Problems In Calculus Of Variations Research Articles

Local generalization of transversality conditions for optimal control problem

In this paper, we introduce the transversality conditions of optimal control problems formulated with the conformable derivative. Since the optimal control theory is based on variational calculus, the transversality conditions for variational calculus problems are first investigated and then supported by some illustrative examples. Utilizing from these formulations, the transversality conditions for optimal control problems are attained by using the Hamiltonian formalism and Lagrange multiplier technique. To illustrate the obtained results, the dynamical system on which optimal control problem constructed is taken as a diffusion process modeled in terms of the conformable derivative. The optimal control law is achieved by analytically solving the time dependent conformable differential equations occurring from the eigenfunction expansions of the state and the control functions. All figures are plotted using MATLAB.

On a degenerate problem in the calculus of variations

We establish the uniqueness of the solutions for a degenerate scalar problem in the multiple integrals calculus of variations. The proof requires as a preliminary step the study of the regularity properties of the solutions and of their level sets. We exploit the uniqueness and the regularity results to explore some of their qualitative properties. In particular, we emphasize the link between the supports of the solutions and the Cheeger problem.

Optimal Fund Menus

We study the optimal design of a menu of funds by a manager who is required to use linear pricing and does not observe the preferences of investors regarding one of the risky assets. The optimal menu involves bundling of assets and can be explicitly constructed from the solution to a calculus of variations problem that optimizes over the indirect utility that each type of investor receives. We provide a complete characterization of the optimal fund menu and show that the need to maintain incentive compatibility leads the manager to behave as a closet indexer by offering funds that are inefficiently tilted towards the asset which is not subject to the information friction.

Trigonometric Tension B-Spline Method for the Solution of Problems in Calculus of Variations

In this paper, the tension B-spline collocation method is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. The problems are reduced to an explicit system of algebraic equations by this approximation. We apply some numerical examples to illustrate the accuracy and implementation of the method.

A collocation method for solving the fractional calculus of variation problems

In this paper we use a family of Muntz polynomials and a computational technique based on the collocation method to solve the calculus variation problem. This approach is utilizedto reduce the solution of linear and nonlinear fractional order dierential equations to the solution of a system of algebraic equations. Thus we can obtain a good approximation evenby using a smaller of collocation points.

On Self Codes: a Case Study within Mathematics and Performance Art

Unicity and existence of bounded solutions are interesting results not just for PDE (Partial Differential Equations) but can also be extended to minimization problems in Calculus of Variations. These results also have a very interesting behavior. The reasoning and research involved in these theoretical results were part of an artistic process in performance art: the performance piece “On Self Codes”. This paper is devoted to the development of possible connections between mathematics research in Calculus of Variations and performance art through a case study. How can one construct a performance art piece inspired on specific research within mathematics, without translation and using intersubjectivities instead? How can we relate two different fields without using them hierarchically? Can we agree on affirming that this type of work is a statement regarding transdisciplinarity, or it maps the ways in which a contemporary body of work can be shaped today? Are these two features incompatible? These are some of the questions that originated this paper.

Regularity of the Eikonal equation with two vanishing entropies

Let Ω⊂R2 be a bounded simply-connected domain. The Eikonal equation |∇u|=1 for a function u:Ω⊂R2→R has very little regularity, examples with singularities of the gradient existing on a set of positive H1 measure are trivial to construct. With the mild additional condition of two vanishing entropies we show ∇u is locally Lipschitz outside a locally finite set. Our condition is motivated by a well known problem in Calculus of Variations known as the Aviles–Giga problem. The two entropies we consider were introduced by Jin, Kohn [26], Ambrosio, DeLellis, Mantegazza [2] to study the Γ-limit of the Aviles–Giga functional. Formally if u satisfies the Eikonal equation and if(1)∇⋅(Σ˜e1e2(∇u⊥))=0 and ∇⋅(Σ˜ϵ1ϵ2(∇u⊥))=0 distributionally in Ω, where Σ˜e1e2 and Σ˜ϵ1ϵ2 are the entropies introduced by Jin, Kohn [26], and Ambrosio, DeLellis, Mantegazza [2], then ∇u is locally Lipschitz continuous outside a locally finite set.Condition (1) is motivated by the zero energy states of the Aviles–Giga functional. The zero energy states of the Aviles–Giga functional have been characterized by Jabin, Otto, Perthame [25]. Among other results they showed that if limn→∞⁡Iϵn(un)=0 for some sequence un∈W02,2(Ω) and u=limn→∞⁡un then ∇u is Lipschitz continuous outside a finite set. This is essentially a corollary to their theorem that if u is a solution to the Eikonal equation |∇u|=1 a.e. and if for every “entropy” Φ (in the sense of [18], Definition 1) function u satisfies ∇⋅[Φ(∇u⊥)]=0 distributionally in Ω then ∇u is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result in that we require only two entropies to vanish.The method of proof is to transform any solution of the Eikonal equation satisfying (1) into a differential inclusion DF∈K where K⊂M2×2 is a connected compact set of matrices without Rank-1 connections. Equivalently this differential inclusion can be written as a constrained non-linear Beltrami equation. The set K is also non-elliptic in the sense of Sverak [32]. By use of this transformation and by utilizing ideas from the work on regularity of solutions of the Eikonal equation in fractional Sobolev space by Ignat [23], DeLellis, Ignat [15] as well as methods of Sverak [32], regularity is established.

