Regular discretizations in optimal control theory

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

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 July 2013 Revised November 2013 Published December 2013

Abstract
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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.

Keywords: variational integrators, Optimal control, discrete Noether invariants., regularity analysis.

Mathematics Subject Classification: 37J60, 37M15, 49J15, 65P1.

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:

[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.

show all references

References:

[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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