Related papers: A General Backwards Calculus of Variations via Dua…
In this paper the structures of the generalised Euler-Lagrange equations and their associated conserved quantities are derived for one-dimensional Herglotz variational problems of order $n$. Their derivations use the framework of moving…
Through duality it is possible to transform left fractional operators into right fractional operators and vice versa. In contrast to existing literature, we establish integration by parts formulas that exclusively involve either left or…
We obtain new regularity conditions for problems of calculus of variations with higher-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main regularity result asserts that autonomous integral…
This article is devoted to obtain new sufficient conditions for an extremum in problems of classical calculus of variations. The concept of a set of integrands is introduced. Using this concept, first and second order sufficient conditions…
We study dynamic minimization problems of the calculus of variations with Lagrangian functionals containing Riemann-Liouville fractional integrals, classical and Caputo fractional derivatives. Under assumptions of regularity, coercivity and…
We present a unified treatment to control problems on an arbitrary time scale by introducing the study of forward-backward optimal control problems. Necessary optimality conditions for delta-nabla isoperimetric problems are proved, and…
We study the variational problem for $N$-parallel curves on a Finslerian surface by means of Exterior Differential Systems using Griffiths' method. We obtain the conditions when these curves are extremals of a length functional and write…
The least action principle, through its variational formulation, possesses a finalist aspect. It explicitly appears in the fractional calculus framework, where Euler-Lagrange equations obtained so far violate the causality principle. In…
We obtain Euler-Lagrange equations, transversality conditions and a Noether-like theorem for Herglotz-type variational problems with Lagrangians depending on generalized fractional derivatives. As an application, we consider a damped…
We analyze the relation between the concept of auxiliary variables and the Inverse problem of the calculus of variations to construct a Lagrangian from a given set of equations of motion. The problem of the construction of a consistent…
We argue that the variational calculus leading to Euler's equations and Noether's theorem can be replaced by equivariance and invariance conditions avoiding the action integral. We also speculate about the origin of Lagrangian theories in…
In this paper we develop the theory of initial and boundary value problems for the self-adjoint nabla fractional difference equation containing a Caputo fractional nabla difference that is given by \[ \nabla[p(t+1)\nabla_{a*}^\nu x(t+1)] +…
In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on…
A variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases. Necessary and sufficient conditions for an infinitesimal…
We consider a variational problem with boundary singularity and Dirichlet condition. We give a blow-up analysis for sequences of solutions of an equation with exponential nonlinearity. Also, we derive a compactness criterion under some…
It is shown that a chain of closed systems of first order ordinary differential equations describing the evolution of moments can be constructed using the Jacobi equation. It is shown that Wronsky determinants for fundamental matrices of…
We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional $p$-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration…
Using min-max inequality we investigate the existence of solutions and thier dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle…
We analyze controlled mass transportation plans with free end-time that minimize the transport cost induced by the generating function of a Lagrangian within a bounded domain, in addition to costs incurred as export and import tariffs at…
Unlike many deterministic PDEs, stochastic equations are not amenable to the classical variational theory of Euler-Lagrange. In this paper, we show how self-dual variational calculus leads to solutions of various stochastic partial…