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 w ith 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 contr ollable matrix pairs. We prove that phi preserves controllability if a nd only if phi(A, B)= beta(SAS(-1) + SBF, SBR) + f(A, B)(I,O) where be ta is a nonzero scalar, S, R are nonsingular, and f is a linear functi onal. Based on this result, we also find ail linear mappings on the li near space of all matrix 4-tuples (A, B, C, D) which preserve controll ability. Characterizations of linear preservers of observability-a con cept dual to controllability-hence follow. Some variations of the abov e problems are also discussed.