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

• Autores: Colombo, Leonardo Jesus; Ferraro, Sebastián José; Martin de Diego, David
• Año de publicación: 2016
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: DISCRETE VARIATIONAL CALCULUS; HIGHER-ORDER MECHANICS; OPTIMAL CONTROL; VARIATIONAL INTEGRATORS
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

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→ R with k≥ 1 , the resulting discrete equations define a generally implicit numerical integrator algorithm on T( k - 1 )Q× 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 Lde 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 Lde, 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.
Fil: Colombo, Leonardo Jesus. University of Michigan; Estados Unidos
Fil: Ferraro, Sebastián José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentina
Fil: Martin de Diego, David. Instituto de Ciencias Matemáticas; España. Consejo Superior de Investigaciones Científicas; España