Regular discretizations in optimal control theory

Antonio Fernández; Pedro L. García

doi:10.3934/jgm.2013.5.415

Article Contents

2013, Volume 5, Issue 4: 415-432. Doi: 10.3934/jgm.2013.5.415

This issue Previous Article Lagrange-Poincaré reduction in affine principal bundles Next Article A geometric approach to discrete connections on principal bundles

Regular discretizations in optimal control theory

Antonio Fernández^1, and
Pedro L. García^2,

1.
Department of Applied Mathematics, University of Salamanca, Salamanca 37008
2.
IUFFyM-USAL and Real Academia de Ciencias, Plaza de la Merced 1-4, 37008 Salamanca

Received: June 30, 2013

Revised: October 31, 2013

Published: November 2013

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Abstract

Given a regular optimal control problem with Lagrangian density $\mathcal{L} (t,x^\alpha,u^i)dt$ and constraints $\phi^\alpha\equiv \dot x^\alpha-f^\alpha(t,x^\beta,u^i)=0$, $1\le \alpha,\beta\le n$, $1\le i\le m$, we study the discretization defined for each pair $I_k=(k-1,k)$, $1\le k\le N$ by the functions: $$ \begin{aligned} L_{I_k}(x^\beta_{k-1},u^i_{k-1},x^\beta_k,u^i_k) = \mathcal{L} (t_{I_k},x^\alpha_{I_k},u^i_{I_k})h\\ \phi ^\alpha_{I_k}(x^\beta_{k-1},u^i_{k-1},x^\beta_k,u^i_k)=&\left(\frac{x^\alpha_{k}-x^\alpha_{k-1}}{h}-f^\alpha(t_{I_k},x^\beta_{I_k},u^i_{I_k}) \right)h \end{aligned} $$ where $t_k-t_{k-1}=h\in\mathbb{R}^+$ is fixed, and where: $$ \begin{aligned} t_{I_k}=&\epsilon t_{k-1}+(1-\epsilon) t_k=t_0+h(k-\epsilon)\\ x^\alpha_{I_k}=&\epsilon x^\alpha_{k-1}+(1-\epsilon)x^\alpha_k\\ u^i_{I_k}=&\epsilon u^i_{k-1}+(1-\epsilon)u^i_k \end{aligned}\quad 0\le \epsilon \le 1. $$ We prove that for $\epsilon\ne 0, 1$, the discrete Lagrange problems so defined are non singular in the sense of the discrete vakonomic mechanics admitting as infinitesimal symmetries the vector fields $D^i_k=\frac1{\epsilon h}\left(-\frac\epsilon{1-\epsilon}\right)^k\frac{\partial}{\partial u^i_k}$, $1\le i\le m$. The Noether invariants associated to these symmetries are used to construct the corresponding variational integrators. Finally, the theory is illustrated with two examples: the optimal regulator problem and the Heisenberg optimal control problem.

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Mathematics Subject Classification: 37J60, 37M15, 49J15, 65P10.

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