2015, 35(9): 4193-4223. doi: 10.3934/dcds.2015.35.4193

High order variational integrators in the optimal control of mechanical systems

1. 

IMUVA, Universidad de Valladolid, 47011 Valladolid, Spain

2. 

Department of Mathematics, University of Paderborn, 33098 Paderborn, Germany

3. 

Sorbonne Universités, UPMC Univ. Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Institut Universitaire de France, F-75005, Paris, France

Received  May 2014 Revised  October 2014 Published  April 2015

In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are momentum preserving and symplectic. In this article, we develop two high order variational integrators which distinguish themselves in the dimension of the underling space of approximation and we investigate their application to finite-dimensional optimal control problems posed with mechanical systems. The convergence of state and control variables of the approximated problem is shown. Furthermore, by analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that, for these particular integrators, dualization and discretization commute.
Citation: Cédric M. Campos, Sina Ober-Blöbaum, Emmanuel Trélat. High order variational integrators in the optimal control of mechanical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4193-4223. doi: 10.3934/dcds.2015.35.4193
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics,, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, (1978).

[2]

G. Ferreyra, R. Gardner and H. Sussmann, A hamitonian approach to strong minima in optimal control,, in Proceedings of Symposia in Pure Mathematics, (1999), 11. doi: 10.1090/pspum/064/1654537.

[3]

J. T. Betts, Survey of numerical methods for trajectory optimization,, J. Guid. Contr. Dynam., 21 (1998), 193. doi: 10.2514/2.4231.

[4]

J. F. Bonnans and J. Laurent-Varin, Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control,, Numer. Math., 103 (2006), 1. doi: 10.1007/s00211-005-0661-y.

[5]

B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control,, ESAIM Control Optim. Calc. Var., 13 (2007), 207. doi: 10.1051/cocv:2007012.

[6]

C. M. Campos, High order variational integrators: A polynomial approach,, in Advances in Differential Equations and Applications, (2014), 249. doi: 10.1007/978-3-319-06953-1_24.

[7]

C. M. Campos, H. Cendra, V. A. Díaz and D. Martín de Diego, Discrete Lagrange-d'Alembert-Poincaré equations for Euler's disk,, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 225. doi: 10.1007/s13398-011-0053-3.

[8]

C. M. Campos, O. Junge and S. Ober-Blöbaum, Higher order variational time discretization of optimal control problems,, in 20th International Symposium on Mathematical Theory of Networks and Systems, (2012).

[9]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems,, SIAM J. Control Optim., 47 (2008), 1078. doi: 10.1137/060663003.

[10]

L. Colombo, F. Jiménez and D. Martín de Diego, Discrete second-order Euler-Poincaré equations. Applications to optimal control,, Int. J. Geom. Methods Mod. Phys., 9 (2012). doi: 10.1142/S0219887812500375.

[11]

J. Cortés and S. Martínez, Non-holonomic integrators,, Nonlinearity, 14 (2001), 1365. doi: 10.1088/0951-7715/14/5/322.

[12]

A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control,, Math. Comp., 70 (2001), 173. doi: 10.1090/S0025-5718-00-01184-4.

[13]

A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order Runge-Kutta approximations in control constrained optimal control,, SIAM J. Numer. Anal., 38 (2000), 202. doi: 10.1137/S0036142999351765.

[14]

R. C. Fetecau, J. E. Marsden, M. Ortiz and M. West, Nonsmooth Lagrangian mechanics and variational collision integrators,, SIAM J. Appl. Dyn. Syst., 2 (2003), 381. doi: 10.1137/S1111111102406038.

[15]

M. Gerdts, Optimal Control of ODEs and DAEs,, de Gruyter Textbook, (2012). doi: 10.1515/9783110249996.

[16]

Q. Gong, I. M. Ross, W. Kang and F. Fahroo, Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control,, Comput. Optim. Appl., 41 (2008), 307. doi: 10.1007/s10589-007-9102-4.

