    December  2013, 5(4): 415-432. doi: 10.3934/jgm.2013.5.415

## Regular discretizations in optimal control theory

 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  July 2013 Revised  November 2013 Published  December 2013

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.
Citation: Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415
##### References:
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##### References:
  M. Aldeen and F. Crusca, Quadratic cost function design for linear optimal control systems, in TENCON '92. "Technology Enabling Tomorrow: Computers, Communications and Automation towards the 21st Century.' 1992 IEEE Region 10 International Conference, IEEE, 1992, 958-962. doi: 10.1109/TENCON.1992.271828. Google Scholar  V. I. Arnol'd, V. V. Kozlov and A. I. Neĭshtadt, Dynamical Systems. III, Encyclopaedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-61551-1.  Google Scholar  R. Benito and D. Martín de Diego, Discrete vakonomic mechanics, J. Math. Phys., 46 (2005), 083521, 18 pp. doi: 10.1063/1.2008214.  Google Scholar  A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376.  Google Scholar  A. Fernández, P. L García and Ana G Sípols, Variational integrators in discrete time-dependent optimal control theory, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 173-189. doi: 10.1007/s13398-011-0037-3.  Google Scholar  P. L. García, A. García and C. Rodrigo, Cartan forms for first order constrained variational problems, J. Geom. Phys., 56 (2006), 571-610. doi: 10.1016/j.geomphys.2005.04.002.  Google Scholar  P. L. García, A. Fernández and C. Rodrigo, Variational integrators for discrete Lagrange problems, J. Geom. Mech., 2 (2010), 343-374. doi: 10.3934/jgm.2010.2.343.  Google Scholar  V. Guibout and A. Bloch, A discrete maximum principle for solving optimal control problems, in CDC. 43rd IEEE Conference on Decision and Control, 2004, Volume 2, IEEE, 2004, 1806-1811. doi: 10.1109/CDC.2004.1430309. Google Scholar  F. Jiménez and D. Martín de Diego, A geometric approach to discrete mechanics for optimal control theory, in Proceedings of the IEEE Conference on Decision and Control, Atlanta, Georgia, USA, 2010, 5426-5431. Google Scholar  F. Jiménez, M. Kobilarov and Martín de Diego, Discrete variational optimal control, accepted in Journal of Nonlinear Science, arXiv:1203.0580, 2012. Google Scholar  M. Kobilarov and J. E. Marsden, Discrete geometric optimal control on Lie groups, IEEE Transactions on Robotics, 27 (2011), 641-655. doi: 10.1109/TRO.2011.2139130. Google Scholar  T. Lee, N. McClamroch and M. Leok, Optimal control of a rigid body using geometrically exact computations on SE(3), in Proceedings of the IEEE Conference on Decision and Control, (2006), 2710-2715. doi: 10.1109/CDC.2006.376687. Google Scholar  M. de León, D. Martín de Diego and A. Santamaría-Merino, Discrete variational integrators and optimal control theory, Adv. Comput. Math., 26 (2006), 251-268. doi: 10.1007/s10444-004-4093-5.  Google Scholar  M. de León, J. C. Marrero and D. Martín de Diego, Vakonomic mechanics versus non-holonomic mechanics: A unified geometrical approach, J. Geom. Phys., 35 (2000), 126-144. doi: 10.1016/S0393-0440(00)00004-8.  Google Scholar  M. Leok, Foundations of Computational Geometric Mechanics, Ph.D. Thesis, California Institute of Technology, 2004. Google Scholar  J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. in Math. Phys., 199 (1998), 351-395. doi: 10.1007/s002200050505.  Google Scholar  J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 317-514. doi: 10.1017/S096249290100006X.  Google Scholar  J. Moser and A. P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494.  Google Scholar  S. Ober-Blöbaum, O. Junge and J. E. Marsden, Discrete mechanics and optimal control: An analysis, ESAIM Control Optim. Calc. Var., 17 (2011), 322-352. doi: 10.1051/cocv/2010012.  Google Scholar  P. Piccione and D. V. Tausk, Lagrangian and Hamiltonian formalism for constrained variational problems, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1417-1437. Google Scholar  M. West, Variational Integrators, Ph.D. Thesis, California Institute of Technology, 2004. Google Scholar
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