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Regular discretizations in optimal control theory

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

    Citation:

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  • [1]

    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.

    [2]

    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.

    [3]

    R. Benito and D. Martín de Diego, Discrete vakonomic mechanics, J. Math. Phys., 46 (2005), 083521, 18 pp.doi: 10.1063/1.2008214.

    [4]

    A. M. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003.doi: 10.1007/b97376.

    [5]

    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.

    [6]

    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.

    [7]

    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.

    [8]

    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.

    [9]

    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.

    [10]

    F. Jiménez, M. Kobilarov and Martín de Diego, Discrete variational optimal control, accepted in Journal of Nonlinear Science, arXiv:1203.0580, 2012.

    [11]

    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.

    [12]

    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.

    [13]

    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.

    [14]

    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.

    [15]

    M. Leok, Foundations of Computational Geometric Mechanics, Ph.D. Thesis, California Institute of Technology, 2004.

    [16]

    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.

    [17]

    J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 317-514.doi: 10.1017/S096249290100006X.

    [18]

    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.

    [19]

    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.

    [20]

    P. Piccione and D. V. Tausk, Lagrangian and Hamiltonian formalism for constrained variational problems, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1417-1437.

    [21]

    M. West, Variational Integrators, Ph.D. Thesis, California Institute of Technology, 2004.

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