Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints

Boris S. Mordukhovich; Ilya Shvartsman

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 3, page 811-827
  • ISSN: 1292-8119

Abstract

top
The paper studies discrete/finite-difference approximations of optimal control problems governed by continuous-time dynamical systems with endpoint constraints. Finite-difference systems, considered as parametric control problems with the decreasing step of discretization, occupy an intermediate position between continuous-time and discrete-time (with fixed steps) control processes and play a significant role in both qualitative and numerical aspects of optimal control. In this paper we derive an enhanced version of the Approximate Maximum Principle for finite-difference control systems, which is new even for problems with smooth endpoint constraints on trajectories and occurs to be the first result in the literature that holds for nonsmooth objectives and endpoint constraints. The results obtained establish necessary optimality conditions for constrained nonconvex finite-difference control systems and justify stability of the Pontryagin Maximum Principle for continuous-time systems under discrete approximations.

How to cite

top

Mordukhovich, Boris S., and Shvartsman, Ilya. "Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints." ESAIM: Control, Optimisation and Calculus of Variations 19.3 (2013): 811-827. <http://eudml.org/doc/272792>.

@article{Mordukhovich2013,
abstract = {The paper studies discrete/finite-difference approximations of optimal control problems governed by continuous-time dynamical systems with endpoint constraints. Finite-difference systems, considered as parametric control problems with the decreasing step of discretization, occupy an intermediate position between continuous-time and discrete-time (with fixed steps) control processes and play a significant role in both qualitative and numerical aspects of optimal control. In this paper we derive an enhanced version of the Approximate Maximum Principle for finite-difference control systems, which is new even for problems with smooth endpoint constraints on trajectories and occurs to be the first result in the literature that holds for nonsmooth objectives and endpoint constraints. The results obtained establish necessary optimality conditions for constrained nonconvex finite-difference control systems and justify stability of the Pontryagin Maximum Principle for continuous-time systems under discrete approximations.},
author = {Mordukhovich, Boris S., Shvartsman, Ilya},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {discrete and continuous control systems; discrete approximations; constrained optimal control; maximum principles; discrete/continuous control systems},
language = {eng},
number = {3},
pages = {811-827},
publisher = {EDP-Sciences},
title = {Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints},
url = {http://eudml.org/doc/272792},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Mordukhovich, Boris S.
AU - Shvartsman, Ilya
TI - Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 3
SP - 811
EP - 827
AB - The paper studies discrete/finite-difference approximations of optimal control problems governed by continuous-time dynamical systems with endpoint constraints. Finite-difference systems, considered as parametric control problems with the decreasing step of discretization, occupy an intermediate position between continuous-time and discrete-time (with fixed steps) control processes and play a significant role in both qualitative and numerical aspects of optimal control. In this paper we derive an enhanced version of the Approximate Maximum Principle for finite-difference control systems, which is new even for problems with smooth endpoint constraints on trajectories and occurs to be the first result in the literature that holds for nonsmooth objectives and endpoint constraints. The results obtained establish necessary optimality conditions for constrained nonconvex finite-difference control systems and justify stability of the Pontryagin Maximum Principle for continuous-time systems under discrete approximations.
LA - eng
KW - discrete and continuous control systems; discrete approximations; constrained optimal control; maximum principles; discrete/continuous control systems
UR - http://eudml.org/doc/272792
ER -

References

top
  1. [1] R. Gabasov and F.M. Kirillova, On the extension of the maximum principle by L.S. Pontryagin to discrete systems. Autom. Remote Control27 (1966) 1878–1882. Zbl0153.12902MR215632
  2. [2] B.S. Mordukhovich, Approximate maximum principle fot finite-difference control systems. Comput. Maths. Math. Phys.28 (1988) 106–114. Zbl0713.49037MR935743
  3. [3] B.S. Mordukhovich, Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim.33 (1995) 882–915. Zbl0844.49017MR1327242
  4. [4] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006). Zbl1100.49002MR2191744
  5. [5] B.S. Mordukhovich and I. Shvartsman, The approximate maximum principle in constrained optimal control. SIAM J. Control Optim.43 (2004) 1037–1062. Zbl1080.49017MR2114388
  6. [6] B.S. Mordukhovich and I. Shvartsman, Nonsmooth approximate maximum principle in optimal control. Proc. 50th IEEE Conf. Dec. Cont. Orlando, FL (2011). 
  7. [7] K. Nitka-Styczen, Approximate discrete maximum principle for the discrete approximation of optimal periodic control problems, Int. J. Control50 (1989) 1863–1871. Zbl0686.49010MR1032439
  8. [8] L.C. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes. Wiley, New York (1962). Zbl0117.31702MR166037
  9. [9] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, NJ (1973). Zbl0932.90001MR1451876
  10. [10] G.V. Smirnov, Introduction to the Theory of Differential Inclusions. American Mathematical Society, Providence, RI (2002). Zbl0992.34001MR1867542
  11. [11] R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000). Zbl1215.49002

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.