arXiv subjects

María Barbero-Liñán

Publications and source records attributed to María Barbero-Liñán.

12 recordsLinked to original sources

arXiv2025

Discretization of Dirac systems and port-Hamiltonian systems: the role of the constraint algorithm

We study the discretization of (almost-)Dirac structures using the notion of retraction and discretization maps on manifolds. Additionally, we apply the proposed discretization techniques to obtain numerical integrators for port-Hamiltonian systems and we discuss how to merge the discretization procedure and the constraint algorithm associated to systems of implicit differential equations.

math.NA↗

arXiv2024

From retraction maps to symplectic-momentum numerical integrators

Retraction maps have been generalized to discretization maps in (Barbero Liñán and and Mart\'ın de Diego, 2022). Discretization maps are used to systematically derive numerical integrators that preserve the symplectic structure, as well as the discrete momemtum map under the assumption of symmetric preservation for the discretization map. The procedure described here gives a geometrical construction that can be easily adapted to discretize dynamics on more general structures and open the door to reduction processes.

math.NA↗

arXiv2017

The inverse problem of the calculus of variations for discrete systems

We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propose variationality as an interesting geometric property to take into account in the design and computer simulation of numerical integrators.

math.DG↗

arXiv2015

Inverse problem for Lagrangian systems on Lie algebroids and applications to reduction by symmetries

The language of Lagrangian submanifolds is used to extend a geometric characterization of the inverse problem of the calculus of variations on tangent bundles to regular Lie algebroids. Since not all closed sections are locally exact on Lie algebroids, the Helmholtz conditions on Lie algebroids are necessary but not sufficient, so they give a weaker definition of the inverse problem. As an application the Helmholtz conditions on Atiyah algebroids are obtained so that the relationship between the inverse problem and the reduced inverse problem by symmetries can be described. Some examples and comparison with previous approaches in the literature are provided.

math.DG↗

arXiv2014

Isotropic submanifolds and the inverse problem for mechanical constrained systems

The inverse problem of the calculus of variations consists in determining if the solutions of a given system of second order differential equations correspond with the solutions of the Euler-Lagrange equations for some regular Lagrangian. This problem in the general version remains unsolved. Here, we contribute to it with a novel description in terms of Lagrangian submanifolds of a symplectic manifold, also valid under some adaptation for the non-autonomous version. One of the advantages of this new point of view is that we can easily extend our description to the study of the inverse problem of the calculus of variations for second order systems along submanifolds. In this case, instead of Lagrangian submanifolds we will use isotropic submanifolds, covering both the nonholonomic and holonomic constraints for autonomous and non-autonomous systems as particular examples. Moreover, we use symplectic techniques to extend these isotropic submanifolds to Lagrangian ones, allowing us to describe the constrained solutions as solutions of a variational problem now without constraints. Mechanical examples such as the rolling disk are provided to illustrate the main results.

math.DG↗

arXiv2012

Morse families in optimal control problems

We geometrically describe optimal control problems in terms of Morse families in the Hamiltonian framework. These geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm. Moreover the integrability algorithm is adapted to optimal control problems in such a way that the trajectories originated by discontinuous controls are also obtained. From the Hamiltonian viewpoint we obtain the equations of motion for optimal control problems in the Lagrangian formalism by means of a proper Lagrangian submanifold. Singular optimal control problems and overdetermined ones are also studied along the paper.

math.OC↗

arXiv2012

$k$-symplectic Pontryagin's Maximum Principle for some families of PDEs

An optimal control problem associated with the dynamics of the orientation of a bipolar molecule in the plane can be understood by means of tools in differential geometry. For first time in the literature $k$-symplectic formalism is used to provide the optimal control problems associated to some families of partial differential equations with a geometric formulation. A parallel between the classic formalism of optimal control theory with ordinary differential equations and the one with particular families of partial differential equations is established. This description allows us to state and prove Pontryagin's Maximum Principle on $k$-symplectic formalism. We also consider the unified Skinner-Rusk formalism for optimal control problems governed by an implicit partial differential equation.

math.OC↗

arXiv2011

Kinematic reduction and the Hamilton-Jacobi equation

A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric structure allows us to describe in a simplified way the mechanics of nonholonomic systems with both control and external forces.

math-ph↗

arXiv2011

Characterization of accessibility for affine connection control systems at some points with nonzero velocity

Affine connection control systems are mechanical control systems that model a wide range of real systems such as robotic legs, hovercrafts, planar rigid bodies, rolling pennies, snakeboards and so on. In 1997 the accessibility and a particular notion of controllability was intrinsically described by A. D. Lewis and R. Murray at points of zero velocity. Here, we present a novel generalization of the description of accessibility algebra for those systems at some points with nonzero velocity as long as the affine connection restricts to the distribution given by the symmetric closure. The results are used to describe the accessibility algebra of different mechanical control systems.

math.OC↗

arXiv2009

High order sufficient conditions for tracking

In this paper, we study under which conditions the trajectories of a mechanical control system can track any curve on the configuration manifold. We focus on systems that can be represented as forced affine connection control systems and we generalize the sufficient conditions for tracking known in the literature. The sufficient conditions are expressed in terms of convex cones of vector fields defined through particular brackets of the control vector fields of the system. The tracking control laws obtained by our constructions depend on several parameters. By imposing suitable asymptotic conditions on such parameters, we construct algorithmically one-parameter tracking control laws. The theory is supported by examples of control systems associated with elliptic hovercrafts and ellipsoidal submarines.

math.OC↗

arXiv2008

Geometric Approach to Pontryagin's Maximum Principle

Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self\textendash{}contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry.

math.OC↗

arXiv2008

Unified formalism for non-autonomous mechanical systems

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and non-regular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk (1983). The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism.

math-ph↗