Pergamon Nonlinear Analysis, Theory, Methods & Applicafiom, Vol. 28, No. 5, pp. 889-915, 1997 Copyright 0 19% Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362446X/% $17.OO+O.CUI 0362-546X(95)00186-7 VARIATIONAL ANALYSIS IN NONREFLEXIVE SPACES AND APPLICATIONS TO CONTROL PROBLEMS WITH L’ PERTURBATIONS JONATHAN M. BORWEINt* and QIJI J. ZHUS t Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC V5A lS6 Canada; and $ Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008 U.S.A. (Received 6 January 1994; received for publication 26 September 1995) Key words andphrases: Weak-Hadamard sub-derivatives, Holder sub-derivatives, variational principles, smooth renorms, Clarke-subdifferentials, G-subdifferentials, sensitivity analysis, control system, infinite horizon problems. 1. INTRODUCTION It has long been recognized that the value function plays an important role in optimization. It measures the sensitivity of the problem to perturbations of the objective function and the various constraints. Particularly interesting is the derivative of the value function, a measure of so called “differential stability”. When the value function is differentiable, it plays the role of a multiplier. In the context of dynamic optimization this observation establishes an heuristic relationship between the maximum principle and the dynamic programming approaches. Generally, however, the value function of a constrained optimization problem is far from being differentiable. To obtain a rigorous treatment of these heuristic relations one needs to apply the techniques of nonsmooth analysis. Consider the constrained finite dimensional optimization problem: (P): minimize f(x) subject to g(x) = 0. Suppose that we perturb the constraint and define the value function of the perturbed problem as V(z) : = inf (f(x) : g(x) = z). Then a precise form of the heuristic relation previously alluded to is (cf. [l]) -a,iqo(O) c clcoM(Z) where a, is the Clarke generalized gradient, C is the solution set of the optimization problem (P) and M(Z) denotes the set of multipliers corresponding to all solutions in C. This inclusion has been extended to optimal control problems with finite dimensional or L2 perturbations [2, 3,4]. However, in the setting of dynamic optimization problems, in particular those discussed in [2-41 that involve control systems, the natural space for perturbations is L’ rather than L2 or a finite dimensional space. The reason that in previous research the perturbation spaces were * Research was supported by NSERC and by the Shrum Endowment at Simon Fraser University. 889