Introduction

In the special case in which A generates a group the above written abstract problem is treated in some previous works: ([9] and [15]); but, for instance in the case of equation (0 .1), A doesn’t fulfil this extra assumption. When the unbounded operator AG is replaced by a bounded one the problem becomes a “diffused control problem” and is studied, under very general assumptions, for example in [4], [5], [10], [14]; finally for the deterministic case (C = 0) see [1], [2], [6], [7], [8], and [13].