[17]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system,, Numer. Math., 87 (2000), 247. doi: 10.1007/s002110000178.

[18]

W. W. Hager, Numerical analysis in optimal control,, in Optimal control of complex structures (Oberwolfach, (2002), 83. doi: 10.1007/978-3-0348-8148-7_7.

[19]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, vol. 31 of Springer Series in Computational Mathematics,, Springer, (2006).

[20]

J. Hall and M. Leok, Spectral variational integrators, 2012,, , ().

[21]

M. Herty, L. Pareschi and S. Steffensen, Implicit-explicit Runge-Kutta schemes for numerical discretization of optimal control problems,, SIAM J. Numer. Anal., 51 (2013), 1875. doi: 10.1137/120865045.

[22]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids,, J. Nonlinear Sci., 18 (2008), 221. doi: 10.1007/s00332-007-9012-8.

[23]

E. R. Johnson and T. D. Murphey, Dangers of two-point holonomic constraints for variational integrators,, in Proceedings of the 2009 conference on American Control Conference, (2009), 4723. doi: 10.1109/ACC.2009.5160488.

[24]

C. Kane, J. E. Marsden, M. Ortiz and M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems,, Internat. J. Numer. Methods Engrg., 49 (2000), 1295. doi: 10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W.

[25]

M. Kobilarov, J. E. Marsden and G. S. Sukhatme, Geometric discretization of nonholonomic systems with symmetries,, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 61. doi: 10.3934/dcdss.2010.3.61.

[26]

M. Leok, Foundations of Computational Geometric Mechanics,, PhD thesis, (2004), 03022004.

[27]

M. Leok and T. Shingel, General techniques for constructing variational integrators,, Frontiers of Mathematics in China, 7 (2012), 273. doi: 10.1007/s11464-012-0190-9.

[28]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators,, Arch. Ration. Mech. Anal., 167 (2003), 85. doi: 10.1007/s00205-002-0212-y.

[29]

A. Lew, J. E. Marsden, M. Ortiz and M. West, An overview of variational integrators,, in Finite Element Methods: 1970's and Beyond, (2004), 98.

[30]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Variational time integrators,, Internat. J. Numer. Methods Engrg., 60 (2004), 153. doi: 10.1002/nme.958.

[31]

S. Leyendecker, J. E. Marsden and M. Ortiz, Variational integrators for constrained dynamical systems,, ZAMM Z. Angew. Math. Mech., 88 (2008), 677. doi: 10.1002/zamm.200700173.

[32]

S. Leyendecker and S. Ober-Blöbaum, A variational approach to multirate integration for constrained systems,, in Multibody dynamics, (2013), 97. doi: 10.1007/978-94-007-5404-1_5.

[33]

S. Leyendecker, S. Ober-Blöbaum, J. E. Marsden and M. Ortiz, Discrete mechanics and optimal control for constrained systems,, Optimal Control Appl. Methods, 31 (2010), 505. doi: 10.1002/oca.912.

[34]

R. S. MacKay, Some aspects of the dynamics and numerics of Hamiltonian systems,, in The dynamics of numerics and the numerics of dynamics (Bristol, (1992), 137.

[35]

J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs,, Comm. Math. Phys., 199 (1998), 351. doi: 10.1007/s002200050505.

[36]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numer., 10 (2001), 357. doi: 10.1017/S096249290100006X.

[37]

H. Maurer and N. P. Osmolovskii, Second order sufficient conditions for time-optimal bang-bang control,, SIAM J. Control Optim., 42 (2004), 2239. doi: 10.1137/S0363012902402578.

[38]

S. Ober-Blöbaum, Galerkin variational integrators and modified symplectic Runge-Kutta methods, 2014,, Submitted., ().

[39]

S. Ober-Blöbaum, O. Junge and J. Marsden, Discrete mechanics and optimal control: An analysis,, ESAIM Control Optim. Calc. Var., 17 (2011), 322. doi: 10.1051/cocv/2010012.

