The resolution problem of the system
where U(t), A, B, D and Uo are bounded linear operators on H and B* denotes the adjoint operator of B, arises in control theory, , transport theory, , and filtering problems, . The finite-dimensional case has been introduced in [6,7], and several authors have studied the infinite-dimensional case, , , . A recent paper, ,studies the finite dimensional boundary problem
where t ∈[0,b].In this paper we consider the more general boundary problem
where all operators which appear in (1.2) are bounded linear operators on a separable Hilbert space H. Note that we do not suppose C = −B* and the boundary condition in (1.2) is more general than the boundary condition in (1.1).