A complete optimally condition in the inverse problem of optimal control

A complete optimality condition in the standard inverse problem of optimal control for the multi-input case is established. The optimality of a feedback control law is characterized completely in terms of a new geometric condition as well as the well-known return difference condition. The new condition not only provides a better insight into the well-known sensitivity reduction property of optimal control for the multi-input case, but also indicates an essential difference between the solutions of the single- and multi-input inverse problem.