It is well known that the optimal control sequence for a linear system with a quadratic cost and linear inequality constraints over a finite optimisation horizon can be computed by means of a quadratic programme (QP). The aim of this article is to investigate when the optimal control sequence for a non-linear single-input single-output system also can be computed via QP. Our main contribution is to show that the optimal control sequence for non-linear systems, with a quadratic cost and linear inequality constraints can be computed in exact form via QP provided the optimisation horizon is no larger than a critical quantity that we name the 'input-output linear horizon'. The results do not require any linearisation technique and are applicable to general non-linear systems, provided their input-output linear horizon is larger than their relative degree.