Aspects of Optimality in the Eigenvalue Assignment Problem

The eigenvalue assignment problem involves finding a state feedback vector k such that the eigenvalues of A-bk^T are in the desired locations. This paper considers the case where the system is not completely controllable, and the solution for the feedback gain vector k is not unique. The set of possible solutions for the feedback gain vector is identified, and optimal solutions are defined in the sense of minimum 2-norm, minimum infinite-norm (minimum maximum gain), and maximum number of zero gain elements in k. In addition, the methods presented here can also be applied to the case where linear state-variable feedback is applied to more than one input.