Abstract: The optimal augmented proportion al navigation for a constant maneuver target was deduced using optimal control theory based on the cost function weighted by a power of time-to-go. The analysis of the exponential of the weighted function shows that there exists a bijection between the navigation ratio and the exponential which gives a reasonable theoretical explanation for the engineering value of the APN navigation ratio. The seeker heading error and a constant maneuver target was introduced into the augmented proportional navigation guidance system while the guidance dynamics was simplified as a single lag. According to the mathematic thought of the adjoint method, the miss distance of the APN guidance system with a single-lag was studied and the closed-form solutions of which were deduced when the heading error and the constant maneuver target enters into the system and the miss distance closed-form solutions was validated through the simulation of the adjoint guidance system.