Duality for Optimal Control-Approximation Problems with Gauges

Abstract

Looking for $m$ state variables and $n$ control variables such that the sum of the distance functions between the state variables and the control variables becomes minimal is called control-approximation problem. This problem is investigated under constraints. Moreover, the distances between the control variables themselves are taken into account. Powers of several gauges are chosen as distance functions. The considerations happen in Hausdorif locally convex topological real vector spaces.
In particular, location problems of very general type (e.g. so-called multifacility problems) turn out to be special cases of such control-approximation problems.
After the formulation of the primal control-approximation problem some considerations concerning gauges follow. Then a dual problem is given and weak and strong duality assertions are obtained.