A framework is developed for the control design and stability analysis of state-feedback systems made out of automaton-controller pairs, here referred to as automata control systems. A single theorem, based on the Bellman-Ford algorithm, provides the conditions for the design of the controllers that make a given automaton optimal and stable. The automata approximation of continuous state-space models is also developed for the design of state-feedback controllers that can drive continuous plants. The approximation of continuous plants through automata makes the design of state-feedback controllers independent of the state-space description. No distinction is made in the treatment of linear and nonlinear plants. Controller synthesis and specification of the domains of attraction for the resulting plant-controller pair are systematically obtained for continuous time-invariant state-space models. The application of this framework for the stability analysis of the exact model of a digital filter is presented. The automata approximation is applied to design a single controller that stabilises a forced pendulum around two equilibria. The design of switching controllers using automata approximation is also developed and applied to the longitudinal motion control of an aircraft.