[40]

S. Ober-Blöbaum and N. Saake, Construction and analysis of higher order galerkin variational integrators, 2014,, Discrete mechanics and optimal control: An analysis, (2014). doi: 10.1007/s10444-014-9394-8.

[41]

S. Ober-Blöbaum, M. Tao, M. Cheng, H. Owhadi and J. E. Marsden, Variational integrators for electric circuits,, J. Comput. Phys., 242 (2013), 498. doi: 10.1016/j.jcp.2013.02.006.

[42]

I. M. Ross, A roadmap for optimal control: The right way to commute,, Ann. N.Y. Acad., 1065 (2005), 210. doi: 10.1196/annals.1370.015.

[43]

I. M. Ross and F. Fahroo, Legendre pseudospectral approximations of optimal control problems,, in New trends in nonlinear dynamics and control, (2003), 327. doi: 10.1007/978-3-540-45056-6_21.

[44]

J. M. Sanz-Serna, Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control and more, 2014,, URL , ().

[45]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, vol. 7 of Applied Mathematics and Mathematical Computation,, Chapman & Hall, (1994).

[46]

A. Stern and E. Grinspun, Implicit-explicit variational integration of highly oscillatory problems,, Multiscale Model. Simul., 7 (2009), 1779. doi: 10.1137/080732936.

[47]

Y. B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpretation,, Mat. Model., 2 (1990), 78.

[48]

M. Tao, H. Owhadi and J. E. Marsden, Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging,, Multiscale Model. Simul., 8 (2010), 1269. doi: 10.1137/090771648.

[49]

E. Trélat, Some properties of the value function and its level sets for affine control systems with quadratic cost,, J. Dynam. Control Systems, 6 (2000), 511. doi: 10.1023/A:1009552511132.

[50]

E. Trélat, Contrôle Optimal,, Mathématiques Concrètes. [Concrete Mathematics], (2005).

[51]

E. Trélat, Optimal control and applications to aerospace: Some results and challenges,, J. Optim. Theory Appl., 154 (2012), 713. doi: 10.1007/s10957-012-0050-5.

[52]

A. Walther, Automatic differentiation of explicit Runge-Kutta methods for optimal control,, Comput. Optim. Appl., 36 (2007), 83. doi: 10.1007/s10589-006-0397-3.

[53]

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods,, SIAM Rev., 47 (2005), 197. doi: 10.1137/S0036144503432862.

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics,, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, (1978).

[2]

G. Ferreyra, R. Gardner and H. Sussmann, A hamitonian approach to strong minima in optimal control,, in Proceedings of Symposia in Pure Mathematics, (1999), 11. doi: 10.1090/pspum/064/1654537.

[3]

J. T. Betts, Survey of numerical methods for trajectory optimization,, J. Guid. Contr. Dynam., 21 (1998), 193. doi: 10.2514/2.4231.

[4]

J. F. Bonnans and J. Laurent-Varin, Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control,, Numer. Math., 103 (2006), 1. doi: 10.1007/s00211-005-0661-y.

[5]

B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control,, ESAIM Control Optim. Calc. Var., 13 (2007), 207. doi: 10.1051/cocv:2007012.

[6]

C. M. Campos, High order variational integrators: A polynomial approach,, in Advances in Differential Equations and Applications, (2014), 249. doi: 10.1007/978-3-319-06953-1_24.

[7]

C. M. Campos, H. Cendra, V. A. Díaz and D. Martín de Diego, Discrete Lagrange-d'Alembert-Poincaré equations for Euler's disk,, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 225. doi: 10.1007/s13398-011-0053-3.

[8]

C. M. Campos, O. Junge and S. Ober-Blöbaum, Higher order variational time discretization of optimal control problems,, in 20th International Symposium on Mathematical Theory of Networks and Systems, (2012).

[9]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems,, SIAM J. Control Optim., 47 (2008), 1078. doi: 10.1137/060663003.

