $$\begin{aligned}\inf _{u\in {\mathcal {W}}} \frac{\displaystyle \int _{-\pi }^{\pi }[(u')^2-u^2]d\theta }{\displaystyle \left[ \int _{-\pi }^{\pi } |u| d\theta \right] ^2} \end{aligned}$$ where a function $$u\in {\mathcal {W}}$$ is a $$H^1(-\pi ,\pi )$$ periodic function, with zero average on $$(-\pi ,\pi )$$ and orthogonal to sine and cosine.

APPLICATION OF THE ”WAVE“ METHOD TO SOLVE THE PROBLEMS OF OPTIMIZATION OF PASSENGER AND CARGO TRANSPORTATION IN MEGACITIES TO IMPROVE THEIR TRANSPORT NETWORK AND INFRASTRUCTURE. Conceptual basis, mathematical framework

The paper proposes a new approach to solving optimization problems arising in engineering and transport logistics in designing and construction of roads (in particular, in megacities, near large transport hubs, near state borders) for cargo and passenger transportation and implementation of international trade. The fundamental problems of modern engineering logistics - the problem of optimal location (transport hubs) and the problem of identification and segmentation of logistics, transport and logistics zones are considered. These problems are solved using methods of variational calculus, in particular, the so-called "wave" method based on the Fermat principle existing in physical optics, which is based on the analogy between finding the global extremum of the integral functional and the propagation of light in an optically heterogeneous medium. A numerical method for the above technique has been developed programatically.
 The idea of the "wave method/approach belongs to V.V. Bashurov, who proposed to use the methods of geometrical and physical optics to investigate applied safety problems and some related issues. The essence of the "wave method" is that initially the safety problem is reduced to the search for the global minimum of a nonlinear functional. In turn, the minimization problem is solved by constructing the trajectory of motion of the front of the "light wave" moving in an optically inhomogeneous medium.
 Finding the minimum of a functional is a classical problem of variational calculus, for the solution of which a significant mathematical apparatus has been developed. However, most of known methods effectively determine only local extrema. "Wave" method allows to solve the problem of finding a global extremum with greater efficiency.
 This paper proposes a conceptual framework and scientifically justified modification of this "wave" method for solving optimization problems arising in engineering and transport logistics, including the problem of optimal location of the transport hub, transport and logistics center (warehouse) and the problem of optimal identification and segmentation of logistical zones (metropolitan areas, large transport hubs).

Necessary conditions for the extremum in non-smooth problems of variational calculus

In the paper, we proposed an approach for studying strong and weak extremums in non-smooth vector problems of calculus of variation, namely, in classic variational problems with fixed ends and with a free right end, and also in a variational problem with higher derivatives. The essence of the proposed approach is to introduce a Weierstrass type variation characterized by a numerical parameter. Necessary conditions for minimum containing as corollaries the Weierstrass condition, its local modification and also the Legendre and transversality conditions are obtained. In the case when the Legendre condition degenerates, equality and inequality type necessary conditions are obtained for the weak local minimum. The examples showing the content-richness of the obtained main results are given.

Direct Restarted Pell Algorithm for Solving Calculus of Variation Problems

The Pell Polynomial series is employed to solve problems in calculus of variations. The solution can be considered as linear combinations of Pell polynomials with unknown coefficients. Then a calculus of variational problem transforms into an optimization problem and will be solved using mathematical programming techniques. The proposed method converges rapidly to the exact solution and gives accurate results using small number of Pell polynomials basis polynomials. Some illustrative examples are included and the approximate results obtained by restarted Pell series are compared with the exact solutions.

Information Projection on Banach Spaces with Applications to State Independent KL-Weighted Optimal Control

This paper studies constrained information projections on Banach spaces with respect to a Gaussian reference measure. Specifically our interest lies in characterizing projections of the reference measure, with respect to the KL-divergence, onto sets of measures corresponding to changes in the mean (or shift measures). As our main result, we give a portmanteau theorem that characterizes the relationship among several different formulations of this problem. In the general setting of Gaussian measures on a Banach space, we show that this information projection problem is equivalent to minimization of a certain Onsager–Machlup (OM) function with respect to an associated stochastic process. We then construct several reformulations in the more specific setting of classical Wiener space. First, we show that KL-weighted optimization over shift measures can also be expressed in terms of an OM function for an associated stochastic process that we are able to characterize. Next, we show how to encode the feasible set of shift measures through an explicit functional constraint by constructing an appropriate penalty function. Finally, we express our information projection problem as a calculus of variations problem, which suggests a solution procedure via the Euler–Lagrange equation. We work out the details of these reformulations for several specific examples.

New Exact Operational Shifted Pell Matrices and Their Application in Astrophysics

In this work, the exact operational matrices for shifted Pell polynomials are achievable; so one can integrate and product the vector of basic functions s. The general form of the matrix of integration P is established, the dual matrix of integration Q is derived with general formulation, and the general form of the matrix derived from the product of two shifted Pell polynomials has been given. This idea is implemented on shifted Pell basis vector. Using such exact matrices, then the resident function of the equation is reached which can be written as R.P(x), where R is an algebraic equation vector and P(x) is the shifted Pell basis vector. The presented matrices can be utilized to find the approximate solution of differential equations, integral equation and the calculus of variations problems. An investigation for the convergence and error analysis of the proposed shifted Pell expansion is performed. Numerical treatment for problems in physics are included in this work to demonstrate the accuracy, easy to implement as well as accurate and satisfactory results with a small number of shifted Pell basis. Using operational matrices and the spectral technique are used together for solving Lane-Emden equation.