Regularity for second order stationary mean-field games

In this paper, we prove the existence of classical solutions for second order stationary mean-field game systems. These arise in ergodic (mean-field) optimal control, convex degenerate problems in calculus of variations, and in the study of long-time behavior of time-dependent mean-field games. Our argument is based on the interplay between the regularity of solutions of the Hamilton-Jacobi equation in terms of the solutions of the Fokker-Planck equation and vice-versa. Because we consider different classes of couplings, distinct techniques are used to obtain a priori estimates for the density. In the case of polynomial couplings, we recur to an iterative method. An integral method builds upon the properties of the logarithmic function in the setting of logarithmic nonlinearities. This work extends substantially previous results by allowing for more general classes of Hamiltonians and mean-field assumptions.

Classes of high-order numerical methods for solution of certain problem in calculus of variations

In this work, we extend the definition of nonic polynomial spline to non-polynomial spline function which depends on arbitrary parameter k. We derived and discussed the formulation and spline relat...

Weak KAM theory for a weakly coupled system of Hamilton–Jacobi equations

Here, we extend the weak KAM and Aubry–Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton–Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtain necessary conditions for minimality, extend Fathi’s weak KAM theorem, and describe the asymptotic limit of the generalized Lax–Oleinik semigroup.

Optimal VLSI Delay Tuning by Wire Shielding

Interconnect shielding is used in VLSI designs to avoid noise interference from the cross-coupling capacitance between adjacent signals. This paper takes advantage of the shields already present in the design and uses them to tune the propagation delay of the clock signals, thus eliminating expensive dedicated delay buffers. The problem of obtaining the desired delay at a minimum shielding cost (silicon area) is formulated as a calculus of variations problem. An analytical solution shows that a square root shield profile is optimal.

On using cubic spline for the solution of problems in calculus of variations

Two different approaches based on cubic B-spline are developed to approximate the solution of problems in calculus of variations. Both direct and indirect methods will be described. It is known that, when using cubic spline for interpolating a function gźC4[a,b] on a uniform partition with the step size h, the optimal order of convergence derived is O(h4). In Zarebnia and Birjandi (J. Appl. Math. 1---10, 2012) a non-optimal O(h2) method based on cubic B-spline has been used to solve the problems in calculus of variations. In this paper at first we will obtain an optimal O(h4) indirect method using cubic B-spline to approximate the solution. The convergence analysis will be discussed in details. Also a locally superconvergent O(h6) indirect approximation will be describe. Finally the direct method based on cubic spline will be developed. Some test problems are given to demonstrate the efficiency and applicability of the numerical methods.

Is n sin θ conserved along the light path?

Snell’s law states that the quantity is unchanged in refraction of light passing from one medium to another. We inquire whether this is true in the general case where the speed of light varies continuously within a medium. It turns out to be an instructive exercise in application of Snell’s law and Fermat’s principle. It also provides good pedagogical problems in calculus of variations to deal with the subtleties of a variable domain of integration and inclusion of constraints. The final result of these exercises is that, contrary to an initial expectation, the answer to the question in the title is negative.

Output Diffusion of the Monopolist Over Time and Space

In this paper, we develop an output diffusion model in which a monopoly firm faces a cost of adjusting output over time and across geographic regions. First, we investigate a dynamic monopoly model and a simple spatial model with only regional adjustment costs. Then, we explore how the firm maximizes profits over time and space, a calculus of variation problem. The Euler equation yields a partial differential equation which forms our output diffusion model. It extends the traditional inter-temporal output path into output diffusion over time and space. As long as regional adjustment costs exist, steady-state output in the diffusion model is less than the static monopoly output level. This model suggests that a policy designed to lower regional adjustment costs can increase supply in all geographic regions.

On The Asymptotic Behaviour of Some Problems of the Calculus of Variations

In this note we analyse the asymptotic behaviour of the solution of some class of calculus of variation problems set in cylindrical domains. A special attention is given to limit the assumptions on the functional at stake to a minimum.

Mathematical model and program system for solving a problem of logistic objects placement

This paper considers a placement problem for logistic objects with logistic zones segmentation in the continuous setting (an object can be placed at any point of a given area). The authors reduce this framework to a calculus of variations problem and suggest a numerical algorithm of solution based on the optical-geometrical approach. And finally, this algorithm is employed to solve the problems of municipal infrastructure optimization.

Core Deviation Minimizing Auctions

We study dominant strategy implementable direct mechanisms that minimize the expected surplus from core deviations. Using incentive compatibility conditions, we formulate the core deviation miminimization problem as a calculus of variations problem and then numerically solve it for some particular cases.

Core deviation minimizing auctions

In a stylized environment with complementary products, we study a class of dominant strategy implementable direct mechanisms and focus on the objective of minimizing the expected surplus from core deviations. For this class of mechanisms, we formulate the core deviation minimization problem as a calculus of variations problem and numerically solve it for some interesting special cases. We then compare the core deviation surplus in the optimal auction (CDMA) to that in Vickrey-Clark-Groves mechanism (VCG) and core-selecting auctions (CSAs). We find that the expected surplus from core deviations can be significantly smaller in CDMA than that in both VCG and CSAs.

Solution of Problems in Calculus of Variations Using Parameterization Technique

Solution of Problems in Calculus of Variations Using Parameterization Technique

Inverse Variational Problem for Nonstandard Lagrangians

The non-standard Lagrangians (NSLs) for dissipative-like dynamical systems were introduced in an ad hoc fashion rather than being derived from the solution of the inverse problem of variational calculus. We begin with the first integral of the equation of motion and solve the associated inverse problem to obtain some of the existing results for NSLs. In addition, we provide a number of alternative Lagrangian representations. The case studies envisaged by us include (i) the usual modified Emden-type equation, (ii) Emden-type equation with dissipative term quadratic in velocity and (iii) Lokta-Volterra model. We point out that our method is quite general for applications to other physical systems.