Stabilizing stationary linear discrete systems: Minimal and balanced expansions in any real base (Q1269694)

Given the real dynamical system: \[ x_{k+1} =ax_k+bu_k,\;u_{k+ 1} =u_k+v_k, \] \(v_k\in\{-m, \dots, +m\}\), where \(x_k\), \(u_k\) and \(v_k\) are scalars, and by using an expansion in base a with coefficients in a finite interval of \(\mathbb{Z}\) containing 0 and a balanced expansion for which the sums of the digits of any initial segment are uniformly bounded, one derives the following main result. If \(m\geq a-1\) the given system can be stabilized for any value of \(a>1\). Conversely, the condition \(m\geq [a]-1\) is necessary for stabilization for all values of \(a\). If a belongs to a particular set of reals (a certain union of intervals), the condition \(m\geq a-1\) is necessary and sufficient for stabilization.