scholarly journals A non-standard class of variational problems of Herglotz type

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Natália Martins
Keyword(s):  
Variational Problem ◽  
Optimality Condition ◽  
Unknown Function ◽  
Variational Problems ◽  
Standard Theory ◽  
Necessary Optimality Condition ◽  
Natural Boundary ◽  
Special Cases ◽  
Independent Variable ◽  
Natural Boundary Conditions

<p style='text-indent:20px;'>In this paper, we extend the variational problem of Herglotz considering the case where the Lagrangian depends not only on the independent variable, an unknown function <inline-formula><tex-math id="M1">\begin{document}$ x $\end{document}</tex-math></inline-formula> and its derivative and an unknown functional <inline-formula><tex-math id="M2">\begin{document}$ z $\end{document}</tex-math></inline-formula>, but also on the end points conditions and a real parameter. Herglotz's problems of calculus of variations of this type cannot be solved using the standard theory. Main results of this paper are necessary optimality condition of Euler-Lagrange type, natural boundary conditions and the Dubois-Reymond condition for our non-standard variational problem of Herglotz type. We also prove a necessary optimality condition that arises as a consequence of the Lagrangian dependence of the parameter. Our results not only provide a generalization to previous results, but also give some other interesting optimality conditions as special cases. In addition, two examples are given in order to illustrate our results.</p>

scholarly journals Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Tatiana Odzijewicz ◽  
Agnieszka B. Malinowska ◽  
Delfim F. M. Torres
Keyword(s):  
Fractional Integral ◽  
Necessary Optimality Conditions ◽  
Variational Problems ◽  
Fractional Integrals ◽  
Natural Boundary ◽  
Free Boundary Value Problems ◽  
Fractional Variational Problems ◽  
Generalized Fractional Integral ◽  
Fractional Calculus Of Variations ◽  
Natural Boundary Conditions

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed.

Noether’s theorem for fractional variational problems of variable order

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Tatiana Odzijewicz ◽  
Agnieszka Malinowska ◽  
Delfim Torres
Keyword(s):  
Optimality Condition ◽  
Variational Problems ◽  
Time Variable ◽  
Variable Order ◽  
Necessary Optimality Condition ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Fractional Variational Problems ◽  
Derivatives Of

AbstractWe prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether’s theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.

scholarly journals Natural Boundary Conditions in Geometric Calculus of Variations

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
Giovanni Moreno ◽  
Monika Ewa Stypa
Keyword(s):  
Boundary Conditions ◽  
Variational Problems ◽  
Boundary Values ◽  
Natural Boundary ◽  
Natural Behavior ◽  
Geometric Calculus ◽  
Jet Bundles ◽  
Independent Variables ◽  
Natural Boundary Conditions ◽  
Dimensional Domain

AbstractIn this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of Y - the manifold of dependent and independent variables underlying a given problem - as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over Y . Explicit examples of natural boundary conditions are obtained when Y is an (n + 1)-dimensional domain in ℝ

scholarly journals A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter

2021 ◽  
Vol 5 (1) ◽  
pp. 24
Author(s):  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Variational Problem ◽  
Fractional Derivatives ◽  
Variational Calculus ◽  
Variational Problems ◽  
Sufficient Optimality Conditions ◽  
Caputo Fractional Derivatives ◽  
Independent Variable ◽  
Left And Right ◽  
Necessary And Sufficient ◽  
Fractional Variational Problem

In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results.

scholarly journals Variational Problems with Moving Boundaries Using Decomposition Method

2007 ◽  
Vol 2007 ◽  
pp. 1-11 ◽  
Author(s):  
Reza Memarbashi
Keyword(s):  
Boundary Conditions ◽  
Decomposition Method ◽  
Lagrange Equation ◽  
Adomian Decomposition Method ◽  
Variational Problems ◽  
Moving Boundaries ◽  
Natural Boundary ◽  
Transversality Conditions ◽  
Adomian Decomposition ◽  
Natural Boundary Conditions

The aim of this paper is to present a numerical method for solving variational problems with moving boundaries. We apply Adomian decomposition method on the Euler-Lagrange equation with boundary conditions that yield from transversality conditions and natural boundary conditions.

Optimality of Approximate Quasi-Weakly Efficient Solutions for Vector Equilibrium Problems via Convexificators

2021 ◽  
Author(s):  
Guolin Yu ◽  
Siqi Li ◽  
Xiao Pan ◽  
Wenyan Han
Keyword(s):  
Optimality Condition ◽  
Generalized Convexity ◽  
Equilibrium Problems ◽  
Efficient Solutions ◽  
Sufficient Optimality Condition ◽  
Necessary Optimality Condition ◽  
Weakly Efficient Solutions ◽  
Convexity Assumption ◽  
Pseudoconvex Function ◽  
Vector Equilibrium

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

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