Second-order Variational Problems Research Articles

Minimizing acceleration on the group of diffeomorphisms and its relaxation

We study a second-order variational problem on the group of diffeomorphisms of the interval [0, 1] endowed with a right-invariant Sobolev metric of order 2, which consists in the minimization of the acceleration. We compute the relaxation of the problem which involves the so-called Fisher–Rao functional, a convex functional on the space of measures. This relaxation enables the derivation of several optimality conditions and, in particular, a sufficient condition which guarantees that a given path of the initial problem is also a minimizer of the relaxed one. Based on these sufficient conditions, the main result is that, when the value of the (minimized) functional is small enough, the minimizers are classical, that is the defect measure vanishes.

Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres

The present paper extends the classical second-order variational problem of Herglotz type to the more general context of the Euclidean sphere S^n following variational and optimal control approaches. The relation between the Hamiltonian equations and the generalized Euler-Lagrange equations is established. This problem covers some classical variational problems posed on the Riemannian manifold S^n such as the problem of finding cubic polynomials on S^n. It also finds applicability on the dynamics of the simple pendulum in a resistive medium.

Second-order L ∞ variational problems and the ∞-polylaplacian

Abstract In this paper we initiate the study of second-order variational problems in L ∞ {L^{\infty}} , seeking to minimise the L ∞ {L^{\infty}} norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler–Lagrange equation. Given H ∈ C 1 ⁢ ( ℝ s n × n ) {\mathrm{H}\in C^{1}(\mathbb{R}^{n\times n}_{s})} , for the functional E ∞ ⁢ ( u , 𝒪 ) = ∥ H ⁢ ( D 2 ⁢ u ) ∥ L ∞ ⁢ ( 𝒪 ) , u ∈ W 2 , ∞ ⁢ ( Ω ) , 𝒪 ⊆ Ω , \mathrm{E}_{\infty}(u,\mathcal{O})=\|\mathrm{H}(\mathrm{D}^{2}u)\|_{L^{\infty}% (\mathcal{O})},\quad u\in W^{2,\infty}(\Omega),\mathcal{O}\subseteq\Omega,{} the associated equation is the fully nonlinear third-order PDE A ∞ 2 ⁢ u := ( H X ⁢ ( D 2 ⁢ u ) ) ⊗ 3 : ( D 3 ⁢ u ) ⊗ 2 = 0 . \mathrm{A}^{2}_{\infty}u:=(\mathrm{H}_{X}(\mathrm{D}^{2}u))^{\otimes 3}:(% \mathrm{D}^{3}u)^{\otimes 2}=0.{} Special cases arise when H {\mathrm{H}} is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞ {\infty} -polylaplacian and the ∞ {\infty} -bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of “critical point” generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

Second-order variational problems on Lie groupoids and optimal control applications

In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques forsecond-order variational problems on Lie groupoids and their applications to the construction of variational integratorsfor optimal control problems of mechanical systems. Next, weshow how Lagrangian submanifolds of a symplectic groupoidgives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we study the geometric properties of the implicit flow which defines the dynamics in a Lagrangian submanifold. We also study the theory ofreduction by symmetries and the corresponding Noether theorem.

Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian $$L:T^{(k)}Q\rightarrow {\mathbb {R}}$$ with $$k\ge 1$$ , the resulting discrete equations define a generally implicit numerical integrator algorithm on $$T^{(k-1)}Q\times T^{(k-1)}Q$$ that approximates the flow of the higher-order Euler–Lagrange equations for L. The algorithm equations are called higher-order discrete Euler–Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. We construct an exact discrete Lagrangian $$L_d^e$$ using the locally unique solution of the higher-order Euler–Lagrange equations for L with boundary conditions. By taking the discrete Lagrangian as an approximation of $$L_d^e$$ , we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem.

Tightening elastic (n, 2)-torus knots

We present a theory for equilibria of elastic torus knots made of a single thin, uniform, homogeneous, isotropic, inextensible, unshearable rod of circular cross-section. The theory is formulated as a special case of an elastic theory of geometrically exact braids consisting of two rods winding around each other while remaining at constant distance. We introduce braid strains in terms of which we formulate a second-order variational problem for an action functional that is the sum of the rod elastic energies and constraint terms related to the inextensibility of the rods. The Euler-Lagrange equations for this problem, partly in Euler-Poincare form, yield a compact system of ODEs suitable for numerical solution. By solving an appropriate boundary- value problem for these equations we study knot equilibria as the dimensionless ropelength parameter is varied. We are particularly interested in the approach of the purely geometrical ideal (tightest) limit. For the trefoil knot the tightest shape we could get has a ropelength of 32.85560666, which is remarkably close to the best current estimate. For the pentafoil we find a symmetry-breaking bifurcation.

