An approximate maximum principle for finite-difference control systems

Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints

“…Throughout the paper we assume that the control sets U (t) in (P N ) are compact subsets of a metric space (U, d) and that the set-valued mapping U : [t 0 , t 1 ] → → U is continuous with respect to the Hausdorff distance. Recall [2,4] that a sequence of discrete-time control {u n } in (P N ) is proper if for every sequence of mesh points…”

“…keeping all other assumptions in place. Perturbations of this type were considered in [2,4], where the AMP was derived in form (1.5)-(1.9) in the case of smooth cost and constraint functions. Note that the previous form of the AMP was also obtained in [2,4] for perturbations of the equality constraints…”

“…Considering constrained control problems of type (P N ) with smooth data required a completely different approach invoking the geometry of endpoint constraints under multineedle variations of optimal controls. It has been initiated by Mordukhovich [2], where a certain finite-difference counterpart of the hidden convexity property for sequences of constrained discrete-time systems was revealed and the first version of the Approximate Maximum Principle (AMP) for smooth control problems (P N ) was established; the reader can find all the details and discussions in [4], Section 6.4 and commentaries therein.…”

“…The version of the AMP given in [2,4] for smooth constrained problems (P N ) can be formulated as follows. Let the pair (ū N ,x N ) be optimal to (P N ) for each N ∈ IN such that the control sequence {ū N } is proper (see [2,4] and Section 3 below for the exact definition and discussions). Then for any ε > 0 there exist multipliers λ iN as i = 0, .…”

“…The presented version of the AMP from [2,4] plays actually the same role as the DMP for studying and solving discrete-time smooth optimal control problems with sufficiently small stepsizes without imposing the restrictive convexity assumption on the sets f (x N (t), U(t), t), t ∈ T N . We refer the reader to, e.g., [4,7] and the bibliographies therein for more discussions and applications of the AMP to chemical engineering, periodic control, biotechnological and ecological processes, etc.…”