Optimal Control of an Innovation Diffusion Model in Smartphone Markets

Competitive smartphone markets are shaped by social influence, ecosystem loyalty, technological obsolescence, and marketing interventions. This study develops a nonlinear innovation diffusion model to describe the interactions among potential consumers, Android users, iPhone users, and dissatisfied consumers in a competitive smartphone market. The model incorporates social influence, ecosystem switching, market saturation, and consumer dissatisfaction effects and is calibrated using United States smartphone market share data from 2016 to 2026 obtained from StatCounter and Statista. Time-dependent advertising controls are then introduced for both smartphone ecosystems, and an optimal control problem is formulated and solved using Pontryagin’s Maximum Principle and a forward–backward sweep algorithm. The calibration results indicate a stronger effective diffusion capability for the iPhone ecosystem. Numerical simulations show that early promotional interventions improve long-term market penetration and that the effectiveness of advertising strategies depends strongly on ecosystem valuation, market saturation, and diffusion parameters. Under the estimated parameter set, Android-oriented interventions generate the largest gains in the objective functional, despite the stronger baseline position of the iPhone ecosystem. These results demonstrate how nonlinear diffusion modelling and optimal control theory can provide quantitative support for strategic marketing decisions in competitive technology markets.