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We discuss the problem where the extremal function in embedding theorem of some (generally speaking, non-integer) order is the constant function. We obtain the necessary and sufficient conditions of this.
This paper is concerned with the derivation of first- and second-order sufficient optimality conditions for optimistic bilevel optimization problems involving smooth functions. First-order sufficient optimality conditions are obtained by…
Within the geometrical framework developed in arXiv:0705.2362, the problem of minimality for constrained calculus of variations is analysed among the class of differentiable curves. A fully covariant representation of the second variation…
We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.
In this paper, we investigate the multi-objective optimal control problem of ordinary differential equations on Riemannian manifolds. We first obtain the second-order necessary conditions for weak Pareto optimal solutions for…
We derive the discrete version of the classical Helmholtz condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide…
In this paper we consider second-order field theories in a variational setting. From the variational principle the Euler-Lagrange equations follow in an unambiguous way, but it is well known that this is not true for the Cartan form. This…
We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and defined on bounded-time intervals. Under assumptions of regularity, convexity and…
The jet bundle description of time-dependent mechanics is revisited. The constraint algorithm for singular Lagrangians is discussed and an exhaustive description of the constraint functions is given. By means of auxiliary connections we…
The aim of this paper is to study certain problems of calculus of variations, that are dependent upon a Lagrange function on a Caputo-type fractional derivative. This type of fractional operator is a generalization of the Caputo and the…
Over the last years, minimization problems over spaces of measures have received increased interest due to their relevance in the context of inverse problems, optimal control and machine learning. A fundamental role in their numerical…
The relevant material on differential calculus on graded infinite order jet manifolds and its cohomology is summarized. This mathematics provides the adequate formulation of Lagrangian theories of even and odd variables on smooth manifolds…
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…
We prove a necessary optimality condition of Euler-Lagrange type for variational problems on time scales involving nabla derivatives of higher-order. The proof is done using a new and more general fundamental lemma of the calculus of…
This paper focuses on optimality conditions for $C^{1,1}$-smooth optimization problems subject to inequality and equality constraints. By employing the concept of limiting (Mordukhovich) second-order subdifferentials to the Lagrangian…
This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach involves reducing a variational problem…
We obtain a generalized Euler-Lagrange differential equation and transversality optimality conditions for Herglotz-type higher-order variational problems. Illustrative examples of the new results are given.
The Hamiltonian formulation for the mechanical systems with reparametrization-invariant Lagrangians, depending on the worldline external curvatures is given, which is based on the use of moving frame. A complete sets of constraints are…
The necessary conditions for an optimal control of a stochastic control problem with recursive utilities is investigated. The first order condition is the the well-known Pontryagin type maximum principle. When the optimal control satisfying…
We prove a necessary optimality condition of Euler--Lagrange type for the calculus of variations with Omega derivatives, which turns out to be sufficient under jointly convexity of the Lagrangian.