Linear preservers of controllability and/or observability

Consider a linear control differential equations system x = Ax + Bu, y = Cx + Du, where x is an element of C-n, u is an element of C-n, y is an element of C-p, and A, B, C, D are matrices of appropriate sizes with entries in C. This system, or the matrix pair(A, B), or the matrix 4-tuple (A, B, C, D), is called controllable if rank (A - lambda I, B) = n for all lambda = 0. Let phi be a linear transformation on C-nx(n+m), the linear space of all matrix pairs (A, B). Then phi is said to preserve controllability if it maps controllable matrix pairs to controllable matrix pairs. We prove that phi preserves controllability if and only if phi(A, B)= beta(SAS(-1) + SBF, SBR) + f(A, B)(I,O) where beta is a nonzero scalar, S, R are nonsingular, and f is a linear functional. Based on this result, we also find ail linear mappings on the linear space of all matrix 4-tuples (A, B, C, D) which preserve controllability. Characterizations of linear preservers of observability-a concept dual to controllability-hence follow. Some variations of the above problems are also discussed.