Isoperimetric Variational Problems Research Articles

On the problem of optimal control of rotation axis reorientation of a spacecraft

Various variants of the formulation of optimal control problem of the reorientation of the axis of dynamic symmetry of the spacecraft (spacecraft), which is the axis of rotation, are considered. It is assumed that the solution to this problem should be found in the class of movements with one directional (flat) rotation, provided that before and after the reorientation of the axis of rotation, the angular velocity of the spacecraft is the same. In this case, the angular motion of the spacecraft is controlled according to the "rotary jet engine" scheme, when the control moment is limited by the ellipsoid of rotation. The corresponding mathematical model of spacecraft motion for the control problem under consideration is given. In addition to the formulation of the problem of the steepest reorientation of the axis of rotation of the spacecraft, the optimal control problem is also formulated, for which optimal control for the full control moment is found. In order to analyze the formulation of the problem under consideration, the results of its reduction to the boundary value problem and to the isoperimetric variational problem are also presented.

Numerical solution technique for solving isoperimetric variational problems

In this paper, we have a zeal for fulfilling the estimated scientific answers for the calculus of variations by using the Sumudu transform method (STM). The main target is to search the numerical arrangement of ordinary differential equations (ODEs) which emerge from the variational problems where first the fundamental condition for the arrangement of the issue is to fulfill the Euler–Lagrange condition and then solve the equations using STM. The valuable properties of the Sumudu change technique are used to downsize the calculation of the issue to a gathering of straight arithmetical conditions. We introduce four variational problems and discover the numerical solution of those problems using STM and plot the curves of those solutions. These models are picked such that there exist systematic answers for them to offer a reasonable diagram and show the effectiveness of the proposed strategy. Numerical outcomes are registered utilizing Maple programming.

Constants of motion for isoperimetric variational problems with time delay

ABSTRACT In the present work we obtain the constants of motion for isoperimetric variational problems with time delay. We consider a constrained optimization problem where the Lagrangian function defining the functional depends on time delayed arguments. We prove the isoperimetric Euler–Lagrange and DuBois–Reymond type optimality conditions and, in order to investigate the constants of motion for this problem, we obtain a nonsmooth extension of Noether's symmetry theorem for isoperimetric variational problems with delayed arguments.

A reliable numerical approach for analyzing fractional variational problems with subsidiary conditions

This paper obtains the necessary optimality conditions for a new class of isoperimetric fractional variational problems depending on indefinite integrals (IFVPDI). A new condition is added to the IFVPDI and a modified direct numerical approach based on the Epsilon–Ritz method and polynomial basis functions is applied. The optimization problem is reduced to the problem of optimizing a real value function. The convergence and error analysis of the proposed approach are also assessed. Illustrative examples are included demonstrating the validity and applicability of the method.

On two symmetries in the theory of $m$-Hessian operators

We show that the modern theory of fully nonlinear operators had been started by the skew symmetry of minors in cooperation with the symmetry of symmetric functions. The paper presents some consequences of this interaction for the m-Hessian operators. One of them is a setting of the isoperimetric variational problem for Hessian integral. The m-admissible minimizer is found, what brings out a new simple proof of the well known Poincare - type inequalities for Hessian integrals. Also a new set of inequalities, generated by a special finite set of functions, is found.

Numerical Scheme for a Quadratic Type Generalized Isoperimetric Constraint Variational Problems With A-Operator

This paper presents a numerical scheme for a class of isoperimetric constraint variational problems (ICVPs) defined in terms of an A-operator introduced recently. In this scheme, Bernstein's polynomials are used to approximate the desired function and to reduce the problem from a functional space to an eigenvalue problem in a finite dimensional space. Properties of the eigenvalues and eigenvectors of this problem are used to obtain approximate solutions to the problem. Results for two examples are presented to demonstrate the effectiveness of the proposed scheme. In special cases, the A-operator reduces to Riemann–Liouville, Caputo, Riesz–Riemann–Liouville, and Riesz–Caputo, and several other fractional derivatives defined in the literature. Thus, the approach presented here provides a general scheme for ICVPs defined using different types of fractional derivatives. Although, only Bernstein's polynomials are used here to approximate the solutions, many other approximation schemes are possible. Effectiveness of these approximation schemes will be presented in the future. While the presented numerical scheme is applied to a quadratic type generalized ICVPs, it can also be applied to other types of problems.

