We consider a continuous time linear multi-inventory system with unknown demands bounded within ellipsoids and controls bounded within ellipsoids or polytopes. We address the problem of epsilon-stabilising the inventory since this implies some reduction of the inventory costs. The main results are certain conditions under which epsilon-stabilisability is possible through a saturated linear state feedback control. All the results are based on a linear matrix inequalities approach and on some recent techniques for the modelling and analysis of polytopic systems with saturations. Numerical simulations are provided.