The goal of this paper is to use geometrical theory of nonlinear partial differential equations and singularities theory to obtain the general scheme for constructing an optimal control for fronts of petroleum displacement by active reagent. We illustrate our geometric approach to evolutionary models of petroleum deposits. Original simulation methods for the frontal displacement of oil by water and solutions of active reagents in porous medium of oilfield reservoirs with and without capillary forces has been developed. For simplicity and clarity of presentation the only one-dimensional models of fluid filtration between parallel batteries of production and injection wells (Buckley-Leverett and Rapoport-Liss models) has been considered. The model for oil displacement by solutions of active reagents in hot water has been also analyzed. The proposed results are generalized for two- and three-dimensional filtration models in reservoir engineering using vertical and horizontal wells, which greatly facilitates the statement and development of simulation and optimal control methods for reservoir engineering in general.

We consider a model of price setting duopolists with capacity constraints originated in papers of Bertrand (1883) and Edgeworth (1925). It is well known that this model may not posses a Nash equilibrium. Our aim is to solve original problem in terms of the equilibrium in secure strategies. It coincides with the Nash Equilibrium when Nash Equilibrium exists and takes into account the intention of players to maximize their profit under the condition of security against the actions of other players. This approach reflects the natural logic of behavior of players in this model. We will show that in some cases when Nash equilibrium does not exist there is an Equilibrium in secure strategies. However for some (big enough) capacities equilibrium in secure strategies does not exist either. A criterion for the equilibrium existence is formulated and proved.

In this research it was considered the particular case of a railway problem, specifically, the construction of orders delivery schedule for one locomotive plying among three railway station. In this paper it was suggested a polynomial algorithm and were shown the results of a computing experiment.