Toward the Landau–Lifshitz version of the Thomson electrostatics theorem

In the classical textbook (Landau and Lifshitz, 1963), Landau and Lifshtz suggested their version of the famous Thomson variational principle (a.k.a Thomson theorem.) So far, their version has not gained the interest it deserves, either among physicists or among applied mathematicians. Partially, the lack of interest can be explained because of the quality of the suggested proof of the principle. It is considerably lower than the standards accepted in classical electrostatics and mathematical physics. Even more importantly, Landau and Lifshitz did not demostrate the minimum property of the electrostatic energy at equilibrium. In this note, we, first, modify and specify the Landau–Lifshitz formulation of the principle presenting it as the isoperimetric variational problem. Then, for this isoperimetric problem we calculate the first and second variations, and we prove that the first variation vanishes, whereas the second variation appears to be positive.

Fractional Isoperimetric Noether's Theorem in the Riemann–Liouville Sense

We prove Noether-type theorems for fractional isoperimetric variational problems with Riemann–Liouville derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples, in the fractional context of the calculus of variations , are discussed.

Exact Solution of a Constraint Optimization Problem for the Thermoelectric Figure of Merit.

In the classical theory of thermoelectricity, the performance integrals for a fully self-compatible material depend on the dimensionless figure of merit . Usually these integrals are evaluated for constraints const. and const., respectively. In this paper we discuss the question from a mathematical point of view whether there is an optimal temperature characteristics of the figure of merit. We solve this isoperimetric variational problem for the best envelope of a family of curves .

Hölderian variational problems subject to integral constraints

We prove a necessary stationary condition for non-differentiable isoperimetric variational problems with scale derivatives defined on the class of Hölder continuous functions.

Robust Optimal Design of Beams Subject to Uncertain Loads

Optimality conditions are derived for the robust optimal design of beams subject to a combination of uncertain and deterministic transverse and boundary loads using a variational min‐max approach. The potential energy of the beam is maximized to compute the worst case loading and minimized to determine the optimal cross‐sectional shape which results in coupled nonlinear differential equations for the unknown functions except for the case of a variable width beam. The uncertain component of the transverse load acting on the beam is not known a priori resulting in load uncertainty subject only to an norm constraint. Similarly the optimal area function is subject to a volume constraint leading to an isoperimetric variational problem. The min‐max approach leads to robust optimal designs which are not susceptible to unexpected load variations as it occurs under operational conditions. The solution methodology is illustrated for the variable width beam by obtaining analytical results for several cases. The efficiency of the optimal designs is computed with respect to a uniform beam under worst case loading taking the maximum deflection as the quantity for comparison. It is observed that the optimal shapes are more than 70% efficient for the examples given in this study.

A unified view of ostensibly disparate isoperimetric variational problems

A class of isoperimetric variational calculus problems is solved. It is shown that the said class ties together seemingly dissimilar optimal control problems in economics and physics, to wit, the classical nonrenewable resource extracting model of the firm, the model of the patient worker, and the principle of maximum entropy. In particular, it is shown that a certain family of wealth maximizing nonrenewable resource extracting firms extract at the rate that maximizes the entropy of extraction, and that patient intertemporal utility maximizing individuals do not procrastinate and maximize the entropy of leisure.

On the first and second variations of a nonlocal isoperimetric problem

We consider a nonlocal perturbation of an isoperimetric variational problem. The problem may be viewed as a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation associated with competing short and long-range interactions. In particular, it arises as a Γ-limit of a model for microphase separation of diblock copolymers. In this article, we establish precise conditions for criticality and stability (i.e. we explicitly compute the first and second variations). We also present some applications.

