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.

Keywords: discrete and continuous control systems, discrete approximations, constrained optimal control, maximum principles

@article{COCV_2013__19_3_811_0, author = {Mordukhovich, Boris S. and Shvartsman, Ilya}, title = {Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {811--827}, publisher = {EDP-Sciences}, volume = {19}, number = {3}, year = {2013}, doi = {10.1051/cocv/2012034}, zbl = {1270.49017}, mrnumber = {3092363}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012034/} }

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 SP - 811 EP - 827 VL - 19 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012034/ DO - 10.1051/cocv/2012034 LA - en ID - COCV_2013__19_3_811_0 ER -

%0 Journal Article %A Mordukhovich, Boris S. %A Shvartsman, Ilya %T Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 811-827 %V 19 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012034/ %R 10.1051/cocv/2012034 %G en %F COCV_2013__19_3_811_0

Mordukhovich, Boris S.; 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, Volume 19 (2013) no. 3, pp. 811-827. doi : 10.1051/cocv/2012034. http://www.numdam.org/articles/10.1051/cocv/2012034/

[1] On the extension of the maximum principle by L.S. Pontryagin to discrete systems. Autom. Remote Control 27 (1966) 1878-1882. | MR | Zbl

and ,[2] Approximate maximum principle fot finite-difference control systems. Comput. Maths. Math. Phys. 28 (1988) 106-114. | MR | Zbl

,[3] Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33 (1995) 882-915. | MR | Zbl

,[4] Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006). | MR | Zbl

,[5] The approximate maximum principle in constrained optimal control. SIAM J. Control Optim. 43 (2004) 1037-1062. | MR | Zbl

and ,[6] Nonsmooth approximate maximum principle in optimal control. Proc. 50th IEEE Conf. Dec. Cont. Orlando, FL (2011).

and ,[7] Approximate discrete maximum principle for the discrete approximation of optimal periodic control problems, Int. J. Control 50 (1989) 1863-1871. | MR | Zbl

,[8] The Mathematical Theory of Optimal Processes. Wiley, New York (1962). | MR | Zbl

, , and ,[9] Convex Analysis. Princeton University Press, Princeton, NJ (1973). | MR | Zbl

,[10] Introduction to the Theory of Differential Inclusions. American Mathematical Society, Providence, RI (2002). | MR | Zbl

,[11] Optimal Control. BirkhĂ¤user, Boston (2000). | Zbl

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