Assignable polynomials to a weakly reachable single-input system over a B´ ezout domain J.A. Hermida-Alonso, A. DeFrancisco-Iribarren, and M.V. Carriegos. Departamento de Matem´ aticas. Universidad de Le´ on. SPAIN Abstract We describe how can we obtain all assignable polynomials to a weakly reachable linear system over a B´ ezout domain in normal form by solving a system of linear diophantine equations. We also point out some applications. AMS classification: 93B10; 93B25; 93B55. PACS: 02.30.Yy; 02.10.Uw; 02.40.Re Keywords: Pole assignability; pole shifting; stabilization. 1 Introduction A single input linear system over a commutative domain R (commutative ring with no zerodivisors) is just a pair Σ=(A, b ) where A ∈ R n×n is a square matrix and b ∈ R n×1 = R n is a column vector. For control theoretic affairs, this pair summarizes the sequential dynamical equation x (t + 1) = Ax (t)+ b u(t) where x (t) is the internal state at discrete time t and u(t) is the scalar input we introduce in the system. The design of u = f t x as a R-linear function of the states is the celebrated ”Feedback Action” which allows to stabilize linear systems in some cases: Closed loop u = f t x yields to the finite difference equation x (t + 1) = (A + b f t )x (t) and the behavior is defined by the characteristic polynomial χ(A + b f t ) = det(z 1 − (A + b f t )) It is interesting to research what are the assignable polynomials to a given system Σ= (A, b ); that is to say, to describe the family Pols((A, b )) = {χ(A + b f t ): f t ∈ R 1×n } 1