Effective numerical—analytical solution of isoperimetric variational problems of mechanics by an accelerated convergence method

A new numerical-analytical method for solving non-linear variational problems of mechanics is presented. The method enables additional isoperimetric conditions and boundary conditions of different types to be taken into account. Unlike existing approaches, the method is based on the use of residuals of the unknown functions corresponding to abscissae at which they satisfy the required conditions. The method has a clear geometrical interpretation and provides more confident idea as to the convergence of the iteration algorithm which is quadratic in nature. The algorithm constructed is used to compute single-mode and multi-mode viscous incompressible flows in a plane convergent channel (Jeffrey-Hamel flow) for a broad range of governing parameters (the aperture angle and Reynolds number).

Discrete versions of continuous isoperimetric problems

Discrete isoperimetric variational problems which model single and double integral isoperimetric problems are formulated and some multiplier rules are derived. For quardratic functionals, the Euler-Lagrange equation is linear and can be analyzed bydeterminant methods. difference equations methods. or numerically bythealgebraic eigenvalue problem. A specific example is given wherethe eigenfuntions and eigenvalue of this discrete problem converge. as the step size goes to zero, to the eigenfunctionsand eigenvalues of the corresponding, continuous problem Recent developments in algebraic geometry offer the hopeof using Groebner (=Gröbner or Grobner) basis methods to numerically solve some systems of equationsgenerated by Lagrange multiplier methods. These methods mayapply to nonlinear systems of equations whenever they can be reduced to systems of polynomial equations. The Groebner basis algorithm of Buchberger is an extension of Gaussian elimination to polynomial systems.

Large strain analysis of rubber-like membranes under dead weight, gas pressure, and hydrostatic loading

The title problem is formulated variationally. Especially membranes being totally or partly under hydrostatic loading are considered and it is assumed that either the liquid volume or the liquid level is constant during the deformation. The most general case is characterized by an isoperimetric variational problem and free boundaries of the loaded (wetted) surface. Particularly we analyse membrane strips, membranes of revolution, and arbitrary spatial membranes consisting of piecewise plane ones. Numerous numerical results are reported and discussed with respect to the stability of equilibrium. Contact problems are addressed also.

On Poincaré's isoperimetric problem for simple closed geodesics

We show in the context of integral currents that Poincaré's isoperimetric variational problem for simple closed geodesics on ovaloids has a smooth extremal C without self-intersection. Provided the smooth Riemannian metric on the ovaloid M in question does not deviate too far from constant curvature, we further show that (i) this extremal C is connected and so is the desired non-trivial geodesic of shortest length on M and (ii) C is close (in the sense of Hausdorff distance) to a great circle.

CROSS-SECTIONAL SHAPE OF AN ANISOTROPIC, HOLLOW BAR WITH MAXIMUM TORSIONAL STIFFNESS∗

A rectilinearly anisotropic and hollow bar is optimized with respect to the shape of its transverse cross-section so as to maximize its torsional stiffness for a given cross-sectional area. Two cases in respect of the wall thickness of the bar are considered. in one case, both inner and outer boundaries of a bar of arbitrary wall thickness are taken as design variables. In the other, the inner boundary of a thin-walled bar is specified to be an ellipse with given semi-axes, and the outer boundary of the bar is optimally determined. The optimization problem is formulated as an isoperimetric variational problem using the relation between the energy of the bar and the objective functional. The necessary condition for optimality, which is derived using the method of Lagrange multipliers, provides an additional condition on a free boundary. The cross-sectional shapes and the efficiencies of the optimal bars are given in graphical form for various problem parameters.

Imperfection sensitivity and isoperimetric variational problems in stability theory

Imperfection sensitivity and isoperimetric variational problems in stability theory

Embedding Theorems and Quasi-Linear Elliptic Boundary Value Problems for Unbounded Domains

The Sobolev-Kondrachov embedding and compactness theorems are extended to cover general unbounded domains, by introducing appropriate weighted ${L_p}$ norms. These results are then applied to the Dirichlet problem for quasi-linear elliptic partial differential equations and isoperimetric variational problems defined on general unbounded domains in ${{\mathbf {R}}^N}$.