Stabilizing Stationary Linear Discrete Systems: Minimal and Balanced Expansions in Any Real Base

Let a be a real number strictly greater than 1 and let D be a finite interval of Z containing 0. In this paper we give necessary and sufficient conditions that guarantee the existence, for any real, of an expansion in base a with coefficients in D. Then we turn to balanced expansions, for which the sums of the digits of any initial segment are uniformly bounded. These issues yield necessary and sufficient conditions for stabilizing stationary linear discrete systems with a particular family of control laws.