Showing papers on "Legendre–Clebsch condition published in 2010"
TL;DR: In this paper, the authors developed some basics of discrete fractional calculus such as Leibniz rule and summation by parts formula and derived Euler-Lagrange equation.
287 citations
TL;DR: In this article, a discrete-time fractional calculus of variations is introduced, and the first and second order necessary optimality conditions are established, showing that the solutions of the fractional problems coincide with the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.
Abstract: We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.
116 citations
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06 Dec 2010
27 citations
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TL;DR: In this paper, the authors prove Euler-Lagrange and natural boundary necessary optimality conditions for fractional problems of the calculus of variations which are given by a composition of functionals.
Abstract: We prove Euler–Lagrange and natural boundary necessary optimality conditions for fractional problems of the calculus of variations which are given by a composition of functionals. Our approach uses the recent notions of Riemann– Liouville fractional derivatives and integrals in the sense of Jumarie. As an application, we get optimality conditions for the product and the quotient of fractional variational functionals.
27 citations
TL;DR: In this article, a new approach to the calculus of variations on time scales is proposed, which consists in applying a differentiation tool called the contingent epiderivative, which allows to unify the delta and nabla approaches.
Abstract: The calculus of variations on time scales is considered. We propose a new approach to the subject that consists in applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to the calculus of variations on time scales is very useful: it allows to unify the delta and nabla approaches previously considered in the literature. Generalized versions of the Euler-Lagrange necessary optimality conditions are obtained, both for the basic problem of the calculus of variations and isoperimetric problems. As particular cases one gets the recent delta and nabla results.
19 citations
TL;DR: In this paper, the authors considered a nonlocal Samarksii-Ionkin condition for a parabolic-hyperbolic equation with the heat and wave operators in a rectangular domain.
Abstract: For a parabolic-hyperbolic equation with the heat and wave operators in a rectangular domain, we consider a problem with a nonlocal Samarksii-Ionkin condition. A criterion for the uniqueness of the solution is established by the spectral expansion method. The classical solution of the problem is constructed in the form of the sum of a biorthogonal series. The solution is proved to be stable with respect to the initial condition.
17 citations
TL;DR: In this paper, the authors studied the problem of minimizing ∫ Ω F(Du(x)) dx over the functions u ∈ W 1,1(Ω) that assume given boundary values φ on Γ ≔ ∂Ω.
Abstract: Abstract This article studies the problem of minimizing ∫ Ω F(Du(x)) dx over the functions u ∈ W 1,1(Ω) that assume given boundary values φ on Γ ≔ ∂Ω. The Lagrangian F and the domain Ω are assumed convex but not necessarily strictly conxex. When φ is continuous and F superlinear, we prove the existence of a minimum which is continuous on the closure of Ω. We also consider the case when F is not superlinear.
16 citations
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TL;DR: In this article, a new discrete-time fractional calculus of variations with Euler-Lagrange and Legendre necessary optimality conditions is proposed, and some illustrative examples provided.
Abstract: We study problems of the calculus of variations and optimal control within the framework of time scales Specifically, we obtain Euler-Lagrange type equations for both Lagrangians depending on higher order delta derivatives and isoperimetric problems We also develop some direct methods to solve certain classes of variational problems via dynamic inequalities In the last chapter we introduce fractional difference operators and propose a new discrete-time fractional calculus of variations Corresponding Euler-Lagrange and Legendre necessary optimality conditions are derived and some illustrative examples provided
11 citations
TL;DR: In this paper, the authors consider a nonlinear optimal control problem with an integral equation as the control object, subject to control constraints and derive the first-order necessary optimality conditions in the form of the Euler-Lagrange equation, and then apply them to obtain a numerical solution of the problem of optimal portfolio selection.
Abstract: We consider a nonlinear optimal control problem with an integral equation as the control object, subject to control constraints. This integral equation corresponds to the fractional moment of a stochastic process involving short-range and long-range dependences. For both cases, we derive the first-order necessary optimality conditions in the form of the Euler–Lagrange equation, and then apply them to obtain a numerical solution of the problem of optimal portfolio selection.
10 citations
TL;DR: In this paper, the brachistochrone for a steerable particle moving on a 1D curved surface in a gravity field is solved using an optimal control formulation with state feedback.
Abstract: The brachistochrone for a steerable particle moving on a 1D curved surface in a gravity field is solved using an optimal control formulation with state feedback. The process begins with a derivation of a fourth-order open-loop plant model with the system input being the body yaw rate. Solving for the minimum-time control law entails introducing four costates and solving the Euler-Lagrange equations, with the Hamiltonian being stationary with respect to the control. Also, since the system is autonomous, the Hamiltonian must be zero. A two-point boundary value problem results with a transversality condition, and its solution requires iteration of the initial bearing angle so the integrated trajectory runs through the final point. For this choice of control, the Legendre-Clebsch necessary condition is not satisfied. However, the k = 1 generalized Legendre-Clebsch necessary condition,from singular control theory is satisfied for all numerical simulations performed, and optimality is assured. Simulations in MATLAB ® exercise the theory developed and illustrate application such as to ski racing and minimizing travel time over either a concave or undulating surface when starting from rest. Lastly, a control law singularity in particle speed is overcome numerically.
3 citations
TL;DR: In this paper, the validity of the Euler-Lagrange equation for the class of functionals of the type ∫ Ω [ L ( ∇ u ) + g ( x, u ) ] d x, where u ↦ g (x, u ) is a concave function, was proved.
Abstract: We prove the validity of the Euler–Lagrange equation for the class of functionals of the type ∫ Ω [ L ( ∇ u ) + g ( x , u ) ] d x , where u ↦ g ( x , u ) is a concave function. Our result is based on a recent paper by Degiovanni and Marzocchi (2009) [6] , where no growth assumptions on L are needed.
TL;DR: In this paper, the Euler-Lagrange equations for the order parameter and the vector potential were developed for the first time using well-known mathematical formulations, and solved them using well known mathematical formulations.
Abstract: The thermodynamics expected of systems undergoing third order phase transition has been investigated by identifying the orders through the analytic continuation of the functional of the free energy, using Ehrenfest thermodynamic theory. We developed the Euler–Lagrange equations for the order parameter and the vector potential and solved them for the first time using well-known mathematical formulations.
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TL;DR: In this article, the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves, and a necessary and sufficient condition for minimality is proved.
Abstract: This paper is a direct continuation of [1]. The Hamiltonian aspects of the theory are further developed. Within the framework provided by the first paper, the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A necessary and sufficient condition for minimality is proved.