The proposed research lies within the area of optimal stochastic control. It deals with optimal control of diffusion processes by means of a free boundary, as well as optimal control at a reflection boundary of the stochastic process. This type of control naturally appears in problems in which there is no natural restriction on the rates of control and the optimal control rate id infinite. The optimal policy is then to reflect the process from the a priori unknown boundary. Such type of action arises in problems with additive control, where there are no a priori limits on the control rates. It is intended to develop of the related partial differential equations with gradient constraints, to study optimal reflection barriers in multi-dimensional cases and to investigate other problems pertinent to singular control in queuing, inventory and flexible manufacturing models. The proposed research lies within the area of optimal stochastic control. This type of control problem arises in models involving automatic cruise control of an airplane subject to uncertain wind conditions, of a space vehicle subject to small perturbations in mechanical units. The solution to this problem suggests the optimal timing of the correction of the course of the airplane of the space vehicle. Singular control also appears in queuing/production/manufacturing models. In a typical model of a flexible manufacturing system one seeks the optimal scheduling of turning on and off the production unit, the timing of buying new equipment or increasing the capacity of the existing unit, finding optimal levels of inventory stocked for supply, etc. In many cases when the production process faces uncertainties due to fluctuation of demand or due to unreliability in the process itself, it is possible to use the theory of singular control to address the questions posed above.
