Higher Order Variational Integrators: a polynomial approach

We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the structural properties of these systems, like the symplectic form, the evolution of the momentum maps or the energy behaviour. Also they are easily applicable to optimal control problems based on mechanical systems as proposed in Ober-Bl\"obaum et al. [2011]. Following the same approach, we develop a family of variational integrators to which we refer as symplectic Galerkin schemes in contrast to symplectic partitioned Runge-Kutta. These two families of integrators are, in principle and by construction, different one from the other. Furthermore, the symplectic Galerkin family can as easily be applied in optimal control problems, for which Campos et al. [2012b] is a particular case.Comment: 12 pages, 1 table, 23rd Congress on Differential Equations and Applications, CEDYA 201

Discrete Variational Optimal Control

This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian setting. It extends the theory of discrete mechanics to enable the solutions of optimal control problems through the discretization of variational principles. The key point is to solve the optimal control problem as a variational integrator of a specially constructed higher-dimensional system. The developed framework applies to systems on tangent bundles, Lie groups, underactuated and nonholonomic systems with symmetries, and can approximate either smooth or discontinuous control inputs. The resulting methods inherit the preservation properties of variational integrators and result in numerically robust and easily implementable algorithms. Several theoretical and a practical examples, e.g. the control of an underwater vehicle, will illustrate the application of the proposed approach.Comment: 30 pages, 6 figure

LOS FUNDAMENTOS DE LA PROSTITUCIÓN COLOMBIANA EN ARUBA

Basados en estadísticas, entrevistas e investigaciones, se revela la situación de las mujeres dedicadas a la prostitución en la isla caribeña Aruba; revelando las oportunidades que se presentan en una población llamada San Nicolás, incluso siendo una profesión penalizada en este país, siendo este el porqué la isla es tan llamativa a la hora de dedicarse a esta “profesión”, presentando igualmente argumentos históricos acerca del mismo tema. La situación y vulnerabilidad de la mujer colombiana, además de la economía, muestran un panorama vulnerable para las mujeres, especialmente aquellas que buscan un sustento económico, grupos vulnerables como: mujeres solteras, casadas, viudas y hasta divorciadas, con o sin hijos, expuesta a través de estadísticas tanto del departamento como nacionales.Viendo este tema desde el punto jurídico y teniendo en cuenta que las mayoría de prostitutas que laboran en la isla caribeña son colombianas, se presentan propuestas para la disminución de esta problemáticasocial, en concordancia entre los dos países, pero queda la duda abierta acerca de las leyes fundamentales y cómo se podrá fundamentar una solución sólida y viable paras las dos partes: las mujeres colombianas así como para Aruba

Variational integrators of mixed order for dynamical systems with multiple time scales and split potentials

The simulation of mechanical systems that act on multiple time scales, caused e.g. by different types or stiffnesses in potentials, is challenging as a stable integration of the fast dynamics requires a highly accurate approximation whereas for the simulation of the slow part a coarser approximation is accurate enough. With regard to the general goals of any numerical method, high accuracy and low computational costs, the presented variational integrators of mixed order couple coarse and fine approximations. The idea builds up on the higher order Galerkin variational integrators in [9] that are derived via Hamilton’s variational principle with a polynomial to approximate the configuration and an appropriate quadrature formula for the approximation of the integral of the Lagrangian. For the variational integration of systems with dynamics on multiple time scales, we use polynomials of different degrees to approximate the components that act on different time scales. Furthermore, quadrature formulas of different order approximate the integrals of the single energy contributions of the Lagrangian. This approach provides great flexibility in the design of the integrators. Their performance is investigated numerically by means of the Fermi-Pasta-Ulam problem and a numerical analysis regarding accuracy versus efficiency is carried out, where we focus on the integrators most promising to resolve the mentioned trade-off

Multiobjective optimal control methods for the Navier-Stokes equations using reduced order modeling

In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under con- sideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to effciently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the two-dimensional incompressible flow around a cylinder

Multiobjective optimal control methods for the Navier-Stokes equations using reduced order modeling

In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under con- sideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to effciently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the two-dimensional incompressible flow around a cylinder