Differential Equations and Nonlinear Mechanics

Optimal Control of Mechanical Systems

Vadim Azhmyakov

Full-text: Open access


In the present work, we consider a class of nonlinear optimal control problems, which can be called “optimal control problems in mechanics.” We deal with control systems whose dynamics can be described by a system of Euler-Lagrange or Hamilton equations. Using the variational structure of the solution of the corresponding boundary-value problems, we reduce the initial optimal control problem to an auxiliary problem of multiobjective programming. This technique makes it possible to apply some consistent numerical approximations of a multiobjective optimization problem to the initial optimal control problem. For solving the auxiliary problem, we propose an implementable numerical algorithm.

Article information

Differ. Equ. Nonlinear Mech., Volume 2007 (2007), Article ID 018735, 16 pages.

Received: 7 February 2007
Accepted: 10 May 2007
First available in Project Euclid: 26 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Azhmyakov, Vadim. Optimal Control of Mechanical Systems. Differ. Equ. Nonlinear Mech. 2007 (2007), Article ID 018735, 16 pages. doi:10.1155/2007/18735. https://projecteuclid.org/euclid.ijde/1485399783

Export citation


  • J. Baillieul, “The geometry of controlled mechanical systems,” in Mathematical Control Theory, J. Baillieul and J. C. Willems, Eds., pp. 322–354, Springer, New York, NY, USA, 1999. MR1661476
  • A. M. Bloch and P. E. Crouch, “Optimal control, optimization, and analytical mechanics,” in Mathematical Control Theory, J. Baillieul and J. C. Willems, Eds., pp. 268–321, Springer, New York, NY, USA, 1999. MR1661475
  • R. W. Brockett, “Control theory and analytical mechanics,” in Conference on Geometric Control Theory (Moffett Field, Calif, 1976), C. F. Martin and R. Hermann, Eds., pp. 1–48, Math. Sci. Press, Brookline, Mass, USA, 1977. MR0484621
  • H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems, Springer, New York, NY, USA, 1990. MR1047663
  • V. Azhmyakov, “A numerically stable method for convex optimal control problems,” Journal of Nonlinear and Convex Analysis, vol. 5, no. 1, pp. 1–18, 2004. MR2049053
  • V. Azhmyakov and W. Schmidt, “Approximations of relaxed optimal control problems,” Journal of Optimization Theory and Applications, vol. 130, no. 1, pp. 61–78, 2006. MR2275354
  • E. Polak, Optimization, vol. 124 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1997. MR1454128
  • R. Pytlak, Numerical Methods for Optimal Control Problems with State Constraints, vol. 1707 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1999. MR1713434
  • K. L. Teo, C. J. Goh, and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, vol. 55 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, UK; John Wiley & Sons, New York, NY, USA, 1991. MR1153024
  • R. E. Bellman and S. E. Dreyfus, Applied Dynamic Programming, Princeton University Press, Princeton, NJ, USA, 1962. MR0140369
  • A. E. Bryson Jr. and Y. C. Ho, Applied Optimal Control, John Wiley & Sons, New York, NY, USA, 1975. MR0446628
  • J. F. Bonnans, “On an algorithm for optimal control using Pontryagin's maximum principle,” SIAM Journal on Control and Optimization, vol. 24, no. 3, pp. 579–588, 1986. MR838058
  • Y. Sakawa, Y. Shindo, and Y. Hashimoto, “Optimal control of a rotary crane,” Journal of Optimization Theory and Applications, vol. 35, no. 4, pp. 535–557, 1981. MR663330
  • S. J. Wright, “Interior point methods for optimal control of discrete time systems,” Journal of Optimization Theory and Applications, vol. 77, no. 1, pp. 161–187, 1993. MR1222789
  • M. Fukushima and Y. Yamamoto, “A second-order algorithm for continuous-time nonlinear optimal control problems,” IEEE Transactions on Automatic Control, vol. 31, no. 7, pp. 673–676, 1986.
  • C. Büskens and H. Maurer, “SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control,” Journal of Computational and Applied Mathematics, vol. 120, no. 1-2, pp. 85–108, 2000. MR1781710
  • F.-S. Kupfer and E. W. Sachs, “Reduced SQP methods for nonlinear heat conduction control problems,” in Optimal Control (Freiburg, 1991), vol. 111 of International Series of Numerical Mathematics, pp. 145–160, Birkhäuser, Basel, Switzerland, 1993. MR1298004
  • R. Abraham, Foundations of Mechanics, W. A. Benjamin, New York, NY, USA, 1967.
  • V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, NY, USA, 1978. MR0690288
  • A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, vol. 6 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1979. MR528295
  • F. R. Gantmakher, Lectures on Analytical Mechanics, Nauka, Moscow, Russia, 1966.
  • J. C. Dunn, “On state constraint representations and mesh-dependent gradient projection convergence rates for optimal control problems,” SIAM Journal on Control and Optimization, vol. 39, no. 4, pp. 1082–1111, 2000. MR1814268
  • R. Fletcher, Practical Methods of Optimization, John Wiley & Sons, New York, NY, USA, 1991.
  • Y. Sawaragi, H. Nakayama, and T. Tanino, Theory of Multiobjective Optimization, vol. 176 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, 1985. MR807529
  • F. H. Clarke, Optimization and Nonsmooth Analysis, vol. 5 of Classics in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 2nd edition, 1990. MR1058436
  • V. Azhmyakov, “On optimal control of mechanical systems,” Tech. Rep. 22, EMA University of Greifswald, Greifswald, Germany, 2003.