Abstract: The extended object cost function was proposed based on the time-to-go exponential function and for constant target maneuver, a family of extended optimal trajectory shaping guidance law was deduced. The missile non-dimensional position and angle miss distance vs non-dimensional terminal guidance time was studied using the non-dimensional method and adjoint technique when the constant/random maneuver target was introduced into the one single lag system. The adjoint simulation results were validated by the direct simulation results. The results show that the position and angle miss distance produced by the final impact angle approaches to zero when the missile terminal guidance time is about 15 times of the system lag time while the constant target maneuver is about 20 times. If the exponential number n and the system navigation ratio increase, the non-dimensional position and angle miss distance is oscillatory more worse and the convergence time of the miss distance is larger than before. With the increasing of the terminal guidance time, the position and angle miss distance of the random target maneuver approaches to a stable value but not near zero and also, the miss of which is larger when the exponential number n increases.