[10]

L. Colombo, F. Jiménez and D. Martín de Diego, Discrete second-order Euler-Poincaré equations. Applications to optimal control,, Int. J. Geom. Methods Mod. Phys., 9 (2012). doi: 10.1142/S0219887812500375.

[11]

J. Cortés and S. Martínez, Non-holonomic integrators,, Nonlinearity, 14 (2001), 1365. doi: 10.1088/0951-7715/14/5/322.

[12]

A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control,, Math. Comp., 70 (2001), 173. doi: 10.1090/S0025-5718-00-01184-4.

[13]

A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order Runge-Kutta approximations in control constrained optimal control,, SIAM J. Numer. Anal., 38 (2000), 202. doi: 10.1137/S0036142999351765.

[14]

R. C. Fetecau, J. E. Marsden, M. Ortiz and M. West, Nonsmooth Lagrangian mechanics and variational collision integrators,, SIAM J. Appl. Dyn. Syst., 2 (2003), 381. doi: 10.1137/S1111111102406038.

[15]

M. Gerdts, Optimal Control of ODEs and DAEs,, de Gruyter Textbook, (2012). doi: 10.1515/9783110249996.

[16]

Q. Gong, I. M. Ross, W. Kang and F. Fahroo, Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control,, Comput. Optim. Appl., 41 (2008), 307. doi: 10.1007/s10589-007-9102-4.

[17]

W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system,, Numer. Math., 87 (2000), 247. doi: 10.1007/s002110000178.

[18]

W. W. Hager, Numerical analysis in optimal control,, in Optimal control of complex structures (Oberwolfach, (2002), 83. doi: 10.1007/978-3-0348-8148-7_7.

[19]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, vol. 31 of Springer Series in Computational Mathematics,, Springer, (2006).

[20]

J. Hall and M. Leok, Spectral variational integrators, 2012,, , ().

[21]

M. Herty, L. Pareschi and S. Steffensen, Implicit-explicit Runge-Kutta schemes for numerical discretization of optimal control problems,, SIAM J. Numer. Anal., 51 (2013), 1875. doi: 10.1137/120865045.

[22]

D. Iglesias, J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids,, J. Nonlinear Sci., 18 (2008), 221. doi: 10.1007/s00332-007-9012-8.

[23]

E. R. Johnson and T. D. Murphey, Dangers of two-point holonomic constraints for variational integrators,, in Proceedings of the 2009 conference on American Control Conference, (2009), 4723. doi: 10.1109/ACC.2009.5160488.

[24]

C. Kane, J. E. Marsden, M. Ortiz and M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems,, Internat. J. Numer. Methods Engrg., 49 (2000), 1295. doi: 10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W.

[25]

M. Kobilarov, J. E. Marsden and G. S. Sukhatme, Geometric discretization of nonholonomic systems with symmetries,, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 61. doi: 10.3934/dcdss.2010.3.61.

[26]

M. Leok, Foundations of Computational Geometric Mechanics,, PhD thesis, (2004), 03022004.

[27]

M. Leok and T. Shingel, General techniques for constructing variational integrators,, Frontiers of Mathematics in China, 7 (2012), 273. doi: 10.1007/s11464-012-0190-9.

[28]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators,, Arch. Ration. Mech. Anal., 167 (2003), 85. doi: 10.1007/s00205-002-0212-y.

[29]

A. Lew, J. E. Marsden, M. Ortiz and M. West, An overview of variational integrators,, in Finite Element Methods: 1970's and Beyond, (2004), 98.

[30]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Variational time integrators,, Internat. J. Numer. Methods Engrg., 60 (2004), 153. doi: 10.1002/nme.958.

[31]

S. Leyendecker, J. E. Marsden and M. Ortiz, Variational integrators for constrained dynamical systems,, ZAMM Z. Angew. Math. Mech., 88 (2008), 677. doi: 10.1002/zamm.200700173.

[32]

S. Leyendecker and S. Ober-Blöbaum, A variational approach to multirate integration for constrained systems,, in Multibody dynamics, (2013), 97. doi: 10.1007/978-94-007-5404-1_5.