Theory of equilibria of elastic 2-braids with interstrand interaction

Motivated by continuum models for DNA supercoiling we formulate a theory for equilibria of 2-braids, i.e., structures formed by two elastic rods winding around each other in continuous contact and subject to a local interstrand interaction. No assumption is made on the shape of the contact curve. The theory is developed in terms of a moving frame of directors attached to one of the strands. The other strand is tracked by including in this frame the normalised closest-approach chord connecting the two strands. The kinematic constant-distance constraint is formulated at strain level through the introduction of what we call braid strains. As a result the total potential energy involves arclength derivatives of these strains, thus giving rise to a second-order variational problem. The Euler–Lagrange equations for this problem give balance equations for the overall braid force and moment referred to the moving frame as well as differential equations that can be interpreted as effective constitutive relations encoding the effect that the second strand has on the first as the braid deforms under the action of end loads. Hard contact models are used to obtain the normal contact pressure between strands that has to be non-negative for a physically realisable solution without the need for external devices such as clamps or glue to keep the strands together. The theory is first illustrated by a number of problems that can be solved analytically and then applied to several new problems that have not hitherto been treated.

Geometric Hamiltonian Formulation of a Variational Problem Depending on the Covariant Acceleration

We consider a second-order variational problem depending on the covariant acceleration, which is related to the notion of Riemannian cubic polynomials. This problem and the corresponding optimal control problem are described in the context of higher order tangent bundles using geometric tools. The main tool, a presymplectic variant of Pontryagin’s maximum principle, allows us to study the dynamics of the control problem.

Free gradient discontinuity and image inpainting

We introduce and study a formulation of the inpainting problem for two-dimensional images that are locally damaged. This formulation is based on the regularization of the solution of a second-order variational problem with Dirichlet boundary condition. A variational approximation algorithm is proposed. Bibliography: 45 titles.

A second-order variational problem with a lack of coercivity

For Ω ⊂ ℝn and 2n/(n − 2) < p ⩽ (2n − 4)/(n − 4), we study the functional E 2 ( u ) = 1 2 ∫ Ω ( Δ u + | u | p − 2 u ) 2 d x . In an appropriate nonlinear space, the minimization problem can be solved using the direct method. We also construct solutions of the gradient flow.

Higher-Order Quasiconvexity Reduces to Quasiconvexity

In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p>1, is the restriction to symmetric matrices of a 1-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems.

Parameter-invariant second-order variational problems in one variable

A projection is defined such that a second-order Lagrangian density factors through this projection modulo contact forms if and only if it is parameter invariant. In this way, a geometric interpretation of the parameter invariance conditions is obtained. The above projection is then used to prove the strict factorization of the Poincare-Cartan form attached to a parameter-invariant variational problem thus leading us to state the Hamilton-Cartan formalism, the complete description of symmetries and regularity for such problems. The case of the squared curvature Lagrangian in the plane is analysed especially.

Conservation laws for second-order invariant variational problems

In the literature, a symmetry requirement in the mixed, second partials appearing in the Lagrangian is needed before conservation laws for second-order variational problems can be obtained. In this paper this assumption is removed and an example of a modified, linear Korteweg-de Vries equation is analysed.

Conformal identities for invariant second-order variational problems depending on a covariant vector field

Necessary and sufficient conditions for the conformal invariance of a multiple integral variational problem whose Lagrangian depends upon second-order derivatives of a covariant vector field are obtained. These conditions take the form of differential identities involving the Lagrangian, its derivatives, and the infinitesimal generators of the special conformal group; they differ from the classical Noether identities in that they involve only second-order derivatives of the field, not fourth-order derivatives. The conditions are not conservation laws, but rather identities which provide a practical test for invariance which, if established, can lead to conservation laws via the Noether theorem. Finally, an application to 'generalised electrodynamics' is given.

An invariance theory for second-order variational problems

This paper investigates the invariance properties of second-order variational problems when the configuration space is subjected to an r-parameter local Lie group of transformations. In particular, the recent results of Hanno Rund on first-order problems are extended to the higher order case: A new set of fundamental invariance identities are derived for single and multiple integral problems, and new proofs of the Zermelo conditions and Noether’s theorem are presented. The results are applied to a variational problem whose second-order Lagrangian depends upon a scalar field in Minkowski space, and some conformal identities are obtained.