Related papers: Euler's optimal profile problem
Based on a method introduced by Leitmann [Internat. J. Non-Linear Mech. {\bf 2} (1967), 55--59], we exhibit exact solutions for some fractional optimization problems of the calculus of variations and optimal control.
We investigate the properties of minimizers of one-dimensional variational problems when the Lagrangian has no higher smoothness than continuity. An elementary approximation result is proved, but it is shown that this cannot be in general…
We consider minimizers of \[ F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) + |\Omega|, \] where $F$ is a function nondecreasing in each parameter, and $\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. This includes, in…
The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of…
A variational principle is introduced which minimizes an action formulated for configurations of vacuum Dirac seas. The action is analyzed in position and momentum space. We relate the corresponding Euler-Lagrange equations to the notion of…
We explicitly solve a variational problem related to upper bounds on the optimal constants in the Cwikel--Lieb--Rozenblum (CLR) and Lieb--Thirring (LT) inequalities, which has recently been derived in [Invent. Math. 231 (2023), no.1,…
We study a vectorial $L^\infty$-variational problem of second order, where the supremal functional depends on the vector function $u$ through a linear elliptic operator in divergence form. We prove existence and uniqueness of the minimiser…
There is currently a great deal of interest in the 4D-Var data assimilation scheme, in which one uses observational data to find the optimal initial condition for a differential equation by minimizing a cost function over the set of all…
This paper examines a variety of classical optimization problems, including well-known minimization tasks and more general variational inequalities. We consider a stochastic formulation of these problems, and unlike most previous work, we…
We propose a novel discretization procedure for the classical Euler equation based on the theory of Galois differential algebras and the finite operator calculus developed by G.C. Rota and collaborators. This procedure allows us to define…
We consider the numerical computation of a variational problem that arises from materials science. The target functional is a type of elastic energy that is influenced by obstacles and adhesion. Owing to its strong nonlinearity and…
An optimization-based approach for the Tucker tensor approximation of parameter-dependent data tensors and solutions of tensor differential equations with low Tucker rank is presented. The problem of updating the tensor decomposition is…
We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of variations with time delays, where the delay in the unknown function is different from the delay in its derivative. Then, a more general…
This work focuses on numerical solutions of optimal control problems. A time discretization error representation is derived for the approximation of the associated value function. It concerns Symplectic Euler solutions of the Hamiltonian…
We discover a new minimality property of the absolute minimisers of supremal functionals (also known as $L^\infty$ Calculus of Variations problems).
The divergence minimization problem plays an important role in various fields. In this note, we focus on differentiable and strictly convex divergences. For some minimization problems, we show the minimizer conditions and the uniqueness of…
Euler's elastica model has a wide range of applications in Image Processing and Computer Vision. However, the non-convexity, the non-smoothness and the nonlinearity of the associated energy functional make its minimization a challenging…
A scheme for generating a family of convex variational principles is developed, the Euler- Lagrange equations of each member of the family formally corresponding to the necessary conditions of optimal control of a given system of ordinary…
This translation has been withdrawn due to certain imperfections and mistakes, which are corrected in the version uploaded at The Euler Archive (see E65 at http://www.eulerarchive.org/)
Whereas in a coordinate-dependent setting the Euler-Lagrange equations establish necessary conditions for solving variational problems in which both the integrands of functionals and the resulting paths are assumed to be sufficiently…