Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T05:53:17.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximate maximum principle for discrete approximations ofoptimal control systems with nonsmooth objectives and endpoint constraints∗

Published online by Cambridge University Press:  17 May 2013

Boris S. Mordukhovich  and
Ilya Shvartsman
Boris S. Mordukhovich
Affiliation:
Department of Mathematics, Wayne State University Detroit, MI 48202, U.S.A.. [email protected]
Ilya Shvartsman
Affiliation:
Department of Mathematics and Computer Science, Pennsylvania State University Harrisburg, Middletown, PA 17110, U.S.A.; [email protected]
Get access

Abstract

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

Keywords

Discrete and continuous control systemsdiscrete approximationsconstrained optimal controlmaximum principles
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 811 - 827
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Gabasov, R. and Kirillova, F.M., On the extension of the maximum principle by L.S. Pontryagin to discrete systems. Autom. Remote Control 27 (1966) 1878–1882. Google Scholar
Mordukhovich, B.S., Approximate maximum principle fot finite-difference control systems. Comput. Maths. Math. Phys. 28 (1988) 106–114. Google Scholar
Mordukhovich, B.S., Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions. SIAM J. Control Optim. 33 (1995) 882–915. Google Scholar
B.S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006).
Mordukhovich, B.S. and Shvartsman, I., The approximate maximum principle in constrained optimal control. SIAM J. Control Optim. 43 (2004) 1037–1062. Google Scholar
B.S. Mordukhovich and I. Shvartsman, Nonsmooth approximate maximum principle in optimal control. Proc. 50th IEEE Conf. Dec. Cont. Orlando, FL (2011).
Nitka-Styczen, K., Approximate discrete maximum principle for the discrete approximation of optimal periodic control problems, Int. J. Control 50 (1989) 1863–1871. Google Scholar
L.C. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes. Wiley, New York (1962).
R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, NJ (1973).
G.V. Smirnov, Introduction to the Theory of Differential Inclusions. American Mathematical Society, Providence, RI (2002).
R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000).