Reachability and controllability of 2-D Roesser model with bounded inputs

The most popular models of two dimensional (2D) linear systems are the discrete models proposed by R.P. Roesser (1975), E. Fornasini and G. Marchesini (1976; 1978) and J. Kurek (1985). It is well known (R.R. Mohler, 1991) that the linear continuous time system x˙=Ax+Bu with u bounded is not completely controllable if the eigenvalues of the system matrix A have negative real parts. Similarly, the linear discrete time system x_i+1=Ax_i+Bu_i with u_i bounded is not completely reachable (controllable) if the eigenvalues of A have modules less than one. A counterpart for 2D linear systems described by the 2D Roesser model with constant and variable coefficients is established. It is shown that the 2D Roesser model with bounded inputs and a bounded norm of its input matrix is not locally reachable and locally controllable if the norm of its system matrix is less than or equal to one.