Optimal control of ship maneuvers for course change, sidestep, and collision avoidance

We consider a ship subject to kinematic, dynamic, and moment equations and steered via the rudder under the assumptions that the rudder angle and rudder angle time rate are subject to upper and lower bounds.; We formulate and solve four Mayer problems of optimal control, the optimization criterion being the minimum time. Also, we formulate and solve four Chebyshev problems of optimal control, the optimization criterion being the maximization with respect to the control history of the minimum value with respect to time of the distance between two identical ships, one maneuvering and one moving in a predetermined way.; Problems P1 and P2 deal with course change maneuvers. In Problem P1, a ship initially in quasi-steady state must reach the final point with a given yaw angle and zero yaw angle time rate. Problem P2 differs from Problem P1 in that quasi-steady state is required at the final point.; Problems P3 and P4 deal with sidestep maneuvers. In Problem P3, a ship initially in quasi-steady state must reach the final point with a given lateral distance, zero yaw angle, and zero yaw angle time rate. Problem P4 differs from Problem P3 in that quasi-steady state is required at the final point.; Problems P5 and P6 deal with collision avoidance maneuvers without cooperation, while Problems P7 and P8 deal with collision avoidance maneuvers with cooperation. In Problems P5 and P7, the maneuvering ship must reach the final point with a given lateral distance, zero yaw angle, and zero yaw angle time rate. Problems P6 and P8 differ from Problems P5 and P7 in that quasi-steady state is required at the final point.; The above Mayer problems and Chebyshev problems, transformed into Lagrange problems via suitable transformations, are solved via the sequential gradient-restoration algorithm in conjunction with a new singularity avoiding transformation which accounts for the bounds on rudder angle and rudder angle time rate. The optimal control histories involve multiple subarcs along which either the rudder angle is kept at one of the extreme positions or the rudder angle time rate is held at one of the two extreme values.; If quasi-steady state is not required at the final point, the total number of subarcs ranges from 4 to 8, depending on the particular problem. If quasi-steady state is required at the final point, the total number of subarcs ranges from 6 to 10, depending on the particular problem: the higher number of subarcs, is due to the additional requirements that the lateral velocity and rudder angle vanish at the final point.