Optimal Control Strategies for an Advertising Competing Model

An advertising competing model is proposed in this article. Two variables denoting informative and predatory advertising efforts are considered to investigate the different effects of the informative and predatory advertising on sales promotion. In dynamic and competing market, close-loop solutions are more reasonable than open-loop ones, for states feedback can resemble the real situation where firms can respond to rivals behavior. Thus, we use the Bellman function to figure out the close-loop Nash-equilibrium in this duopoly game, i.e., optimal advertising strategies. Besides, the parameters conditions such that state variables have economic sense and Nash equilibrium converges are obtained. In the end of this article, a numerical instance illustrates the analyzed results as well as the impact of the parameters on the market shares and the advertising strategies.