On the control stabilizing the Nash optimal equilibrium in some model of competitive price forming

In this paper we consider a model of price forming in the case of monopolistic competition with $N$ sellers supplying «almost homogeneous commodity». The model is investigated numerically for $N=2$. If each seller changes his price proportionally to the derivative of his profit with respect to the price, then a trajectory may come to the Nash non-optimal equilibrium (which is also not Pareto-optimal) or dynamical chaos. Additional price control based on the sign of the second derivative of the profit with respect to the price may lead the system to the Nash locally optimal equilibrium (which is also Pareto-optimal) for various initial conditions.