Inserted: 23 jul 2001
Last Updated: 23 oct 2002
Journal: J. Nonlinear and Convex Anal.
The paper studies the representation of relaxed settings of optimal control problems for nonlocal differential equations with deviating argument involving possibly many shifts in control variables. As opposed to the more general approach of J. Rosenblueth, J. Warga, R. Vinter and Q. Zhu, the relaxation is accomplished in original function spaces rather than in much larger spaces of special Young measures. To find relaxed setting, one uses the straightforward convexification method similar to that used in optimal control problems for ordinary (local) differential equations. It is shown that such an approach is essentially limited, since it can work only in the cases when the shifts in control variables satisfy the special ``unifiability'' condition, which can be regarded as some commensurability or ``collective nonergodicity'' property. Slightly restricting further the unifiability requirement one is able to obtain the exact representation of relaxed control problem setting. The latter will be again an optimal control problem but involving, generally speaking, infinitely many new shifts both in the state equation and in the cost functional.