Admissible Extremals in the Problems of Variational Calculus with Constraints in the Form of a System of Linear Inhomogeneous Differential Equations of the First Order with Rectangular Matrices

We establish necessary and sufficient conditions for the existence of smooth admissible extremals in the problems of variational calculus with constraints in the form of a system of linear inhomogeneous differential equations of the first order with rectangular matrices and different types of boundary conditions.

Transformation preserving controllability for nonlinear optimal control problems with joint boundary conditions

In this paper we develop a new approach for optimal control problems with general jointly varying state endpoints (also called coupled endpoints). We present a new transformation of a nonlinear optimal control problem with jointly varying state endpoints and pointwise equality control constraints into an equivalent optimal control problem of the same type but with separately varying state endpoints in double dimension. Our new transformation preserves among other properties the controllability (normality) of the considered optimal control problems. At the same time it is well suited even for the calculus of variations problems with joint state endpoints, as well as for optimal control problems with free initial and/or final time. This work is motivated by the results on the second order Sturm–Liouville eigenvalue problems with joint endpoints by Dwyer and Zettl (1994) and by the sensitivity result for nonlinear optimal control problems with separated state endpoints by the authors (2018).

Cosmic Analogues of Classic Variational Problems

Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.

Gamkrelidze Convexification and Bogolyubov’s Theorem

The interconnection between an optimal control problem and its convexification in the sense of Gamkrelidze and Bogolyubov is studied. Bogolyubov’s classical result for the simplest problem of variational calculus is obtained as a corollary.

Generalized conformable variational calculus and optimal control problems with variable terminal conditions

This paper provides generalized transversality conditions for the problems of variational calculus and optimal control, constructed by the conformable derivative. The generalized term is used to emphasize the problems with performance indexes containing the conformable derivative and defined by the classical integral and to distinguish them from the problems with performance indexes defined by the conformable integral. Special cases of the generalized transversality conditions both for variational calculus and optimal control are exhibited and supported by illustrative examples.

Sufficiency for singular trajectories in the calculus of variations

For calculus of variations problems of Bolza with variable end-points, nonlinear inequality and equality isoperimetric constraints and nonlinear inequality and equality mixed pointwise constraints, sufficient conditions for strong minima are derived. The main novelty of the new sufficiency results presented in this article concerns their applicability to cases in which the derivatives of the extremals to be optimal solutions are not necessarily continuous nor piecewise continuous but only essentially bounded and they do not necessarily satisfy the standard strengthened Legendre condition but only the corresponding necessary condition.

Bernoulli Collocation Polynomials Algorithm for Calculus of Variational Problems

<p>This paper presents an approximate method that depends on the Bernoulli Polynomials as basic functions. The method is concerned with collocation technique for solving problems in calculus of variation. Some interesting properties of Bernoulli polynomials are used to reduce the original problem to mathematical problem. Some illustrative examples are described to show the applicability of the proposed method.</p>

A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems

In this paper, we present a new method based on the Clenshaw-Curtis formula to solve a class of fractional optimal control problems. First, we convert the fractional optimal control problem to an equivalent problem in the fractional calculus of variations. Then, by utilizing the Clenshaw-Curtis formula and the Chebyshev-Gauss-Lobatto points, we transform the problem to a discrete form. By this approach, we get a nonlinear programming problem by solving of which we can approximate the optimal solution of the main problem. We analyze the convergence of the obtained approximate solution and solve some numerical examples to show the efficiency of the method.

A generalized multidirectional mean value inequality and dynamic optimization

We consider a general calculus of variations problem with lower semicontinuous data which includes an optimal control problem with a dynamic differential inclusion constraint as a particular case. We provide a new approach to the derivation of necessary optimality conditions for such a problem, based on a new variant of the multidirectional mean-value inequality for smooth Banach spaces. The multidirectional mean-value inequality is a result in nonlinear and nonsmooth analysis which provides estimates for the minimum difference in function values between a closed convex subset of a Banach space E and a point .

The Bernoulli wavelets operational matrix of integration and its applications for the solution of linear and nonlinear problems in calculus of variations

In this paper, a new method for solving some classes of linear and nonlinear problems in calculus of variations is presented. The method is based upon Bernoulli wavelets. The operational matrices of integration and product of Bernoulli wavelets are calculated. These matrices are then utilized to reduce the calculus of variations problems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Некоторые модели оптимальных траекторий сближения космического аппарата с астероидом

We consider the optimal trajectories in the rendezvous problem calculated by calculus of variations techniques. We formulate the problem of translational motion of the spacecraft taking the asteroid rotation into consideration. The motion is supposed to occur outside of a sphere of influence of any planet, the gravity of the asteroid is defined like the one in the restricted two-body problem. We represent the equations of controlled motion in central gravity field and construct the calculus of variations problem, given the control constraints. The problem is solved applying Valentine’s method. We solve the Euler–Lagrange equations generated from the functional related to the fuel consumption and show the solutions in the special case of neglecting the asteroid gravity, both taking and not taking the Coriolis force into consideration. The optimal control in these solutions is occurred to be twofold: continuous (either smooth and piecewise smooth one) and piecewise continuous one, having one or more points of jump discontinuities. In case of these discontinuities the Weierstrass–Erdmann condition is applied. The optimality is validated by the Legendre condition. We note the way the control function and the trajectory change while varying some of the parameters of the problem: the initial speed, the flight duration, the control limit, the angular velocity of the asteroid. The descripted method allows one to evaluate some parameters of the flight which may be used in the rendezvous mission planning.