When is the naive quantized control law globally optimal?

We investigate the properties of a control law which quantizes the unconstrained solution to a unitary horizon quadratic programme. From a quantized receding horizon point of view, this naive quantized control law is globally optimal for horizon one. However, the question arises as to whether it is also globally optimal for horizons greater than one, i.e. whether it solves (in a receding horizon sense) a multi-step quadratic programme, where decision variables are restricted to belong to a quantized set. By using dynamic programming, we develop necessary and sufficient conditions for this to hold for the case of first order plants. The results can be applied to arbitrary horizons and quantized sets, which may contain a finite or an infinite (though countable) number of elements.