[33]

S. Leyendecker, S. Ober-Blöbaum, J. E. Marsden and M. Ortiz, Discrete mechanics and optimal control for constrained systems,, Optimal Control Appl. Methods, 31 (2010), 505. doi: 10.1002/oca.912.

[34]

R. S. MacKay, Some aspects of the dynamics and numerics of Hamiltonian systems,, in The dynamics of numerics and the numerics of dynamics (Bristol, (1992), 137.

[35]

J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs,, Comm. Math. Phys., 199 (1998), 351. doi: 10.1007/s002200050505.

[36]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numer., 10 (2001), 357. doi: 10.1017/S096249290100006X.

[37]

H. Maurer and N. P. Osmolovskii, Second order sufficient conditions for time-optimal bang-bang control,, SIAM J. Control Optim., 42 (2004), 2239. doi: 10.1137/S0363012902402578.

[38]

S. Ober-Blöbaum, Galerkin variational integrators and modified symplectic Runge-Kutta methods, 2014,, Submitted., ().

[39]

S. Ober-Blöbaum, O. Junge and J. Marsden, Discrete mechanics and optimal control: An analysis,, ESAIM Control Optim. Calc. Var., 17 (2011), 322. doi: 10.1051/cocv/2010012.

[40]

S. Ober-Blöbaum and N. Saake, Construction and analysis of higher order galerkin variational integrators, 2014,, Discrete mechanics and optimal control: An analysis, (2014). doi: 10.1007/s10444-014-9394-8.

[41]

S. Ober-Blöbaum, M. Tao, M. Cheng, H. Owhadi and J. E. Marsden, Variational integrators for electric circuits,, J. Comput. Phys., 242 (2013), 498. doi: 10.1016/j.jcp.2013.02.006.

[42]

I. M. Ross, A roadmap for optimal control: The right way to commute,, Ann. N.Y. Acad., 1065 (2005), 210. doi: 10.1196/annals.1370.015.

[43]

I. M. Ross and F. Fahroo, Legendre pseudospectral approximations of optimal control problems,, in New trends in nonlinear dynamics and control, (2003), 327. doi: 10.1007/978-3-540-45056-6_21.

[44]

J. M. Sanz-Serna, Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control and more, 2014,, URL , ().

[45]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, vol. 7 of Applied Mathematics and Mathematical Computation,, Chapman & Hall, (1994).

[46]

A. Stern and E. Grinspun, Implicit-explicit variational integration of highly oscillatory problems,, Multiscale Model. Simul., 7 (2009), 1779. doi: 10.1137/080732936.

[47]

Y. B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpretation,, Mat. Model., 2 (1990), 78.

[48]

M. Tao, H. Owhadi and J. E. Marsden, Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging,, Multiscale Model. Simul., 8 (2010), 1269. doi: 10.1137/090771648.

[49]

E. Trélat, Some properties of the value function and its level sets for affine control systems with quadratic cost,, J. Dynam. Control Systems, 6 (2000), 511. doi: 10.1023/A:1009552511132.

[50]

E. Trélat, Contrôle Optimal,, Mathématiques Concrètes. [Concrete Mathematics], (2005).

[51]

E. Trélat, Optimal control and applications to aerospace: Some results and challenges,, J. Optim. Theory Appl., 154 (2012), 713. doi: 10.1007/s10957-012-0050-5.

[52]

A. Walther, Automatic differentiation of explicit Runge-Kutta methods for optimal control,, Comput. Optim. Appl., 36 (2007), 83. doi: 10.1007/s10589-006-0397-3.

[53]

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods,, SIAM Rev., 47 (2005), 197. doi: 10.1137/S0036144503432862.

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B. Bonnard, J.-B. Caillau, E. Trélat. Geometric optimal control of elliptic Keplerian orbits. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 929-956. doi: 10.3934/dcdsb.2005.5.929

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Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001

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