On the Optimal Regulation of an Axi-symmetric Rigid Body with Two Controls

We present a partial solution to the problem of opti mal feedback reorientation of the symmetry axis of an axially symmetric rigid body The performance index is quadratic in the state and the control vari able and the optimal reorientation maneuver requires the use of only two control torques Because of the passivity characteristics and the cascade structure of the system we rst state two optimal regulation problems for the dynamics and the kinematics sub systems separately In this case one is able to nd explicit solutions to the associated Hamilton Jacobi equations For the complete system the optimal regulation problem is not solvable in general We present solutions for two partial cases The rst case is when there is no penalty on the control in put In this case one can asymptotically recover the cost for the kinematics by making the dynamics su ciently fast The second case investigates re strictions imposed by optimality considerations on the aforementioned control law to avoid high gain