This paper deals with an optimization problem with monotone controls. The problem studied is deterministic and non-stationary; there are no restrictions on the evolution of the state of the dynamical system, but there is a restriction on the total amount of available control. The optimal cost function V(t,x,y), \(x\in R^ n\), \(y\in [0,1]\), \(t\in [0,T]\) has an associated quasi-variational inequality (QVI): (1) \(\max (\frac{\partial V}{\partial t}+f\cdot \nabla_ xV+h,\partial V/\partial y)==0\) and the paper uses various approaches to study the ''viscosity'' solution of this QVI. It is proved that V is a uniformly Lipschitz continuous function and that it is the unique viscosity solution of (1). In addition, V is characterized as the limit of the functions \(V^ M\), where \(V^ M\) is the optimal cost function of an optimization problem with monotone and Lipschitz continuous controls (with a Lipschitz constant bounded by M). Also discussed is the extension of this method to obstacle problems and to differential games with monotone controls.