# 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

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topMordukhovich, 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

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- [6] B.S. Mordukhovich and I. Shvartsman, Nonsmooth approximate maximum principle in optimal control. Proc. 50th IEEE Conf. Dec. Cont. Orlando, FL (2011).
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- [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] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, NJ (1973). Zbl0932.90001MR1451876
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