Related papers: The variational discretization of the constrained …
We present a unified approach to constrained implicit Lagrangian and Hamiltonian systems based on the introduced concept of Dirac algebroid. The latter is a certain almost Dirac structure associated with the Courant algebroid on the dual…
We study variational problems for curves approximated by B-spline curves. We show that, one can obtain discrete Euler-Lagrange equations, for the data describing the approximated curves. Our main application is to the curve completion…
The gauge symmetries of a general dynamical system can be systematically obtained following either a Hamiltonean or a Lagrangean approach. In the former case, these symmetries are generated, according to Dirac's conjecture, by the first…
A multi-agent system designed to achieve distance-based shape control with flocking behavior can be seen as a mechanical system described by a Lagrangian function and subject to additional external forces. Forced variational integrators are…
We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic partial differential equation (PDE) as state equation and a control variable…
We consider Lagrangians in Hamilton's principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincar\'e…
Retractions maps are used to define a discretization of the tangent bundle of the configuration manifold as two copies of the configuration manifold where the dynamics take place. Such discretization maps can be conveniently lifted to the…
A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie…
This work deals with a numerical method for solving a mean-field type control problem with congestion. It is the continuation of an article by the same authors, in which suitably defined weak solutions of the system of partial differential…
We investigate higher-order geometric $k$-splines for template matching on Lie groups. This is motivated by the need to apply diffeomorphic template matching to a series of images, e.g., in longitudinal studies of Computational Anatomy. Our…
The aim of this paper is to study the relationship between Hamiltonian dynamics and constrained variational calculus. We describe both using the notion of Lagrangian submanifolds of convenient symplectic manifolds and using the so-called…
We consider a control-constrained parabolic optimal control problem without Tikhonov term in the tracking functional. For the numerical treatment, we use variational discretization of its Tikhonov regularization: For the state and the…
The equations for the critical points of the action functional defined by a Lagrangian depending on higher-order derivatives of admissible curves on a Lie algebroid are found. The relation with Euler-Poincar\'e and Lagrange Poincar\'e type…
The Dirac-Bergmann algorithm is a recipe for converting a theory with a singular Lagrangian into a constrained Hamiltonian system. Constrained Hamiltonian systems include gauge theories -- general relativity, electromagnetism, Yang Mills,…
This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and…
In this work, we address the numerical identification of entanglement in dynamical scenarios. To this end, we consider different programs based on the restriction of the evolution to the set of separable (i.e., non-entangled) states,…
This study deals with continuous limits of interacting one-dimensional diffusive systems, arising from stochastic distortions of discrete curves with various kinds of coding representations. These systems are essentially of a…
The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate.…
After recalling standard nonlinear port-Hamiltonian systems and their algebraic constraint equations, called here Dirac algebraic constraints, an extended class of port-Hamiltonian systems is introduced. This is based on replacing the…
A gauge-invariant formulation of constrained variational calculus, based on the introduction of the bundle of affine scalars over the configuration manifold, is presented. In the resulting setup, the Lagrangian is replaced by a section of a…