Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 811-827.

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.

DOI : https://doi.org/10.1051/cocv/2012034
Classification : 49K15,  49M25,  49J52,  49J53,  93C55
Mots clés : discrete and continuous control systems, discrete approximations, constrained optimal control, maximum principles
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author = {Mordukhovich, Boris S. and Shvartsman, Ilya},
title = {Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints},
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pages = {811--827},
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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, Tome 19 (2013) no. 3, pp. 811-827. doi : 10.1051/cocv/2012034. http://www.numdam.org/articles/10.1051/cocv/2012034/

[1] R. Gabasov and F.M. Kirillova, On the extension of the maximum principle by L.S. Pontryagin to discrete systems. Autom. Remote Control 27 (1966) 1878-1882. | MR 215632 | Zbl 0153.12902

[2] B.S. Mordukhovich, Approximate maximum principle fot finite-difference control systems. Comput. Maths. Math. Phys. 28 (1988) 106-114. | MR 935743 | Zbl 0713.49037

[3] B.S. Mordukhovich, Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33 (1995) 882-915. | MR 1327242 | Zbl 0844.49017

[4] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006). | MR 2191744 | Zbl 1100.49002

[5] B.S. Mordukhovich and I. Shvartsman, The approximate maximum principle in constrained optimal control. SIAM J. Control Optim. 43 (2004) 1037-1062. | MR 2114388 | Zbl 1080.49017

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

[7] K. Nitka-Styczen, Approximate discrete maximum principle for the discrete approximation of optimal periodic control problems, Int. J. Control 50 (1989) 1863-1871. | MR 1032439 | Zbl 0686.49010

[8] L.C. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes. Wiley, New York (1962). | MR 166037 | Zbl 0117.31702

[9] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, NJ (1973). | MR 1451876 | Zbl 0932.90001

[10] G.V. Smirnov, Introduction to the Theory of Differential Inclusions. American Mathematical Society, Providence, RI (2002). | MR 1867542 | Zbl 0992.34001

[11] R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000). | Zbl 1215.49002

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