Variational integrators in discrete time-dependent optimal control theory

  1. Fernández, Antonio 3
  2. García, Pedro L. 1
  3. Sípols, Ana G. 2
  1. 1 Department of Mathematics, University of Salamanca
  2. 2 Department of Statistics and Operation Research, University Rey Juan Carlos, Campus de Fuenlabrada, Madrid
  3. 3 Department of Applied Mathematics, University of Salamanca
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Revista:
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

ISSN: 1578-7303,  1579-1505

Año de publicación: 2012

Volumen: 106

Número: 1

Páginas: 173-189

Tipo: Artículo

DOI: 10.1007/S13398-011-0037-3 SCOPUS: 2-s2.0-84856906084 DIALNET GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

Resumen

The discrete optimal control problem with discrete Lagrangian function L(tk,xαk,uik)(tk+1−tk) and constraintsφα≡xαk+1−xαktk+1−tk−fα(tk,xβk,uik)=0,1≤α,β≤n,1≤i≤mwhere xαk are the dynamical variables, uik are the control variables and t k is the time is studied. This problem is the discretization by the initial point of the differentiable optimal control problem with Lagrangian density L(t,xα,ui)dt and constraints φα≡x˙α−fα(t,xβ,ui)=0. The most remarkable fact of this discrete problem is that a part of the Euler–Lagrange equations of the unconstrained extended discrete Lagrangian L^(tk+1−tk)=(L+∑nα=1λαIk+1φα)(tk+1−tk) , I k+1 = (k, k + 1), λαIk+1 : Lagrange multipliers, degenerates into a constraint condition on the variables (tk,xαk,uik,λαIk+1) and that the associated Cartan 1-form Θ+L^(tk+1−tk) projects into a certain discrete bundle, in which this constraint condition in addition to the initial constraints of the problem define, under certain regularity hypothesis, a submanifold M. In this situation, a notion of variational integrator on M is introduced, that is characterized by a Cartan equation that assures its symplecticity. In the case of dΘ+L^(tk+1−tk)∣∣∣M being non singular (regular problems), we prove that these integrators can be locally constructed from a generating function which is expressed in terms of a discrete Pontryagin Hamiltonian. Finally, the theory is illustrated with two elementary examples for which we will construct variational integrators from generating functions.

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