A Differential-Game Model of Unfair Competitive Pressure with Bounded Nonlinear Dynamics: An Application of Boltyanski–Poznyak’s Robust Maximum Principle

An interpretable nonlinear differential game is proposed for a firm operating under unfair competitive pressure. Four bounded state variables represent market share, reputation, service level, and alliance strength. The firm controls price posture, quality investment, and supply effort, while an adversarial player aggregates reputational attacks and channel disruptions. A finite-horizon minimax tracking criterion is posed and necessary optimality conditions are obtained via Boltyanski–Poznyak’s robust maximum principle. A projected forward–backward sweep scheme computes open-loop saddle trajectories under box constraints. A numerical experiment illustrates bounded trajectories, saturation patterns, and convergence diagnostics.