We propose a numerical approach to compute Stationary Markov perfect Nash equilibrium advertising strategies for the extended form of the Lanchester model. The proposed algorithm has the merit of considering a positive discount rate. We also propose an interesting comparison between the simple and extended form of the Lanchester model.

We study a reciprocity principle for optimal control problems and apply the results to some classical economic models. Thus we extend some calculus of variations results on reciprocal isoperimetric problems, which have already been exploited and specified by Caputo to develop an interesting comparative dynamics analysis for Hotelling's model.

An infinite-horizon differential game between a manufacturer and a retailer is considered. The players control their marketing efforts and the sales share of the online channel is the state of the system. The manufacturer seeks to maximize her profit made on both the indirect and direct channels and faces, aside from her marketing effort, a logistics cost of selling online. The retailer seeks to keep consumers buying offline through her effort. A feedback Nash equilibrium is identified and results are discussed.