Paper Info
Title
The Approximate Maximum Principle in Constrained Optimal Control
Abstract
This paper concerns optimal control problems for dynamical systems described by a parametric family of discrete/finite-difference approximations of continuous-time control systems. Control theory for parametric systems governed by discrete approximations plays an important role in both qualitative and numerical aspects of optimal control and occupies an intermediate position in dynamic optimization: between optimal control of discrete-time (with fixed steps) and continuous-time control systems. The central result in optimal control of discrete approximation systems is the approximate maximum principle (AMP), which gives the necessary optimality condition in a perturbed maximum principle form with no a priori convexity assumptions and thus ensures the stability of the Pontryagin maximum principle (PMP) under discrete approximation procedures. The AMP has been justified for optimal control problems of smooth dynamical systems with endpoint constraints under some properness assumption imposed on the sequence of optimal controls. In this paper we show, by a series of counterexamples, that the properness assumption is essential for the validity of the AMP, and that the AMP does not hold, in its expected (lower) subdifferential form, for nonsmooth problems. Moreover, a new upper subdifferential form of the AMP is established for ordinary and time-delay control systems. The results obtained surprisingly solve (in both negative and positive directions) a long-standing and well-recognized question about the possibility of extending the AMP to nonsmooth control problems, for which the affirmative answer has been expected in the conventional lower subdifferential form.
Year
DOI
Venue
2004
10.1137/S0363012903433012
SIAM J. Control and Optimization
Keywords
Field
DocType
optimal control problem,continuous-time control system,constrained optimal control,properness assumption,control theory,conventional lower subdifferential form,time-delay control system,optimal control,approximate maximum principle,paper concerns optimal control,control problem,discrete approximation,maximum principle,constrained optimization
Mathematical optimization,Optimal control,Maximum principle,Optimality criterion,Linear-quadratic-Gaussian control,Calculus of variations,Subderivative,Dynamical systems theory,Dynamical system,Mathematics
Journal
Volume
Issue
ISSN
43
3
0363-0129
Citations 
PageRank 
References 
0
0.34
0
Authors
2
Name
Order
Citations
PageRank
Boris S. Mordukhovich155664.85
Ilya Shvartsman213.29